Problems for Meeting M01

You'll work collaboratively on these problems in groups of 2 – 4 students, writing your solutions clearly on a shared set of papers. Each group will turn in one solution set that has all of their names printed on it. Instructor Isaiah Dempsey will grade those solutions and return them to you.

You will also write some of your solutions on the chalkboard. Instructor Dempsey will discuss those chalkboard solutions with the whole class.

Note that your Recitation score is based not just on the thoroughness of your written solutions, but also on how you collaborated with your group and how you presented your work on the chalkboard.

[1]: Consider the rational function $$f(x)=\frac{x^2+3x-10}{x-2}$$

(a) Factor \(f(x)\).

(b) Cancel terms in the factored form of \(f(x)\), assuming \(x\neq 2\)

[2]: Consider the expression $$\frac{(-7+h)^2-49}{h}$$

(a) (Just work on the numerator for now.) Expand the terms in the numerator and simplify the numerator.

(b) (Now work on the ratio of terms.) Cancel what can be cancelled, assuming \(h\neq 0\)

[3]: Consider the expression $$\frac{\sqrt{49+h}-7}{h}$$

(a) Rationalize the numerator. Hint: See Section 1.4 Example 5 for reference.

(b) Cancel what can be cancelled, assuming \(h\neq 0\)

[4]: Background: Recall that the difference of two squares can always be factored.

(a) Factor the expression $$a^2-b^2$$ Circle the resulting equation. This difference of two squares factorization will be useful throughout MATH 2301 and life.

(b) Now consider the function $$f(x)=\frac{7-\sqrt{x}}{49x-x^2}$$ Rewrite \(f(x)\) by factoring the denominator.

(c) The denominator of \(f(x)\) should now contain a term \((49-x)\). This does not look like the difference of two squares, but it can be treated as the diffence of two squares, and so can be factored. Rewrite \(f(x)\) by replacing \((49-x)\) with a factored version of \((49-x)\).

(d) Cancel terms in \(f(x)\), assuming \(x \neq 49\).

[5]: Consider the expression $$f(x)=\frac{\frac{1}{7}+\frac{1}{x}}{7+x}$$ Part of what makes this expression difficult to work with is the fact that it is a fraction that also has fractions in its numerator. Visually, you can see that there are two base lines: There is the big base line that separates the numerator from the denominator, and there is a second base line that runs through the middle of the two fractions that are in the numerator. It is usually helpful to rewrite expressions that have multiple base lines so that they have a single baseline.

(a) Use the idea that we can rewrite $$\frac{numerator}{denominator}=\frac{1}{denominator}\left(numerator\right)$$ to rewrite \(f(x)\) as the product of two terms that share a single base line. (One of the terms will need to be in parentheses!)

(b) The term in parentheses is the sum of two fractions that have different denominators. Find a common denominator for those two fractions, and use it to rewrite the sum of those two fractions as a single fraction.

(c) Use the result from (b) to rewrite f(x) as a product of two fractions. (They should share a base line. Note that so far, you have only rewritten \(f(x)\) in a form that is easier to work with. You have not altered \(f(x)\) in any way that changes what the function means.)

(d) Rewrite f(x) by cancelling terms, assuming \(x \neq -7\).

[6]: Consider the expression $$\frac{(x+h)^3-x^3}{h}$$

(a) (Just work on the numerator for now.) Expand the terms in the numerator and simplify the numerator.

(b) (Now work on the ratio of terms.) Cancel what can be cancelled, assuming \(h\neq 0\)

[7]: Consider the expression $$f(x)=\frac{2x+12}{|x+6|}$$

(a) Rewrite \(f(x)\) by factoring its numerator.

Case 1: When it is known that \(x \gt -6 \)

(b) Suppose it is known that \(x \gt -6 \) Then will the quantity \((x+6)\) be a positive quantity or a negative quantity?

(c) Suppose it is known that \(x \gt -6 \). Use the information from (b) to simplify \(|x+6|\). That is, figure out a way to write an expression that will have the same value as \(|x+6|\), but that does not have the absolute value symbol.

(c) Suppose it is known that \(x \gt -6 \). Use the information from (c) to simplify \(f(x)\). That is, figure out a way to write an expression that will have the same value as \(f(x)\), but that does not have the absolute value symbol. This expression can be simplified by cancelling terms (because you know that those terms are non-zero.) Do that simplification.

Case 2: When it is known that \(x \lt -6 \)

(d) Suppose it is known that \(x \lt -6 \) Then will the quantity \((x+6)\) be a positive quantity or a negative quantity?

(e) Suppose it is known that \(x \lt -6 \). Use the information from (d) to simplify \(|x+6|\). That is, figure out a way to write an expression that will have the same value as \(|x+6|\), but that does not have the absolute value symbol.

(f) Suppose it is known that \(x \lt -6 \). Use the information from (e) to simplify \(f(x)\). That is, figure out a way to write an expression that will have the same value as \(f(x)\), but that does not have the absolute value symbol. This expression can be simplified by cancelling terms (because you know that those terms are non-zero.) Do that simplification.

Problems for Meeting M02

You'll work collaboratively on these problems in groups of 2 – 4 students, writing your solutions clearly on a shared set of papers. Each group will turn in one solution set that has all of their names printed on it. Instructor Isaiah Dempsey will grade those solutions and return them to you.

You will also write some of your solutions on the chalkboard. Instructor Dempsey will discuss those chalkboard solutions with the whole class.

Note that your Recitation score is based not just on the thoroughness of your written solutions, but also on how you collaborated with your group and how you presented your work on the chalkboard. There is NOT an expectation that you will be able to finish all of these problems. The expectation is just that you collaborate with your group to solve as many as you can.

Part 1: An Algebra Calculation Needed for Section 1.5 Continuity

[1] What value of \(c\) will make the following equation true when \(x=2\)?

Part 2: Some Algebra Calculations Needed for Section 1.6 Limits Involving Infinity

[2] The goal is to rewrite the expression $$\frac{\sqrt{t}+t^2}{2t-t^2}$$ by dividing every term in the numerator and denominator of the expression by the highest power of \(t\) that appears in the denominator. In other words, divide every term in the numerator and denominator by \(t^2\). That is, $$\frac{\sqrt{t}+t^2}{2t-t^2}=\frac{\frac{1}{t^2}\left(\sqrt{t}+t^2\right)}{\frac{1}{t^2}\left(2t-t^2\right)}$$ (Keep in mind that dividing all of the terms in this way does not change the value of the expression.) You'll have to pay close attention when computing $$\frac{1}{t^2}\left(\sqrt{t}\right)$$

[3] The goal is to rewrite the expression $$\sqrt{9x^2+x}-3x$$ in an equivalent form that does not use subtraction. (This might seem like a strange request, but when you work on your homework, you'll see an example where that kind of rewriting is useful.) The trick is to multiply and divide the expression by a conjugate expression. That is, an expression that is just like the expression above, but that has a plus sign instead of a minus sign. $$\sqrt{9x^2+x}-3x=\left(\sqrt{9x^2+x}-3x\right)\cdot \frac{\sqrt{9x^2+x}+3x}{\sqrt{9x^2+x}+3x}$$ Show how the multiplication works. (Remember that when multiplying a fraction by something that is not a fraction, you only multiply the numerator!) Simplify your answer.

[4] The goal is to rewrite the expression $$\frac{x}{\sqrt{9x^2+x}+3x}$$ by dividing every term in the numerator and denominator of the expression by the highest power of \(x\) that appears in the denominator. You might think that that highest power would be \(x^2\), but notice that the \(x^2\) appears inside a radical. So the highest power of \(x\) that appears in the denominator is actually just \(x\). So divide every term in the numerator and denominator by \(x\). In other words, rewrite the expression by computing $$\frac{x}{\sqrt{9x^2+x}+3x}=\frac{\frac{1}{x}\cdot\left(x\right)}{\frac{1}{x}\cdot\left(\sqrt{9x^2+x}+3x\right)}$$ You'll have to pay close attention when computing $$\frac{1}{x}\cdot\left(\sqrt{9x^2+x}\right)$$ (Remember that you cannot break the radical! That is, $$\sqrt{a+b}\neq \sqrt{a}+\sqrt{b}$$

[5] In this problem, you'll factor the expression \(x^4+x^5\) two ways.

1. Factor \(x^4+x^5\) by pulling out the common factor \(x^4\).
2. Start over. This time factor \(x^4+x^5\) by pulling out a factor of \(x^5\). That is, rewrite the expression as $$x^4+x^5=x^5 \cdot\left(\text{something}+\text{something}\right )$$ This way of factoring probably seems strange: You're used to factoring when there is an obvious common factor, and the \(x^5\) doesn't look like it is a common factor. But you'll see in one of our homework problems that this second kind of factoring is useful.

[6] For the rational function $$y=\frac{2x^2+x-1}{x^2+x-2}$$

1. Rewrite the function in factored form.
2. Start over. This time, rewrite the function by dividing every term in the numerator and denominator of the expression by the highest power of \(x \)that appears in the denominator.

[7] For the rational function $$C(t)=\frac{30t}{200+t}$$ rewrite the function in an equivalent form by dividing every term in the numerator and denominator by the highest power of \(t\) that appears in the denominator.

Part 3: Some Algebra Calculations Needed for Section 2.1 Derivatives and Rates of Change

[8] For the function \(f(x)=4x-x^2\)

1. Build the expression for \(f(1+h)\)
2. Build the expression for $$f(1+h)-f(1)$$ Simplify the expression by
   • cancelling all terms that can be cancelled
   • factoring out the common factor of \(h\)
3. Build the expression for $$\frac{f(1+h)-f(1)}{h}$$ Cancel what can be cancelled, assuming \(h \neq 0\)

[9] Again working with the function \(f(x)=4x-x^2\)

1. Find the value of \(f(1)\)
2. Build the expression for $$f(x)-f(1)$$
3. Build the expression for $$\frac{f(x)-f(1)}{x-1}$$
4. Do the division indicated by the expression in (c). There are two ways to do this.
   • Do long division of polynomials, if you remember that from high school. (unlikely)
   • Or factor the numerator and cancel. Hint: \((x-1)\) is one of the factors of the numerator.

[10] The goal is to rewrite the expression $$\frac{\sqrt{1+h}-1}{h}$$ in an equivalent form that does not use subtraction. (This might seem like a strange request, but when you work on your homework, you'll see an example where that kind of rewriting is useful.) The trick is to multiply both the numerator and denominator by a conjugate expression. That is, an expression that is just like the expression in the numerator, but that has a plus sign instead of a minus sign. Figure out the conjugate expression, and then multiply the numerator and denominator by that conjugate expression.

Instructions for Meeting M03

You'll work collaboratively on these problems in groups of 2 – 4 students, writing your solutions clearly on a shared set of papers. Each group will turn in one solution set that has all of their names printed on it. Instructor Isaiah Dempsey will grade those solutions and return them to you.

You will also write some of your solutions on the chalkboard. Instructor Dempsey will discuss those chalkboard solutions with the whole class.

Note that your Recitation score is based not just on the thoroughness of your written solutions, but also on how you collaborated with your group and how you presented your work on the chalkboard. There is NOT an expectation that you will be able to finish all of these problems. The expectation is just that you collaborate with your group to solve as many as you can.

Some Algebra Calculations Needed for Section 2.2 The Derivative as a Function

Introduction

Recall the Definition of the Derivative: $$f'(x)=\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ In problems where you are asked to find a derivative using the Definition of the Derivative, your task is to build the limit and evaluate it, showing the steps clearly.

In Section 2.2 of Stewart's book (and in the accompanying WebAssign problems), you are asked to use the Definition of the Derivative to compute the derivative of a variety of kinds of functions. These problems are all hard, but they are hard because of the messy algebra involved.

(Remark: Some of you may have had a Calculus course before, and therefore may know some shortcuts to finding derivatives, shortcuts that avoid the hard work of using the Definition of the Derivative. You'll learn these kinds of shortcuts in MATH 2301, but not yet. The shortcuts start showing up in Section 2.3. For now, and even after you have studied Section 2.3, when the instructions for a problem say to use the Definition of the Derivative, you have to build the limit and evaluate it, NOT using the derivative shortcuts.)

The Empty Version of a Function

One task in using the Definition of the Derivative is to build the expression \(f(x+h)\). This task often confuses students. One helpful approach is to learn to identify what could be called the empty version of \(f\). That is, learn to write an expression for \(f\) with empty parentheses instead of \(x\). Then, writing the correct expression for \(f(x+h)\) amounts to simply putting \(x+h\) inside every empty pair of parentheses.

For Example: $$\begin{align*} f(x) &=\frac{x^2-1}{2x-3} \ \ \ (\text{original function}) \\ f( \ )&=\frac{( \ )^2-1}{2( \ )-3} \ \ \ (\text{empty version}) \\ f(x+h)&=\frac{(x+h)^2-1}{2(x+h)-3} \ \ \ (\text{one of the expressions needed in the Definition of the Derivative} ) \end{align*}$$ In the problems for today, you will be asked to take this approach to building \(f(x+h)\).

[1] For the function $$f(x)=\frac{1}{5}x-\frac{1}{7}$$

1. Write the empty version of \(f\).
2. Using your result from (a), write the expression for \(f(x+h)\).
3. Using your result from (b), write the expression for \(f(x+h)-f(x)\). Simplify the expression.
4. Using your result from (c), write the expression for \(\frac{f(x+h)-f(x)}{h}\). Simplify the expression, assuming that \(h \neq 0\). Note that this will enable you to cancel the \(\frac{h}{h}\).

[2] For the function $$f(x)=2.3x^2-5x+4.8$$

1. Write the empty version of \(f\).
2. Using your result from (a), write the expression for \(f(x+h)\).
3. Using your result from (b), write the expression for \(f(x+h)-f(x)\). Simplify the expression.
4. Using your result from (c), write the expression for \(\frac{f(x+h)-f(x)}{h}\). Simplify the expression, assuming that \(h \neq 0\). Note that this will enable you to cancel the \(\frac{h}{h}\).

[3] For the function $$f(x)=\frac{5}{\sqrt{x}}$$

1. Write the empty version of \(f\).
2. Using your result from (a), write the expression for \(f(x+h)\).
3. Using your result from (b), write the expression for \(f(x+h)-f(x)\). Your expression should be the difference of two fractions, where each has a radical expression in its denominator. This expression is fairly simple-looking as it is, but it is not in a helpful form. In coming steps, you will rewrite the expression. It will not get simpler, at least not at first, but it will end up in a form that is more useful.
4. Rewrite the expression from (c) by getting a common denominator. The resulting expression should be a fraction with a difference of two radicals in the numerator and the product of two radicals in the denominator.
5. The expression from (d) turns out to be problematic because of the subtraction. The goal in this step is to rewrite the expression from (d) in an equivalent form that does not use subtraction. The trick is to multiply both the numerator and denominator by a conjugate expression. The conjugate expression is an expression that is just like the expression in the numerator, but that has an addition of radicals, instead of a subtraction of radicals. Figure out the conjugate expression, and then multiply the numerator and denominator by that conjugate expression, and simplify your result. (You did a similar thing in a problem in last week's meeting.)
6. Using your result from (e), write the expression for \(\frac{f(x+h)-f(x)}{h}\).
7. The expression from (f) is problematic because it is a fraction with a fraction in the numerator. That means that there are two baselines in the expression. In general, fraction expressions that have more than one baseline are confusing. Rewrite your expression for \(\frac{f(x+h)-f(x)}{h}\) from (f) by rewriting $$\frac{(\text{fraction})}{h}=\frac{1}{h}\cdot (\text{fraction})$$ This new expression will have a single baseline.
8. Simplify the expression, from (g) assuming that \(h \neq 0\). Note that this will enable you to cancel the \(\frac{h}{h}\).

[4] For the function $$f(x)=\sqrt{81-x}$$

1. Write the empty version of \(f\).
2. Using your result from (a), write the expression for \(f(x+h)\).
3. Using your result from (b), write the expression for \(f(x+h)-f(x)\). Your expression should be the difference of two radicals.
4. The expression from (c) turns out to be problematic because of the subtraction. The goal in this step is to rewrite the expression from (c) in an equivalent form that does not use subtraction. The trick is to multiply both the numerator and denominator by a conjugate expression. The conjugate expression is an expression that is just like the expression in the numerator, but that has an addition of radicals, instead of a subtraction of radicals. Figure out the conjugate expression, and then multiply and divide by that conjugate expression, and simplify your result. (You did a similar thing in Problem [3].)
5. Using your result from (d), write the expression for \(\frac{f(x+h)-f(x)}{h}\).
6. The expression from (e) is problematic because it is a fraction with a fraction in the numerator. That means that there are two baselines in the expression. Rewrite your expression for \(\frac{f(x+h)-f(x)}{h}\) from (f) by rewriting $$\frac{(\text{fraction})}{h}=\frac{1}{h}\cdot (\text{fraction})$$ This new expression will have a single baseline.
7. Simplify the expression, from (f) assuming that \(h \neq 0\). Note that this will enable you to cancel the \(\frac{h}{h}\).

[5] For the function $$f(t)=\frac{2-3t}{5+t}$$

1. Write the empty version of \(f\).
2. Using your result from (a), write the expression for \(f(x+h)\).
3. Using your result from (b), write the expression for \(f(x+h)-f(x)\). Your expression should be the difference of two fractions.
4. Rewrite the expression from (c) by getting a common denominator. Simplify the expression.
5. Using your result from (d), write the expression for \(\frac{f(x+h)-f(x)}{h}\).
6. The expression from (e) is problematic because it is a fraction with a fraction in the numerator. That means that there are two baselines in the expression. Rewrite your expression for \(\frac{f(x+h)-f(x)}{h}\) from (e) by rewriting $$\frac{(\text{fraction})}{h}=\frac{1}{h}\cdot (\text{fraction})$$ This new expression will have a single baseline.
7. Simplify the expression, from (f) assuming that \(h \neq 0\). Note that this will enable you to cancel the \(\frac{h}{h}\).

Instructions for Meeting M04

You'll work collaboratively on these problems in groups of 2 – 4 students, writing your solutions clearly on a shared set of papers. Each group will turn in one solution set that has all of their names printed on it. Instructor Isaiah Dempsey will grade those solutions and return them to you.

You will also write some of your solutions on the chalkboard. Instructor Dempsey will discuss those chalkboard solutions with the whole class.

Note that your Recitation score is based not just on the thoroughness of your written solutions, but also on how you collaborated with your group and how you presented your work on the chalkboard. There is NOT an expectation that you will be able to finish all of these problems. The expectation is just that you collaborate with your group to solve as many as you can.

Part 1: Some Algebra Calculations Needed for Section 2.3 Basic Differentiation Properties

Introduction: Rewriting Functions in Power Function Form

In Section 2.3, you will learn six fairly simple derivative Rules:

• the Derivative of a Constant Function
• the Power Rule
• the Constant Multiple Rule
• the Sum Rule (and Difference Rule, not worth stating separately!)
• the Sine Rule and the Cosine Rule

Generally, this means rewriting a function in what is called power function form. What this means is that you rewrite the function as a sum of terms that are each of the form $$\text{constant}\cdot\text{power function}$$ That is, $$c \cdot x^p$$ This rewriting is an algebra skill. If your skill at rewriting functions in different forms is strong (an algebra skill), then your work in finding derivatives (a calculus task) will be much easier.

Rewriting functions in power function form is best done in two steps.

• Step 1: Separate the function into a sum of functions and separate out the constants from the parts involving \(x\).
• Step 2: Rewrite the parts involving \(x\) as power functions.

That’s vague. Here’s an example:

Original presentation of Function $$f(x)=\frac{2x^3-7x^{2/3}+6}{\sqrt[3]{x}}$$ After Step 1: $$f(x)=2\cdot\frac{x^3}{x^{1/3}}-7\cdot\frac{x^{2/3}}{x^{1/3}}+6\cdot\frac{1}{x^{1/3}}$$ After Step 2: $$f(x)=2\cdot x^{8/3}-7 \cdot x^{1/3}+6 \cdot x^{-1/3}$$

Following below are fourteen functions taken from exercises in Section 2.3 of the textbook for MATH 2301. In those exercises, you are asked to find the derivative of each function, using the simple derivative rules presented in Section 2.3. But for every one of these fourteen functions, you would need to first do the algebra task of rewriting the function in a form where the derivative rules can be used. Your job today is to do just that algebra task. That is, rewrite each function in power function form. DO NOT TAKE THE DERIVATIVE in this assignment. Just do the preliminary algebra. (In your calculus homework, you would go on and take the derivative.) $$\begin{align*} [1] \ \ &g(x)=x^2(1-2x) \\ \\ [2] \ \ &h(x)=(x-2)(2x+3) \\ \\ [3] \ \ &A(s)=-\frac{7}{s^{13}} \\ \\ [4] \ \ &R(a)=(3a+1)^2 \\ \\ [5] \ \ &y=5\sqrt{x}(3x-7) \\ \\ [6] \ \ &S(p)=\sqrt{p}-p \\ \\ [7] \ \ &f(x)=\frac{7x^3-5x+13}{\sqrt{x}} \\ \\ [8] \ \ &g(u)=\sqrt{2}u+\sqrt{3u} \\ \\ [9] \ \ &v=t^2-\frac{1}{\sqrt[4]{t^3}} \\ \\ [10] \ \ &f(x)=\frac{7\sqrt{x}+5x}{x^2} \\ \\ [11] \ \ &z=\frac{A}{y^{10}} +B\cos{(y)} \\ \\ [12] \ \ &y=\frac{\sin{(\theta)}}{2}+\frac{c}{\theta} \\ \\ [13] \ \ &H(x)=(x+x^{-1} )^3 \\ \\ [14] \ \ &u=\sqrt[3]{t^2}+2\sqrt{t^3} \end{align*}$$

Part 2: Some Algebra Calculations Needed for Section 2.4 The Product and Quotient Rules

Simplify each expression

Instructions for Meeting M05

You'll work collaboratively on these problems in groups of 2 – 4 students, writing your solutions clearly on a shared set of papers. Each group will turn in one solution set that has all of their names printed on it. Instructor Isaiah Dempsey will grade those solutions and return them to you.

You will also write some of your solutions on the chalkboard. Instructor Dempsey will discuss those chalkboard solutions with the whole class.

Note that your Recitation score is based not just on the thoroughness of your written solutions, but also on how you collaborated with your group and how you presented your work on the chalkboard. There is NOT an expectation that you will be able to finish all of these problems. The expectation is just that you collaborate with your group to solve as many as you can.

Part 1: Some Algebra Calculations Needed for Section 2.4 The Chain Rule

[1] Simplify the expression $$3\left(\frac{x^2+1}{x^2-1}\right)^2\cdot \frac{2x(x^2-1)-(x^2+1)2x}{(x^2-1)^2} $$ Hint: Don’t expand a term like \((x^2-1)^2\) or \((x^2-1)^4\), because the expanded form will be messier. Keep the simpler, factored form. But there are some things in the expression that can be simplified.

[2] Simplify the expression $$\frac{(1)\sqrt{r^2-1}-r\cdot \frac{1}{2\sqrt{r^2-1}}\cdot 2r}{\left(\sqrt{r^2-1}\right)^2} $$

[3] For each equation, find all values of \(x\) that make the equation true.

1. \( \cos (x) = 0\)
2. \( 1+ \sin (x) = 0\)
3. \( 2\cos (x)+ 2\cos (x)\sin (x) = 0\)

Part 2: Some Algebra Calculations Needed for Section 2.6 Implicit Differentiation

[4] Solve each equation for \(y’\). Simplify your answers.

1. $$2x+y+xy'-2yy'=0$$
2. $$y' \ cos{(x)}-y \sin {(x)}=2x+2yy’$$
3. $$-4\sin{(x)} \sin{(y)}+4\cos{(x)}\cos{(y)} y'=0$$
4. $$\sec^2\left(\frac{x}{y}\right)\left(\frac{1}{y}-\frac{x}{y^2}y’\right)=1+y'$$
5. $$\frac{1}{2\sqrt{xy}}\left(y+xy'\right)=2xy+x^2y'$$
6. $$2x+y+xy'+2yy'=0$$
7. $$2x+2yy’=2(2x^2+2y^2-x)(4x+4yy’-1)$$

Part 3: Some Algebra Calculations Needed for Section 2.7 Related Rates

Famous Triangles

[5] A right triangle has hypotenuse of length \(2\)ft and vertical leg of length \(1\)ft.

1. Draw the triangle.
2. Find the length of the horizontal leg. Give an exact answer in symbols, not a decimal approximation.
3. This is a famous triangle. Label its angle measures in both degrees and radians.

[6] A right triangle has a horizontal leg of length \(3\)ft and vertical leg of length \(4\)ft.

1. Draw the triangle.
2. Find the length of the hypotenuse. Give an exact, simplified. This is a famous triangle. Its sides have lengths that are convenient numbers, but its angle measures are not nice numbers, so you don’t need to give the angle measures.

[7] A right triangle has a horizontal leg of length \(5\)ft and vertical leg of length \(12\)ft.

1. Draw the triangle.
2. Find the length of the hypotenuse. Give an exact, simplified. This is a famous triangle. Its sides have lengths that are convenient numbers, but its angle measures are not nice numbers, so you don't need to give the angle measures.

Spheres

[8] Suppose that a sphere has given radius \(r\).

1. Find the Volume \(V\) of the sphere in terms of the given radius \(r\).
2. Find the Surface Area \(A\) of the sphere in terms of the given radius \(r\).

[9] Suppose that a sphere has given surface area \(A\).

1. Find the radius \(r\) of the sphere in terms of the given surface area \(A\).
2. Find the diameter of the sphere in terms of the given surface area \(A\).
3. Find the volume V of the sphere in terms of the given surface area \(A\).

[10] suppose a sphere has given volume \(V\).

1. Find the radius of the sphere in terms of the given volume \(V\).
2. Find the diameter of the sphere in terms of the given volume \(V\).
3. Find the surface area in terms of the given volume \(V\).

Other Shapes

[11] A cone has base radius \(r\) and height \(y\).

1. Draw the cone.
2. What is the volume of the cone, in terms of \(r\) and \(y\)?
Now suppose that it is also known that the diameter is equal to the height.
3. Draw the cone.
4. What is the volume of the cone, in terms of \(y\)?

[12] A trough is \(L\) ft long, and its ends have the shape of isosceles triangles (pointing down) that have a height of \(h\) feet and that are \(3\) times as wide across the top as they are tall. That is, the base of the triangle (which is on top) has width \(w=3h\).

1. Draw the trough.
2. What is the volume \(V_{\text{trough}}\) of the trough, in terms of \(L\) and \(h\)?
Now, suppose that the trough is filled of water to a depth of \(y\) ft, where \(0 \lt y \lt h\)
3. Add a drawing of the water in the trough to your drawing of the trough.
4. What is the volume \(V_{\text{water}}\) of water in the trough, in terms of \(L\) and \(y\)?

Instructions for Meeting M06

You'll work collaboratively on these problems in groups of 2 – 4 students, writing your solutions clearly on a shared set of papers. Each group will turn in one solution set that has all of their names printed on it. Instructor Isaiah Dempsey will grade those solutions and return them to you.

You will also write some of your solutions on the chalkboard. Instructor Dempsey will discuss those chalkboard solutions with the whole class.

Note that your Recitation score is based not just on the thoroughness of your written solutions, but also on how you collaborated with your group and how you presented your work on the chalkboard. There is NOT an expectation that you will be able to finish all of these problems. The expectation is just that you collaborate with your group to solve as many as you can.

Part 1: Some Algebra Calculations Needed for Section 2.8

[1]

1. If \(f(x)=x^{2/3}\), find \(f(8)\) without using technology. Simplify your answer.
2. If \(g(x)=\frac{2}{3}x^{-1/3}\), find \(g(8)\) without using technology. Simplify your answer.

[2]

1. If \(f(x)=\tan (x)\), find \(f\left(\frac{\pi}{4}\right)\) without using technology. Simplify your answer.
2. If \(g(x)=\sec^2(x)\), find \(g\left(\frac{\pi}{4}\right)\) without using technology. Simplify your answer.

Part 2: Some Algebra Calculations Needed for Section 3.1

[3] Consider the exponential function \(y=a^x\), where \(a \gt 0\).

1. What is the domain of this function.
2. If \(a \neq 1\), what is the range of this function?
3. Without using technology, sketch the general shape of the graph of \(y=a^x\) when \(a \gt 1\). Put \((x,y)\) coordinates on two known easy points on the graph.
4. Without using technology, sketch the general shape of the graph of \(y=a^x\) when \(0 \lt a \lt 1\). Put \((x,y)\) coordinates on two known easy points on the graph.

[4]

1. Without using technology, sketch the general shape of the graph of \(y=10^x\). Put \((x,y)\) coordinates on three known easy points on the graph.
2. Without using technology, sketch the general shape of the graph of \(y=10^{x+2}\) by making a transformation of the graph from (a). Put \((x,y)\) coordinates on three known easy points on the graph.

[5]

1. Without using technology, sketch the graph of the basic function \(y=e^x\). Put \((x,y)\) coordinates on two known easy points on the graph.

2. shifting the basic graph \(2\) units downward. Put \((x,y)\) coordinates on two known easy points on the graph.
3. shifting the basic graph \(2\) units to the right. Put \((x,y)\) coordinates on two known easy points on the graph.
4. reflecting the basic graph around the \(x\) axis. Put \((x,y)\) coordinates on two known easy points on the graph.
5. reflecting the basic graph around the \(y\) axis. Put \((x,y)\) coordinates on two known easy points on the graph.
6. reflecting the basic graph around the \(x\) axis and then about the \(y\) axis.

[6] Find the domain of each function

1. \(y=\frac{1-e^{x^2}}{1-e^{\left(1-x^2\right)}}\)
2. \(y=\frac{1+x}{e^{\cos (x)}}\)
3. \(y=\sin \left(e^{-t}\right)\)
4. \(y=\sqrt{1-2^t}\)

[7] An exponential function \(f(x)=Ca^x\) has a graph that passes through the points \((1,6)\) and \((6,24)\). Find the values of \(C\) and \(a\), and use them to write the formula for \(f(x)\).

Part 3: Some Algebra Calculations Needed for Section 3.2

[8] Suppose that \(f\) is a function that is known to be one-to-one.

1. If \(f(6)=17\), what is \(f^{-1}(17)\)?
2. If \(f^{-1}(3)=2\), what is \(f(2)\)?

[9]

1. The function \(f(x)=3+x+e^x\) is known to be one-to-one. Find \(f^{-1}(4)\).
2. The function \(g(x)=2x^3+3x^2+7x+4\) is known to be one-to-one. Find \(g^{-1}(4)\).
3. The function \(h(x)=x^3+3 \sin(x)+2 \cos(x)\) is known to be one-to-one. Find \(h^{-1}(2)\).

[10] The graph of \(f\) is shown.

graph of \(f\) (Click the image to enlarge.)

1. Explain why \(f\) is one-to-one.
2. What are the domain and range of \(f\)?
3. What are the domain and range of \(f^{-1}\)?
4. What is the value of f\(f^{-1}(2)\).?
5. What is the value of \(f^{-1}(0)\).?

[11] Find a formula for the inverse of each function

1. \(f(x)=\frac{2x-1}{4x+3}\).
2. \(g(x)=e^{(2x-1)}\)
3. \(h(x)=\ln (x+3)\)

[12] Solve each inequality for \(x\).

1. \(\ln(x) \lt 0\)
2. \(e^x \gt 5\)
3. \(1 \lt e^{(3x-1)} \lt 2\)
4. \((1-2 \ln(x)) \lt 3\)

Instructions for Meeting M07

Part 1: Some Algebra Calculations Needed for Section 3.3

[1] The function $$G(y)=\ln\left(\frac{(2y+1)^5}{\sqrt{y^2+1}}\right)$$ is extremely complicated. It is a composition of functions–functions nested inside of other functions–in which the nesting is four layers deep: $$G(y)=\ln\left(\text{fraction}\left(\text{power function}\left(\text{polynomial function}\right)\right)\right)$$ Use rules of logarithms to rewrite \(G(y)\) in a simpler form: $$G(y)=a\cdot \ln\left(\text{polynomial function}\right)-b\cdot \ln\left(\text{polynomial function}\right)$$

[2] The function $$y=\log_{10}{\left(\sqrt{x}\right)}$$ is a composition of functions. Use a rule of logarithms to rewrite the function in a simpler form.

Part 2: Some Algebra Calculations Needed for Section 3.4

[3] In problems involving exponential growth and decay, some quantity \(y(t)\) is known to obey the equation $$\frac{dy}{dt}=ky$$ where \(k\) is a constant. The generic form for the solution is $$y(t)=y(0)e^{kt}$$ Observe that the base \(b\) of the exponential function in this generic form is the number \(b=e\). That is, the generic form is written using the natural exponential function.

It is common for solutions to result in a \(k\) that looks like this: $$k=\frac{\ln(r)}{T}$$ Then the generic solution looks like this: $$y(t)=y(0)e^{\left( \frac{\ln(r)}{T}\cdot t\right)}$$ For example, if a population of bacteria has the property that the population size triples every \(7\) hours, then the value of \(k\) will turn out to be $$k=\frac{\ln(3)}{7}$$ So the solution looks like this: $$y(t)=y(0)e^{\left( \frac{\ln(3)}{7} \cdot t\right)}$$

In another example, if some radioactive substance has the property that it decays in a way that the mass halves every \(7000\) years, then the value of \(k\) will turn out to be $$k=\frac{\ln\left(\frac{1}{2}\right)}{7000}$$ So the solution looks like this: $$y(t)=y(0) e^{ \left( \frac{\ln\left(\frac{1}{2}\right)}{7000}\cdot t \right) }$$

Although it is very nice to have one form for the solution that describes every situation, in any particular situation, it is ultimately more useful to rewrite the generic form into a form tailored to that particular situation. One helpful way to rewrite the solution is to convert the exponential function to one that has a different base.

1. Using rules of exponents and logarithms, rewrite the generic solution $$y(t)=y(0)e^{\left( \frac{\ln(r)}{T} \cdot t\right)}$$ in an equivalent form that has an exponential function with base \(b=r\).
2. Using rules of exponents and logarithms, rewrite the generic solution $$y(t)=y(0)e^{\left( \frac{\ln(3)}{7} \cdot t\right)}$$ in an equivalent form that has an exponential function with base \(b=3\).
3. Using rules of exponents and logarithms, rewrite the generic solution $$y(t)=y(0) e^{ \left( \frac{\ln\left(\frac{1}{2}\right)}{7000}\cdot t \right) }$$ in an equivalent form that has an exponential function with base \(b=\frac{1}{2}\).
4. Using rules of exponents and logarithms, rewrite the function $$y(t)=75+100e^{ \left( \frac{ \ln \left( \frac{75}{110}\right)}{30} \cdot t \right)}$$ in an equivalent form that has an exponential function with base \(b=\frac{75}{110}\).

[4] The goal is to graph the function $$y=2-e^{(-t)}$$ without technology, using transformations.

1. Graph the basic function \(y=e^t\). Label all asymptotes with their line equations, and put (x,y) coordinates on two easy points.
2. On a new set of axes, graph the function \(y=e^{(-t)}\) by transforming your graph from (a). Label all asymptotes with their line equations, and put \((x,y)\) coordinates on two easy points.
3. On a new set of axes, graph the function \(y=-e^{(-t)}\) by transforming your graph from (b). Label all asymptotes with their line equations, and put \((x,y)\) coordinates on two easy points.
4. On a new set of axes, graph the function \(y=2-e^{(-t)}\) by transforming your graph from (c). Label all asymptotes with their line equations, and put \((x,y)\) coordinates on two easy points.

Part 3: Some Algebra Calculations Needed for Section 3.5

[5] Without using technology, find the exact value of each expression.

1. \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
2. \(\cos^{-1}\left(-1 \right)\)
3. \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
4. \(\sec^{-1}\left(2 \right)\)
5. \(\arctan\left(1 \right)\)
6. \(\arcsin\left(\frac{1}{\sqrt{2}} \right)\)
7. \(\tan\left(\arctan(10)\right)\)
8. \(\sin^{-1}\left(\sin\left(\frac{7\pi}{3}\right)\right)\) (This one is tricky!)
9. \(\sin\left(\tan^{-1}(x)\right)\) (Draw a right triangle to illustrate.)

Instructions for Meeting M08

Part 1: Some Algebra Calculations Needed for Section 3.7

[1] Rewrite $$x^3e^{-x^2}$$ as a quotient.

[2] Find \(\tan^{-1}(0)\).

[3] Find the value of $$\frac{1}{\frac{1}{1+(9x)^2}\cdot 9}$$ when \(x=0\).

[4] Find the value of $$\cos(x)-1+\frac{1}{2}x^2$$ when \(x=0\).

[5] Find the value of $$-\sin(x)+x$$ when \(x=0\).

[6] Find the value of $$-\cos(x)+1$$ when \(x=0\).

[7] Rewrite the expression $$4x-\ln(x)$$ by factoring out a \(4x\). (The result will look weird.)

[8] Use rules of logarithms to rewrite the expressions. $$\ln\left(a^b\right) \\ \ln\left(\left(f(x)\right)^2\right) \\ \ln\left(\sqrt{f(x)}\right) \\ \ln\left(\left(f(x)\right)^{\frac{1}{x}}\right) \\ \ln\left(\left(1-5x \right)^{\frac{1}{x}}\right) $$

Part 2: Some famous numbers that always trip students up on exams

[9] Sketch a graph of \(y=e^x\). Put \((x,y)\) coordinates on two famous points on the graph. One is the point where \(x=1\). The other is the point where \(x=1\). Write the \(y\) values exactly, not as decimal approximations. Label the asymptote with its line equation.

[10] Sketch a graph of \(y=\ln(x)\) by flipping your graph of \(y=e^x\) across the line \(y=x\). Put \((x,y)\) coordinates on the two famous point on the graph. (They should be points obtained by flipping the two famous points on your graph of \(y=e^x\).) Write the \((x,y)\) values exactly, not as decimal approximations. Label the asymptote with its line equation.

[11] Use your graphs of \(y=e^x\) and \(y=\ln(x)\) to find these values:

1. \(e^0\)
2. \(e^1\)
3. \(\ln(1)\)
4. \(\ln(e)\)

Part 3: Famous Shapes and their Areas and Volumes

[12] Sphere

1. Sketch it
2. Write the formula for volume as a function of radius.
3. Write the formula for volume as a function of diameter.
4. Write the formula for surface area as a function of radius.
5. Write the formula for surface area as a function of diameter.

[13] Right circular cone with base radius \(r\) and height \(h\)

1. Sketch it
2. Write the formula for volume.

[14] Pyramid with square base of sides \(x\) and height \(h\)

1. Sketch it
2. Write the formula for volume.

Instructions for Meeting M09

Review: When is a fraction equal to zero? When is it undefined?

Remember that the equation \(\displaystyle \frac{a}{b}=0\) is only true when both of the following things are true:

• The numerator \(a=0\).
• The denominator \(b\neq 0\).

Part 1: Some Algebra Skills Needed for Section 4.1

For each expression below, answer the following two questions:

1. For what values of \(x\) is the expression equal to zero?
2. For what values of \(x\) is the expression undefined?

[1] \( \displaystyle \ \frac{-x^2+2x}{(x^2-x+1)^2 } \)

[2] \( \displaystyle \ \frac{-x^2+2x+4}{(x^2+4)^2 } \)

[3] \( \displaystyle \ -e^{(-3x)}(3x^2-2x) \)

[4] \( \displaystyle \ 6x^2-6x+12 \) on the interval \([-2,3] \)

[5] \( \displaystyle \ -\frac{2(x^2-2)}{\sqrt{4-x^2}} \) on the interval \([-1,2] \)

[6] \( \displaystyle \ -\frac{1}{4}e^{-\frac{x^2}{8}}(x^2-4) \) on the interval \([-1,4] \)

[7] \( \displaystyle \ \frac{2x+1}{x^2+x+1} \) on the interval \([-1,1] \)

Part 2: Some Algebra Skills Needed for Section 4.2

For each expression below, find all values of \(x\) for which the expression is equal to zero.

[8] \( \displaystyle \ -12+6x \text{ on the interval }(1,3)\)

[9] \( \displaystyle \ \frac{1}{2\sqrt{x}}-\frac{1}{3} \text{ on the interval }(0,9)\)

[10] \( \displaystyle \ \frac{2}{3\cdot \sqrt[3]{x}} \text{ on the interval }(-1,1)\)

[11]

1. Graph \(y=\cos(x)\) and \(y=-2x\) together on the same set of axes, on the interval \([-\pi/2,\pi/2]\).
   Label all important points with their \((x,y)\) coordinates.
   • Label the \((x,y)\) coordinates of the points on the graphs at the endpoints of the interval (with \(x,y\) values written exactly, in symbols, not written as decimal approximations)
   • Label the \((x,y)\) coordinates of all axis crossings (exact values).
   • Estimate (by eyeballing) the \((x,y)\) coordinates of the point where the graphs seem to cross. Label that point with its estimated \((x,y)\) coordinates.
2. What is the exact value of the expression \(2x+\cos(x) \) when \(x=-\frac{\pi}{2}\)?
3. What is the exact value of the expression \(2x+\cos(x) \) when \(x=\frac{\pi}{2}\)?

Part 3 (The Hard Part!): Sign Chart Skills Needed for Section 4.3

Recall the procedure for making a sign chart for a function \(y=f(x)\)

• Find the partition numbers for \(f(x)\). These are \(x\) values \(x=c\) where either of the following conditions is true:
  • \(f(c)=0\).
  • \(f(x)\) is discontinuous at \(x=c\). (In MATH 2301, the \(x\) values where \(f(x)\) is discontinuous are usually just the \(x\) values where \(f(x)\) does not exist.)
• Mark the partition numbers on a number line (in increasing order from left to right). At each partition number, put the \(x\) value of the partition number below the \(x\) axis.
• At each partition number, put the behavior of \(f(x)\) (whether \(f(c)=0\) or \(f(c) \ DNE\)) above the number line.

Remark: The partition numbers create open intervals on the number line. On each of these open intervals, function \(f(x)\) is continuous and non-zero. Therefore if you determine the sign (positive or negative) of \(f(x)\) at any \(x\) value within one of the open intervals, you will know that \(f(x)\) will have that same sign on the entire open interval.


• Within each open interval, choose a test \(x\) value, \(x_{\text{test}}\). Compute the sign (positive or negative) of \(f(x_{\text{test}})\) at that sample \(x\) value. The sign of \(f(x_{\text{test}})\) is the sign that \(f(x)\) has on that entire open interval. Write \(f \ pos\) or \(f \ neg\) above the number line on that interval.

[12] Make a sign chart for \(y=(x+1)^2 (x-3)^5 (x-6)^4\).
Hint: Remember that at each sample number \(x=x_{\text{test}}\), you don’t need to know the exact value of \(f(x_{\text{test}})\) . You just need to determine whether \(f(x_{\text{test}})\) is positive or negative.

[13]

1. Make a sign chart for \(y=6x^2+6x-36\).
2. Make a sign chart for \(y=12x+6\).

[14]

1. Make a sign chart for \(y=\cos(x)-\sin(x)\) on the interval \([0,2\pi]\).
2. Make a sign chart for \(y=-\sin(x)-\cos(x)\) on the interval \([0,2\pi]\).

[15]

1. Make a sign chart for \(y=2e^{2x}-e^{-x}\).
2. Make a sign chart for \(y=4e^{2x}+e^{-x}\).

[16]

1. Make a sign chart for \(y=e^{-x} (-x^4+4x^3 )\).
2. Make a sign chart for \( y=e^{-x} (x^4-8x^3+12x^2)\).

[17]

1. Make a sign chart for \(y=4x-4x^3\).
2. Make a sign chart for \(y=4-12x^2\).

[18]

1. Make a sign chart for \(y=-2\sin(\theta)-2\cos(\theta)\sin(\theta)\) on the interval \([0,2\pi]\).
2. Make a sign chart for \(y=-2\left(-\sin^2(\theta)+\cos^2(\theta)+\cos(\theta)\right)\) on the interval \([0,2\pi]\).

[19]

1. Make a sign chart for \(\displaystyle \ y=-\frac{1}{x^2} +\frac{2}{x^3}\).
2. Make a sign chart for \(\displaystyle \ y=\frac{1}{x^3} -\frac{6}{x^4}\).

Meeting M10 Algebra Skills Needed for Section 4.4

Section 4.4 is about Curve Sketching, using the Guidelines for Sketching a Curve presented in the section. Much of the work in those guidelines involves algebra skills.

One important algebra skill used in curve sketching is determining whether the graph of a function will be symmetric.

Testing a Function for Symmetry

1. Write the formula for \(f(x)\).
2. Build the expression for \(f(-x)\) by replacing all of the x with -x in the formula for \(f(x)\). Simplify the expression for \(f(-x)\).
3. Build the expression for \(-f(x)\) by just parking a minus sign in front of the formula for \(f(x)\). Simplify the expression for \(-f(x)\).
4. You now have three expressions:
   • \(f(x)\)
   • \(f(-x)\)
   • \(-f(x)\)
   Scrutinize them to see if any of them match, and make a conclusion.
   • If \(f(-x)=f(x)\) then f is an even function. Its graph will have even symmetry: the left side of the graph will be a mirror image of the right side of the graph.
   • If \(f(-x)=-f(x)\) , then f is an odd function. Its graph will have odd symmetry: the left side of the graph will be an upside down mirror image of the right side of the graph.

Another important algebra skill used in curve sketching is making sign charts.

Procedure for Making a Sign Chart for a Function \(y=f(x)\)

1. Find the partition numbers for \(f(x)\). These are \(x\) values \(x=c\) where either of the following conditions is true:
   • \(f(c)=0\).
   • \(f(x)\) is discontinuous at \(x=c\). (In MATH 2301, the \(x\) values where \(f(x)\) is discontinuous are usually just the \(x\) values where \(f(x)\) does not exist.)
2. Mark the partition numbers on a number line (in increasing order from left to right). At each partition number, put the \(x\) value of the partition number below the \(x\) axis.
3. At each partition number, put the behavior of \(f(x)\) (whether \(f(c)=0\) or \(f(c) \ DNE\)) above the number line.

Remark: The partition numbers create open intervals on the number line. On each of these open intervals, function \(f(x)\) is continuous and non-zero. Therefore if you determine the sign (positive or negative) of \(f(x)\) at any \(x\) value within one of the open intervals, you will know that \(f(x)\) will have that same sign on the entire open interval.


4. Within each open interval, choose a test \(x\) value, \(x_{\text{test}}\). Compute the sign (positive or negative) of \(f(x_{\text{test}})\) at that sample \(x\) value. The sign of \(f(x_{\text{test}})\) is the sign that \(f(x)\) has on that entire open interval. Write \(f \ pos\) or \(f \ neg\) above the number line on that interval.

Four Problems for Meeting M10

[1] (Algebra skills needed for HW problem 4.4#11)

1. What is the domain of the function \(y=\frac{1}{x^2-9}\)?
2. Will the graph of the function \(y=\frac{1}{x^2-9}\) have any vertical asymptotes? If so, what are their line equations?
3. Will the graph of the function \(y=\frac{1}{x^2-9}\) have any horizontal asymptotes? If so, what are their line equations?
4. Test the function \(y=\frac{1}{x^2-9}\) for symmetry.
5. Can the expression \(x^2-9\) be factored?
6. Can the expression \(x^2+3\) be factored?

In the sign charts for this problem, be sure to indicate the behavior at the partition numbers. That is, say whether the value is zero or the value is undefined.

7. Make a sign chart for \(y=\frac{1}{x^2-9}\)
8. Make a sign chart for \(y=-\frac{2x}{(x^2-9)^2}\).
9. Make a sign chart for \(y=\frac{6(x^2+3)}{(x^2-9)^2}\).

[2] (Algebra skills needed for HW problem 4.4#13)

1. What is the domain of the function \(y=\frac{x}{x^2+9}\)?
2. Will the graph of the function \(y=\frac{x}{x^2+9}\) have any vertical asymptotes? If so, what are their line equations?
3. Will the graph of the function \(y=\frac{x}{x^2+9}\) have any horizontal asymptotes? If so, what are their line equations?
4. Test the function \(y=\frac{x}{x^2+9}\) for symmetry.
5. Can the expression \(x^2+9\) be factored?
6. Can the expression \(x^2-9\) be factored?
7. The expression \(x^2-27\) can be factored. Factor it.

In the sign charts for this problem, be sure to indicate the behavior at the partition numbers. That is, say whether the value is zero or the value is undefined.

8. Make a sign chart for \(y=\frac{x}{x^2+9}\).
9. Make a sign chart for \(y=-\frac{x^2-9}{(x^2+9)^2}\).
10. Make a sign chart for \(y=\frac{2x(x^2-27)}{(x^2+9)^3}\).

[3] (Algebra skills needed for HW problem 4.4#34)

1. What is the domain of the function \(y=\frac{\sin(x)}{2+\cos(x)}\)?
2. Is the function \(y=\frac{\sin(x)}{2+\cos(x)}\) periodic? If so, what is the period?
3. Will the graph of the function \(y=\frac{\sin(x)}{2+\cos(x)}\) have any vertical asymptotes? If so, what are their line equations?
4. Will the graph of the function \(y=\frac{\sin(x)}{2+\cos(x)}\) have any horizontal asymptotes? If so, what are their line equations?
5. Test the function \(y=\frac{\sin(x)}{2+\cos(x)}\) for symmetry.
6. Make a sign chart for \(y=\frac{\sin(x)}{2+\cos(x)}\).
7. Make a sign chart for \(y=\frac{1+2\cos(x)}{(2+\cos(x))^2}\).
8. Make a sign chart for \(y=\frac{2\sin(x)(\cos(x)-1)}{(2+\cos(x))^3}\).

[4] (Algebra skills needed for HW problem 4.4#39)

1. What is the domain of the function \(y=xe^{-x}\)?
2. Will the graph of the function \(y=xe^{-x}\) have any vertical asymptotes? If so, what are their line equations?
3. Will the graph of the function \(y=xe^{-x}\) have any horizontal asymptotes? If so, what are their line equations?
4. Test the function \(y=xe^{-x}\) for symmetry.
5. Make a sign chart for \(y=xe^{-x}\).
6. Make a sign chart for \(y=-(x-1)e^{-x}\).
7. Make a sign chart for \(y=(x-2)e^{-x}\).

Five Problems for Meeting M11

Part 1: some Algebra Skills Needed for Section 4.6

The main algebra skill needed in Section 4.6 is the skill of populating given formulas with the correct numbers and simplifying.

[1] The formula for Newton’s Method is $$x_{n+1}=x_n-\frac{f(x_n)}{f’(x_n)}$$

1. Suppose that it is known that $$\begin{align*} f(x) &=x^3-x^2-1 \\ f’(x)&=3x^2-2x \\ x_1&=1 \end{align*}$$ Use Newton’s Method to find \(x_2\) and \(x_3\). Work in fractions, not decimals.
2. Same function, this time use \(x_1=\frac{3}{2}\) and find \(x_2\). Don’t bother finding \(x_3\). (It would be too messy.) But again, work in fractions, not decimals.

Part 2: Review: Various HW problems from Chapter 4

For problems [2], [3], [4], do the following three things:

1. Determine the values of \(x\) where the expression is undefined.
2. Determine the values of \(x\) where the expression is equal to zero.
3. Make a sign chart for the expression. be sure to indicate the behavior at the partition numbers. That is, say whether the value is zero or the value is undefined.

[2] \(y=-e^{-3x} (3x^2-2x)\)

[3] \(y=\frac{2(x-3)}{x^4}\)

[4] \(y=\cos(x)-\sin(x)\) on the interval \([0,2\pi]\).

Review problem about Curve Sketching, using the Guidelines for Sketching a Curve presented in Section 4.4. Much of the work in those guidelines involves algebra skills. If you need a refresher on how to test a function for symmetry, or how to make a sign chart, look back at the instructions for Meeting M10, Thursday March 26.

[5]

1. What is the domain of the function \(y=\frac{x}{x-5}\)?
2. Give the \((x,y)\) coordinates of all axis intercepts.
3. Will the graph of the function \(y=\frac{x}{x-5}\) have any vertical asymptotes? If so, what are their line equations?
4. Will the graph of the function \(y=\frac{x}{x-5}\) have any horizontal asymptotes? If so, what are their line equations?
5. Test the function \(y=\frac{x}{x-5}\) for symmetry.
6. Make a sign chart for \(y=\frac{x}{x-5}\). Be sure to indicate the behavior at the partition numbers. That is, say whether the value is zero or the value is undefined.

Seven Problems for Meeting M12

Some Algebra Skills Needed for Section 4.7

The main algebra skill needed in Section 4.7 is the skill of rewriting functions in power function form.

[1] It would be possible to find the derivative of a function like $$g(t)=\frac{1+t+t^2}{\sqrt{t}}$$ using the Quotient Rule for Derivatives. But that would not be the smart way to the derivative: It would be messy and prone to mistakes. A much smarter way to find the derivative would be to rewrite \(g(t)\) in power function form. That is, rewrite the function as a sum of terms of the form $$c \cdot t^p$$ where \(c\) is a real number constant and \(t^p\) is a power function. Then, the derivative can be found very simply, using the constant multiple rule for derivatives and the power rule for derivatives. So, when finding \( g’(t) \), you have a choice: the hard way (quotient rule for derivatives) or the easy way (convert to power function form first, then use easier derivative rules).

But realize that when finding an antiderivative of \(g(t)\), you have no choice: there is no quotient rule for antiderivatives. In order to find the antiderivative, you must first rewrite \(g(t)\) in power function form.

Rewrite each function below in power function form. (Do not find any antiderivatives or derivatives! Just rewrite the function.) $$\begin{align*} &(\text{a}) \ \ g(t) =\frac{1+t+t^2}{\sqrt{t}} \\ &(\text{b}) \ \ f(t)=\frac{3t^4-t^3+6t^2}{t^4 } \\ &(\text{c}) \ \ f(x)=\frac{x^5-x^3+2x}{x^4} \\ &(\text{d}) \ \ g(t) =\frac{3}{\sqrt{t}} \\ &(\text{e}) \ \ f(x)=3\sqrt{x}-2\sqrt[3]{x} \\ &(\text{f}) \ \ f(x)=\sqrt[3]{x^2}+x\sqrt{x} \\ \end{align*}$$

Some Algebra Skills Needed for Section 5.1

[2] (no technology) For the given graph, use six rectangles to find estimates of each type for the area under the given graph of \(f(x)\) from \(x=0\) to \(x=12\).

graph of \(f\) (Click the image to enlarge.)

1. Compute \(L_6\) (sample points are left endpoints)
2. Compute \(R_6\) (sample points are right endpoints)
3. Compute \(M_6\) (sample points are midpoints)
4. Is \(L_6\) an underestimate or overestimate of the true area?
5. Is \(R_6\) an underestimate or overestimate of the true area?
6. Which of the numbers \(L_6\), \(R_6\), or \(M_6\) gives the best estimate? Explain.

[3] (no technology)

1. Estimate the area under the graph of \(f(x)=1+x^2\) from \(x=-3\) to \(x=5\) using three rectangles and right endpoints. (Get a number for the estimate, without using a calculator.) Sketch the curve and the approximating rectangles.
2. Repeat part (a) using midpoints.

[4] (no technology) The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the list below. Find lower and upper estimates for the distance that she traveled during those three seconds.

• At time \(t=0\) sec, her speed was \(v=0\) ft/sec.
• At time \(t=0.5\) sec, her speed was \(v=6.2\) ft/sec.
• At time \(t=1.0\) sec, her speed was \(v=10.8\) ft/sec.
• At time \(t=1.5\) sec, her speed was \(v=14.9\) ft/sec.
• At time \(t=2.0\) sec, her speed was \(v=18.1\) ft/sec.
• At time \(t=2.5\) sec, her speed was \(v=19.4\) ft/sec.
• At time \(t=3.0\) sec, her speed was \(v=20.2\) ft/sec.

Some Algebra Skills Needed for Section 5.2

[5]

1. For \(f(x)=e^x-2\) on the interval \(0 \leq x \leq 2\), find a value for \(M_4\), correct to six decimal places. (Use technology to get decimal approximations for the heights of the rectangles, and to do the sum.)
2. Illustrate with a diagram.

[6] For the given graph of \(f(x)\), compute the signed areas listed below.

graph of \(f\) (Click the image to enlarge.)

1. The signed area between the graph of \(f(x)\) and the \(x\) axis from \(x=0\) to \(x=2\).
2. The signed area between the graph of \(f(x)\) and the \(x\) axis from \(x=0\) to \(x=5\).
3. The signed area between the graph of \(f(x)\) and the \(x\) axis from \(x=5\) to \(x=7\).
4. The signed area between the graph of \(f(x)\) and the \(x\) axis from \(x=0\) to \(x=9\).

[7] For the given graph of \(g(x)\), compute the signed areas listed below.

graph of \(g\) (Click the image to enlarge.)

1. The signed area between the graph of \(g(x)\) and the \(x\) axis from \(x=0\) to \(x=2\).
2. The signed area between the graph of \(g(x)\) and the \(x\) axis from \(x=2\) to \(x=6\).
3. The signed area between the graph of \(g(x)\) and the \(x\) axis from \(x=0\) to \(x=7\).

Instructions for Meeting M13 (Short Meeting)

Some Algebra Skills Needed for Section 5.3

Converting Integrands Into Forms That Can Be Integrated

[1] One of your homework problems involves finding an integral where the integrand is the expression $$x\left(\sqrt[3]{x}+\sqrt[4]{x}\right)$$ None of our integral rules apply to that form of integrand. It must first be rewritten in a form that we can integrate. Rewrite the expression in power function form.

[2] One of your homework problems involves finding an integral where the integrand is the expression $$\frac{(x-1)^3}{x^2}$$ None of our integral rules apply to that form of integrand. Simplify the expression into a form that we can integrate.

[3] One of your homework problems involves finding an integral where the integrand is the expression $$\frac{\sin(\theta)+\sin(\theta)\tan^2(\theta)}{\sec^2(\theta)}$$ None of our integral rules apply to that form of integrand. Simplify the expression into a form that we can integrate.

Dealing With Absolute Value

[4] One of your homework problems involves finding an integral where the integrand is the expression $$x-2|x|$$ We have no integral rules for \(|x|\). Recognize that \(|x|\) is a piecewise-defined function. That is, the symbol \(|x|\) means different things on different parts of its domain. $$|x| = \begin{cases} x & \text{when } x \geq 0,\\ -x & \text{when } x \lt 0. \end{cases}$$

1. What does the expression \(x-2|x|\) mean when \(x \geq 0\)? (That is, write a new expression that has the same meaning on that part of the domain but that does not use the absolute value symbol.)
2. What does the expression \(x-2|x|\) mean when \(x \lt 0\)?

[5] Another homework problem involves finding a definite integral where the integrand is $$|2x-1|$$ We have no integral rules for \(|2x-1|\). But again, recognize that it is a piecewise-defined function. That is, the symbol \(|2x-1|\) means different things on different parts of its domain. $$|x| = \begin{cases} x & \text{when } x \geq 0,\\ -x & \text{when } x \lt 0. \end{cases}$$

1. What are the two important parts of the domain for the function \(|2x-1|\)?
2. On each part of the domain, rewrite \(|2x-1|\) as a new expression that has the same meaning on that part of the domain but that does not use the absolute value symbol.

Fractional Powers and Evaluation Notation

[6] For \(f(x)=x^{3/2}\), $$\begin{align*} &\text{(a) Find }f(9) \\ &\text{(b) Find }f(x)\Big|_{x=4} \\ &\text{(c) Find }f(x)\Big|_{x=4}^{x=25} \\ \end{align*}$$

Instructions for Meeting M14

Review Day

In the days since Meeting M13, MATH 2301 has covered Sections 5.4 (The Fundamental Theorem of Calculus) and 5.5 (The Substitution Rule). Because there is no significant algebra content in those sections, and because next week will be your final exam, today will be a review day, revisiting problems that you have worked on throughout the semester.

From Meeting M03: Some Algebra Calculations Needed for Section 2.2 The Derivative as a Function

[5] For the function $$f(t)=\frac{2-3t}{5+t}$$

1. Write the empty version of \(f\).
2. Using your result from (a), write the expression for \(f(x+h)\).
3. Using your result from (b), write the expression for \(f(x+h)-f(x)\). Your expression should be the difference of two fractions.
4. Rewrite the expression from (c) by getting a common denominator. Simplify the expression.
5. Using your result from (d), write the expression for \(\frac{f(x+h)-f(x)}{h}\).
6. The expression from (e) is problematic because it is a fraction with a fraction in the numerator. That means that there are two baselines in the expression. Rewrite your expression for \(\frac{f(x+h)-f(x)}{h}\) from (e) by rewriting $$\frac{(\text{fraction})}{h}=\frac{1}{h}\cdot (\text{fraction})$$ This new expression will have a single baseline.
7. Simplify the expression, from (f) assuming that \(h \neq 0\). Note that this will enable you to cancel the \(\frac{h}{h}\).

From Meeting M04 Part 1: Some Algebra Calculations Needed for Section 2.3 Basic Differentiation Properties

Following below are some functions taken from exercises in Section 2.3 of the textbook for MATH 2301. In those exercises, you are asked to find the derivative of each function, using the simple derivative rules presented in Section 2.3. But for every one of these fourteen functions, you would need to first do the algebra task of rewriting the function in a form where the derivative rules can be used. Your job today is to do just that algebra task. That is, rewrite each function in power function form. DO NOT TAKE THE DERIVATIVE in this assignment. Just do the preliminary algebra. (In your calculus homework, you would go on and take the derivative.) $$\begin{align*} [3] \ \ &A(s)=-\frac{7}{s^{13}} \\ \\ [5] \ \ &y=5\sqrt{x}(3x-7) \\ \\ [7] \ \ &f(x)=\frac{7x^3-5x+13}{\sqrt{x}} \\ \\ [9] \ \ &v=t^2-\frac{1}{\sqrt[4]{t^3}} \\ \\ [10] \ \ &f(x)=\frac{7\sqrt{x}+5x}{x^2} \\ \\ \end{align*}$$

From Meeting M04 Part 2: Some Algebra Calculations Needed for Section 2.4 The Product and Quotient Rules

Simplify each expression

From Meeting M05 Part 2: Some Algebra Calculations Needed for Section 2.6 Implicit Differentiation

[4] Solve each equation for \(y’\). Simplify your answers.

1. $$2x+y+xy'-2yy'=0$$
2. $$2x+2yy’=2(2x^2+2y^2-x)(4x+4yy’-1)$$

From Meeting M05 Part 3: Some Algebra Calculations Needed for Section 2.7 Related Rates

Famous Triangles

[5] A right triangle has hypotenuse of length \(2\)ft and vertical leg of length \(1\)ft.

1. Draw the triangle.
2. Find the length of the horizontal leg. Give an exact answer in symbols, not a decimal approximation.
3. This is a famous triangle. Label its angle measures in both degrees and radians.

[6] A right triangle has a horizontal leg of length \(3\)ft and vertical leg of length \(4\)ft.

1. Draw the triangle.
2. Find the length of the hypotenuse. Give an exact, simplified. This is a famous triangle. Its sides have lengths that are convenient numbers, but its angle measures are not nice numbers, so you don’t need to give the angle measures.

[7] A right triangle has a horizontal leg of length \(5\)ft and vertical leg of length \(12\)ft.

1. Draw the triangle.
2. Find the length of the hypotenuse. Give an exact, simplified. This is a famous triangle. Its sides have lengths that are convenient numbers, but its angle measures are not nice numbers, so you don't need to give the angle measures.

Spheres

[8] Suppose that a sphere has given radius \(r\).

1. Find the Volume \(V\) of the sphere in terms of the given radius \(r\).
2. Find the Surface Area \(A\) of the sphere in terms of the given radius \(r\).

[9] Suppose that a sphere has given surface area \(A\).

1. Find the radius \(r\) of the sphere in terms of the given surface area \(A\).
2. Find the diameter of the sphere in terms of the given surface area \(A\).
3. Find the volume V of the sphere in terms of the given surface area \(A\).

[10] suppose a sphere has given volume \(V\).

1. Find the radius of the sphere in terms of the given volume \(V\).
2. Find the diameter of the sphere in terms of the given volume \(V\).
3. Find the surface area in terms of the given volume \(V\).

Other Shapes

[11] A cone has base radius \(r\) and height \(y\).

1. Draw the cone.
2. What is the volume of the cone, in terms of \(r\) and \(y\)?
Now suppose that it is also known that the diameter is equal to the height.
3. Draw the cone.
4. What is the volume of the cone, in terms of \(y\)?

[12] A trough is \(L\) ft long, and its ends have the shape of isosceles triangles (pointing down) that have a height of \(h\) feet and that are \(3\) times as wide across the top as they are tall. That is, the base of the triangle (which is on top) has width \(w=3h\).

1. Draw the trough.
2. What is the volume \(V_{\text{trough}}\) of the trough, in terms of \(L\) and \(h\)?
Now, suppose that the trough is filled of water to a depth of \(y\) ft, where \(0 \lt y \lt h\)
3. Add a drawing of the water in the trough to your drawing of the trough.
4. What is the volume \(V_{\text{water}}\) of water in the trough, in terms of \(L\) and \(y\)?

From Meeting M06 Part 1: Some Algebra Calculations Needed for Section 2.8

[1]

1. If \(f(x)=x^{2/3}\), find \(f(8)\) without using technology. Simplify your answer.
2. If \(g(x)=\frac{2}{3}x^{-1/3}\), find \(g(8)\) without using technology. Simplify your answer.

[2]

1. If \(f(x)=\tan (x)\), find \(f\left(\frac{\pi}{4}\right)\) without using technology. Simplify your answer.
2. If \(g(x)=\sec^2(x)\), find \(g\left(\frac{\pi}{4}\right)\) without using technology. Simplify your answer.

From Meeting M07 Part 3: Some Algebra Calculations Needed for Section 3.5

[5] Without using technology, find the exact value of each expression.

1. \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
2. \(\cos^{-1}\left(-1 \right)\)
3. \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
4. \(\sec^{-1}\left(2 \right)\)
5. \(\arctan\left(1 \right)\)
6. \(\arcsin\left(\frac{1}{\sqrt{2}} \right)\)
7. \(\tan\left(\arctan(10)\right)\)
8. \(\sin^{-1}\left(\sin\left(\frac{7\pi}{3}\right)\right)\) (This one is tricky!)
9. \(\sin\left(\tan^{-1}(x)\right)\) (Draw a right triangle to illustrate.)

From Meeting M08 Part 2: Some famous numbers that always trip students up on exams

[9] Sketch a graph of \(y=e^x\). Put \((x,y)\) coordinates on two famous points on the graph. One is the point where \(x=1\). The other is the point where \(x=1\). Write the \(y\) values exactly, not as decimal approximations. Label the asymptote with its line equation.

[10] Sketch a graph of \(y=\ln(x)\) by flipping your graph of \(y=e^x\) across the line \(y=x\). Put \((x,y)\) coordinates on the two famous point on the graph. (They should be points obtained by flipping the two famous points on your graph of \(y=e^x\).) Write the \((x,y)\) values exactly, not as decimal approximations. Label the asymptote with its line equation.

[11] Use your graphs of \(y=e^x\) and \(y=\ln(x)\) to find these values:

1. \(e^0\)
2. \(e^1\)
3. \(\ln(1)\)
4. \(\ln(e)\)

From Meeting M09 Part 3 (The Hard Part!): Sign Chart Skills Needed for Section 4.3

[16]

1. Make a sign chart for \(y=e^{-x} (-x^4+4x^3 )\).

From Meeting M11 Part 1: some Algebra Skills Needed for Section 4.6

The main algebra skill needed in Section 4.6 is the skill of populating given formulas with the correct numbers and simplifying.

[1] The formula for Newton’s Method is $$x_{n+1}=x_n-\frac{f(x_n)}{f’(x_n)}$$

1. Suppose that it is known that $$\begin{align*} f(x) &=x^3-x^2-1 \\ f’(x)&=3x^2-2x \\ x_1&=1 \end{align*}$$ Use Newton’s Method to find \(x_2\) and \(x_3\). Work in fractions, not decimals.
2. Same function, this time use \(x_1=\frac{3}{2}\) and find \(x_2\). Don’t bother finding \(x_3\). (It would be too messy.) But again, work in fractions, not decimals.

From Meeting M12 Some Algebra Skills Needed for Section 5.1

[2] (no technology) For the given graph, use six rectangles to find estimates of each type for the area under the given graph of \(f(x)\) from \(x=0\) to \(x=12\).

graph of \(f\) (Click the image to enlarge.)

1. Compute \(L_6\) (sample points are left endpoints)
2. Compute \(R_6\) (sample points are right endpoints)
3. Compute \(M_6\) (sample points are midpoints)
4. Is \(L_6\) an underestimate or overestimate of the true area?
5. Is \(R_6\) an underestimate or overestimate of the true area?
6. Which of the numbers \(L_6\), \(R_6\), or \(M_6\) gives the best estimate? Explain.