Students Solving Problems and Discussing Their Solutions

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

Scoring: If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their R02 score will be 5/5. (For this Sep 5 Recitation, students will get 5/5 regardless of whether their solutions are correct. In the future, the scoring will be more stringent.) If they do not present a solution, their R02 score will be 0/5.

Students find their Student Number in the lists below. The problems to be solved are listed farther down the page.

Student Numbers for Tue Sep 5 Recitation Meetings

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So) Show/Hide Student Numbers

The Problems to Be Done in Tue Sep 5 Recitation Meetings

Limits that are Indeterminate Forms and that require no trick, just messy work

Students 1,2:
(This problem is Exercise 1.4#15, similar to Book Section 1.4 Example 2 and similar to an example done in class on Fri Sep 1)
Find the limit

Students 3,4:
(This problem is Exercise 1.4#17, similar to Book Section 1.4 Example 4)

Students 5,6:
(This problem is Exercise 1.4#25, an example done in class on Fri Sep 1)

Limits that Involve Rationalizing

Students 7,8:
(This problem is Exercise 1.4#21, similar to Book Section 1.4 Example 5 and similar to an example done in class on Fri Sep 1)
Find the limit

Students 9,10:
(This problem is Exercise 1.4#23, similar to Book Section 1.4 Example 5 and similar to an example done in class on Fri Sep 1)
Find the limit

Limits that Involve Absolute Value

Students 11,12:
(This problem is Exercise 1.4#38, similar to Book Section 1.4 Example 7 and similar to an example done in class on Fri Sep 1)
Find the limit

Limits that Involve the Squeeze Theorem

Students 13,14: (1.4#33) Given that for all \(x\), $$4x-9 \leq f(x) \leq x^2-4x+7$$ find the limit $$\lim_{x\rightarrow 4}f(x)$$

Students 15,16:
(This problem is Exercise 1.4#35, similar to Book Section 1.4 Example 9)
Show that

Limits that Use Famous Fact that $$\lim_{x\rightarrow 0}\frac{\sin{(x)}}{x}=1$$

Students 17,18:
(This problem is Exercise 1.4#41, similar to Book Section 1.4 Example 10)
Find the limit

Students 19,20:
(This problem is Exercise 1.4#51, similar to Book Section 1.4 Example 10)
Find the limit

Quiz Q1 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• Three Problems, 10 points each, printed on front & back of one sheet of paper
  • One problem based on Suggested Exercises from Section 1.3.
  • Two problems based on Suggested Exercises from Section 1.4.

Students Solving Problems and Discussing Their Solutions

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

Scoring: If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their R02 score will be in the range 1/5 to 5/5 (depending on how whether they have prepared ahead of time). If they do not present a solution, their R02 score will be 0/5.

Students find their Student Number in the lists below. The problems to be solved are listed farther down the page.

Student Numbers for Tue Sep 5 Recitation Meetings

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So) Show/Hide Student Numbers

The Problems to Be Done in Tue Sep 12 Recitation Meetings

Students #1, 2:

• Similar Problem from Exercise List: 1.6 #13
• Similar Book Example: Section 1.6 Example 1 is similar to part (b)
• Similar Class Example: Fri Sep 8 Example is similar to part (a) and (b)

We are interested in the following three limits: $$\lim_{x\rightarrow -3^-}\frac{x+2}{x+3} \\ \lim_{x\rightarrow -3^+}\frac{x+2}{x+3} \\ \lim_{x\rightarrow -3}\frac{x+2}{x+3}$$

1. Find the limits using the expanded definition of limit presented in Section 1.6. That is, limits can now include the terminology and notation of infinity. The expanded definition of limit is used in Section 1.6 Example 2. Show all details clearly and use correct notation.
2. What does the result of (a) tell you about the graph of the rational function?

Students #3,4:

• Similar Problem from Exercise List: 1.6 #13
• Similar Book Example: Section 1.6 Example 1 is similar to part (b)
• Similar Class Example: Fri Sep 8 Example is similar to part (a) and (b)

We are interested in the following three limits: $$\lim_{x\rightarrow 5^-}\frac{x^2-5x+6}{x-5} \\ \lim_{x\rightarrow 5^+}\frac{x^2-5x+6}{x-5} \\ \lim_{x\rightarrow 5}\frac{x^2-5x+6}{x-5}$$

1. Find the limits using the expanded definition of limit presented in Section 1.6. That is, limits can now include the terminology and notation of infinity. The expanded definition of limit is used in Section 1.6 Example 2. Show all details clearly and use correct notation.
2. What does the result of (a) tell you about the graph of the rational function?

Students #5,6:

• Similar Problem from Exercise List: Exercise 1.4#42 is similar to one of the limits in part (a) and (b)
• Similar Book Example: Section 1.6 Example 1 is similar to one of the limits in part (b)
• Similar Class Example:

We are interested in the following three limits: $$\lim_{x\rightarrow 0^-}\left(\frac{1}{x} - \frac{1}{|x|}\right)\\ \lim_{x\rightarrow 0^+}\left(\frac{1}{x} - \frac{1}{|x|}\right)\\ \lim_{x\rightarrow 0}\left(\frac{1}{x} - \frac{1}{|x|}\right)$$

1. Find the limits using the expanded definition of limit presented in Section 1.6. That is, limits can now include the terminology and notation of infinity. The expanded definition of limit is used in Section 1.6 Example 2. Show all details clearly and use correct notation.
2. What does the result of (b) tell you about the graph of the function?

Students #7,8

• Similar Problem from Exercise List: 1.6 # 19
• Similar Book Example: Section 1.6 Examples 5, 11
• Similar Class Example:

1. Find the limit of the rational function using the methods of Section 1.6 Examples 5,9. Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty}\frac{7x^5-3x^2+13}{4x^5+199x^3-17}$$
2. What does the result of (a) tell you about the graph of the rational function?

Students #9,10

• Similar Problem from Exercise List: 1.6 # 19
• Similar Book Example: Section 1.6 Examples 5, 11
• Similar Class Example:

1. Find the limit of the rational function using the methods of Section 1.6 Examples 5,9. Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty}\frac{7x^5-3x^2+13}{4x^6+199x^3-17}$$
2. What does the result of (a) tell you about the graph of the rational function?

Students: #11,12:

• Similar Problem from Exercise List: 1.6 # 19
• Similar Book Example: Section 1.6 Examples 5, 11
• Similar Class Example:

1. Find the limit of the rational function using the methods of Section 1.6 Examples 5,9. Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty}\frac{7x^8-3x^2+13}{4x^5+199x^3-17}$$
2. What does the result of (a) tell you about the graph of the rational function?

Students: #13,14

• Similar Problem from Exercise List: 1.6 # 25
• Similar Book Example: Section 1.6 Example 6
• Similar Class Example:

1. Find the limit of the function using the methods of Section 1.6 Example 6. Show all details clearly and use correct notation. $$\lim_{x\rightarrow \infty} \left( \sqrt{9x^2+x}-3x\right)$$
2. What does the result of (a) tell you about the graph of the function?

Students: #15,16

• Similar Problem from Exercise List: 1.6 # 29
• Similar Book Example: Section 1.6 Examples 7,8
• Similar Class Example:

1. Find the limit $$\lim_{x\rightarrow -\infty} \cos{(x)}$$
2. What does the result of (a) tell you about the graph of the function?

Students: #17,18

• Similar Problem from Exercise List: 1.6 # 35
• Similar Book Example:
• Similar Class Example:

1. Find the horizontal and vertical asymptotes of the rational function. (Give their line equations and say if they are horizontal or vertical.) Explain how you determined the asymptotes. $$y=\frac{2x^2+x-1}{x^2+x-2}$$
2. Illustrate your results with a sketch of the graph of the function.

Students: #19,20

• Similar Problem from Exercise List: 1.6 # 40
• Similar Book Example:
• Similar Class Example:

1. Find a formula for a function that has vertical asymptotes at \(x=2\) and \(x=5\) and horizontal asymptote \(y=3\). Explain how you determined your function.
2. Illustrate your results with a sketch of the graph of the function that you found in (a).

Quiz Q2 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• Three Problems, 10 points each, printed on front & back of one sheet of paper
  • One problem based on Suggested Exercises from Section 1.5.
  • One problem based on Suggested Exercises from Section 1.6.
  • One problem based on Suggested Exercises from Section 1.6.

Students Solving Problems and Discussing Their Solutions

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

Scoring: If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their R02 score will be in the range 1/5 to 5/5 (depending on how whether they have prepared ahead of time). If they do not present a solution, their R02 score will be 0/5.

Students find their Student Number in the lists below. The problems to be solved are listed farther down the page.

Student Numbers for Tue Sep 19 Recitation Meetings

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So) Show/Hide Student Numbers

The Problems to Be Done in Tue Sep 19 Recitation Meetings

Students 1,2: (2.1#16) Suppose that a function \(g(x)\) is known to have these properties:

• \(g(5)=-3\)
• \(g'(5)=4\)

Students 3,4: (2.1#18) Suppose that the line that is tangent to the graph of a function \(f(x)\) at the point \((4,3)\) also passes through the point \((0,2)\).

1. Find \(f(4)\)
2. Find \(f'(4)\)

Students 5,6: The graph of a function \(f(x)\) can be shown by clicking on the button below. Also shown is a tangent line and a secant line, with some given points on those lines. (Notice that the graph is not drawn to scale.) Use the graph to answer the questions below. Project the graph on the screen. (If the projection system is not working, draw the graph on the whiteboard.)

1. What is the Average Rate of Change of \(f(x)\) from \(x=2\) to \(x=7\)? Explain.
2. What is \(f'(2)\)? Explain.

Hide the graph

Students 7,8: (2.1#1) For the function \(f(x)=4x-x^2\)

1. Find the slope of the line tangent to the graph at \(x=1\). Show all details clearly and explain key steps.
   Remark: When finding derivatives, use the Definition of the Derivative $$f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$ That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use Derivative Rules that you may have learned in previous courses.
2. Find the equation of the line tangent to the graph at \(x=1\). Show all details clearly and explain key steps.
3. Sketch a graph of \(f(x)\) and draw the tangent line. Label key points with their \((x,y)\) coordinates.

Students 9,10: (2.1#5) For the function \(f(x)=\sqrt{x}\)

1. Find the slope of the line tangent to the graph at \(x=4\). Show all details clearly and explain key steps.
   Remark: When finding derivatives, use the Definition of the Derivative $$f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$ That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use Derivative Rules that you may have learned in previous courses.
2. Find the equation of the line tangent to the graph at \(x=4\). Show all details clearly and explain key steps.
3. Sketch a graph of \(f(x)\) and draw the tangent line. Label key points with their \((x,y)\) coordinates.

Students 11,12: (2.1#1) A ball is thrown into the air. Its height (in feet) after \(t\) seconds is given by the equation $$y=40t-16t^2$$ Find the velocity when \(t=2\). Show all details clearly and explain key steps.
Remark: When finding derivatives, use the Definition of the Derivative $$f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$ That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use Derivative Rules that you may have learned in previous courses.

Students 13,14: (This is the messiest problem. Sorry!) (2.1#27) For the function $$f(t)=\frac{2t+1}{t+3}$$

1. Find \(f'(2)\).
   Remark: When finding derivatives, use the Definition of the Derivative $$f'(a)=\lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$$ That is, build the limit and find its value. Show all steps clearly and explain key steps. Do not use Derivative Rules that you may have learned in previous courses.
2. What is the slope of the line tangent to the graph of \(f(t)\) at \(t=2\)? Explain.

Students 15,16: (2.2#19) For the function $$f(x)=3x-5$$

1. Find \(f'(x)\) using the Definition of the Derivative $$f'(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ That is, build the limit and find its value. (Do not use Derivative Rules that you may have learned in previous courses.) Show all steps clearly and explain key steps.
2. What is the slope of the line tangent to the graph of \(f(x)\) at \(x=7\)? Explain, using a graph of \(f(x)\).

Students 17,18: (2.2#22) For the function $$g(t)=\frac{1}{\sqrt{t}}$$

1. Find \(g'(t)\) using the Definition of the Derivative $$g'(t)=\lim_{h\rightarrow 0} \frac{g(t+h)-g(t)}{x}$$ That is, build the limit and find its value. (Do not use Derivative Rules that you may have learned in previous courses.) Show all steps clearly and explain key steps.
2. What is the slope of the line tangent to the graph of \(f(x)\) at \(x=9\)? Explain.

Students 19,20 (2.2#23) For the function $$g(x)=\frac{1}{x}$$

1. Find \(g'(x)\) using the Definition of the Derivative $$g'(x)=\lim_{h\rightarrow 0} \frac{g(x+h)-g(x)}{x}$$ That is, build the limit and find its value. (Do not use Derivative Rules that you may have learned in previous courses.) Show all steps clearly and explain key steps.
2. What is the slope of the line tangent to the graph of \(g(x)\) at \(x=5\)? Explain.

Exam X1 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• The Exam will last the full duration of the class period.
• No books, notes, calculators, or phones
• Eight problems, 25 points each, printed on front & back of three sheets of paper. All problems are based on suggested exercises.
  1. A problem about limits, based on suggested exercises from Section 1.3
  2. A problem about calculating limits, based on suggested exercises from Section 1.4
  3. A problem using the concept of continuity, based on suggested exercises from Section 1.5
  4. A problem about calculating a limit involving infinity, based on suggested exercises from Section 1.6
  5. A problem about calculating a limit involving infinity, based on suggested exercises from Section 1.6
  6. A problem about derivatives and rates of change, based on suggested exercises from Section 2.1
  7. A problem about calculating a derivative, based on suggested exercises from Section 2.2
  8. A problem involving a tangent line, based on suggested exercises from Sections 2.1 and 2.2

Students Solving Problems and Discussing Their Solutions

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

Scoring: If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their R05 score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their R05 score will be 0/5.

Students find their Student Number in the lists below. The problems to be solved are listed farther down the page.

Student Numbers for Tue Sep 26 Recitation Meetings

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So) Show/Hide Student Numbers

Basic Derivative Formulas

Derivative of a Constant Function If \(c\) is a constant, then $$\frac{d}{dx}(c)=0$$

The Power Rule If \(n\) is any real number, then $$\frac{d}{dx}\left(x^n \right)=nx^{n-1}$$

The Sum Constant Multiple Rule If \(a\) and \(b\) are constants and \(f\) and \(g\) are differentiable functions, then $$\frac{d}{dx}\left[af(x) +bg(x)\right]=a\frac{d}{dx}f(x)+b\frac{d}{dx}g(x)$$

The Sine and Cosine Rules (Not discussed in class Monday, but simple enough.) $$\frac{d}{dx}\sin{(x)}=\cos{(x)}$$ $$\frac{d}{dx}\cos{(x)}=-\sin{(x)}$$

First Problems to Be Done in Tue Sep 26 Recitation Meetings: Derivatives

Students 1,2 (first problem)(You'll have another problem later.) (2.3#2) Find the derivative of the function $$f(x) = \pi^2$$ Show all details clearly and use correct notation.

Students 3,4 (first problem)(You'll have another problem later.) (2.3#3) Find the derivative of the function $$f(t)=2-\frac{2}{3}t$$ Show all details clearly and use correct notation.

Students 5,6 (first problem)(You'll have another problem later.) (2.3#4) For the function \(F(x)=\frac{3}{4}x^8\)

1. Find \(F(2)\)
2. Find \(F'(x)\)
3. Find \(F'(2)\)
4. Find the height of the graph of \(F(x)\) at \(x=2\).
5. Find the slope of the graph of \(F(x)\) at \(x=2\).

Students 7,8 (first problem)(You'll have another problem later.) (2.3#5) For the function \(f(x)=x^3-4x+6\)

1. Find \(F(3)\)
2. Find \(F'(x)\)
3. Find \(F'(3)\)
4. Find the height of the graph of \(f(x)\) at \(x=3\).
5. Find the slope of the graph of \(f(x)\) at \(x=3\).

Students 9,10 (first problem)(You'll have another problem later.) (2.3#7) For the function \(f(x)=3x^2-2\cos{(x)}\)

1. Find \(F(\pi)\)
2. Find \(F'(x)\)
3. Find \(F'(\pi)\)
4. Find the height of the graph of \(f(x)\) at \(x=\pi\). (Give an exact answer in symbols, not a decimal approximation.)
5. Find the slope of the graph of \(f(x)\) at \(x=\pi\). (Give an exact answer in symbols, not a decimal approximation.)

Students 11,12 (first problem)(You'll have another problem later.) (2.3#9) Find the derivative of the function $$g(x)=x^2(1-2x)$$ Show all details clearly and use correct notation.

Students 13,14 (first problem)(You'll have another problem later.) For the function \(f(x)=2x^{1/3}\)

1. Find \(f(8)\) (no calculators!)
2. Find \(f'(x)\)
3. Find \(f'(8)\) (no calculators!)
4. Find the height of the graph of \(f(x)\) at \(x=8\).
5. Find the slope of the graph of \(f(x)\) at \(x=8\).

Students 15,16 (first problem)(You'll have another problem later.) (2.3#11) Find the derivative of the function $$f(t)=\frac{2}{t^{3/4}}$$ Show all details clearly and use correct notation

Students 17,18 (first problem)(You'll have another problem later.) (2.3#19) For the function $$f(x)=\frac{x^2+4x+3}{\sqrt{x}}$$

1. Rewrite \(f(x)\) in power function form. That is, write it in the form $$f(x)=ax^p+bx^q+cx^r$$ where \(a,b,c,p,q,r\) are real numbers.
2. Find \(f'(x)\)

Students 19,20 (first problem)(You'll have another problem later.) (2.3#21) For the function $$v=t^2-\frac{1}{\sqrt[4]{t^3}}$$

1. Rewrite the function in power function form. That is, write it in the form $$v(t)=at^p+bt^q$$ where \(a,b,p,q\) are real numbers.
2. Find \(v'(t)\)

Second Problems to Be Done in Tue Sep 26 Recitation Meetings: Tangent Lines and Normal Lines

Remember that the line tangent to the graph of \(f(x)\) at \(x=a\) is the line that has these two properties

• The line touches the graph of \(f(x)\) at \(x=a\). So the line contains the point \((x,y)=(a,f(a))\), called the point of tangency
• The line has slope \(m=f'(a)\)

A new thing, the line normal to the graph of \(f(x)\) at \(x=a\), is the line that has these two properties

• The line touches the graph of \(f(x)\) at \(x=a\). So the line contains the point \((x,y)=(a,f(a))\)
• The line is perpendicular to the line that is tangent to the graph at that point. That is,
  • If the tangent line has slope \(m_T\neq 0\), then the normal line has slope $$m_N=-\frac{1}{m_T}$$
  • If the tangent line has slope \(m_T = 0\), which indicates that the tangent is horizontal, then the normal line is vertical.

Students 1- 12 (second problem) (2.3#27) For the function $$f(x)=2\sin{(x)}$$

• Students 1,2: Find the equation for the line tangent to the graph of \(f(x)\) at \(x=\frac{\pi}{3}\). Draw the graph and draw your tangent line. Label important stuff.
• Students 3,4: Find the equation for the line normal to the graph of \(f(x)\) at \(x=\frac{\pi}{3}\). Draw the graph and draw your normal line. Label important stuff.
• Students 5,6: Find the equation for the line tangent to the graph of \(f(x)\) at \(x=\frac{\pi}{2}\). Draw the graph and draw your tangent line. Label important stuff.
• Students 7,8: Find the equation for the line normal to the graph of \(f(x)\) at \(x=\frac{\pi}{2}\). Draw the graph and draw your normal line. Label important stuff.
• Students 9,10: Find the equation for the line tangent to the graph of \(f(x)\) at \(x=\frac{3\pi}{4}\). Draw the graph and draw your tangent line. Label important stuff.

Students 11 - 20 (second problem) (2.3#29) For the function $$f(x)=-x^2+8x=-x(x-8)$$

• Students 11,12: Find the equation for the line tangent to the graph of \(f(x)\) at \(x=2\). Draw the graph and draw your tangent line. Label important stuff.
• Students 13,14: Find the equation for the line normal to the graph of \(f(x)\) at \(x=2\). Draw the graph and draw your normal line. Label important stuff.
• Students 15,16: Find the equation for the line tangent to the graph of \(f(x)\) at \(x=4\). Draw the graph and draw your tangent line. Label important stuff.
• Students 17,18: Find the equation for the line normal to the graph of \(f(x)\) at \(x=4\). Draw the graph and draw your normal line. Label important stuff.
• Students 19,20: Find the equation for the line tangent to the graph of \(f(x)\) at \(x=5\). Draw the graph and draw your tangent line. Label important stuff.

Quiz Q3 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• Three Problems, 10 points each, about using the Basic Differentiation Formulas, based on Suggested Exercises from Section 2.3, printed on front & back of one sheet of paper.

Student Numbers for Tue Oct 3 Recitation Meetings

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So) Show/Hide Student Numbers

Basic Derivative Formulas Show/Hide Formulas

Part 1: Students Solving Problems and Discussing Their Solutions Show/Hide Part 1

Part 2: Conceptual Questions about Tangent Lines Show/Hide Part 2

Quiz Q4 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• four Problems, printed on front & back of one sheet of paper
  • One Product Rule problem based on Suggested Exercises from Section 2.4.
  • One Quotient Rule problem based on Suggested Exercises from Section 2.4.
  • One Chain Rule problem based on Suggested Exercises from Section 2.5.
  • One Implicit Differentiation problem based on Suggested Exercises from Section 2.6.

Student Numbers for Tue Oct 10 Recitation Meetings

Recitation Section 101 (meets Tue 8:00am – 8:55am in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 102 (meets Tue 9:30am – 10:25am in Ellis 107 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 103 (meets Tue 12:30pm – 1:25pm in Morton 122 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 104 (meets Tue 2:00pm – 2:555pm in Morton 318 with Instructor Isaac Agyei) Show/Hide Student Numbers

Recitation Section 111 (meets Tue 9:30am – 10:25am in Morton 218 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 112 (meets Tue 11:00am – 11:55am in Ellis 107 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 113 (meets Tue 2:00pm – 2:55pm in Morton 126 with Instructor Kenny So) Show/Hide Student Numbers

Recitation Section 114 (meets Tue 3:30pm – 4:25pm in Morton 122 with Instructor Kenny So) Show/Hide Student Numbers

Basic Derivative Formulas Show/Hide Formulas

Students Solving Problems and Discussing Their Solutions

Each student will solve two problems: One problem in Round 1, and another problem in Round 2.

Problems for Oct 10 Round 1 Show/Hide Round 1

Problems for Oct 10 Round 2 Show/Hide Round 2

Quiz Q5 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• Two Problems, 15 points each, printed on front & back of one sheet of paper
  • One Related Rates problem based on Suggested Exercises from Section 2.7.
  • One Linearization problem problem based on Suggested Exercises from Section 2.8.

Students Solving Problems and Discussing Their Solutions

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

Scoring: If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their R07 score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their R07 score will be 0/5.

Students find their Student Number in the lists below. The problems to be solved are listed farther down the page.

Student Numbers for Tue Oct 17 Recitation Meetings Show/Hide Student Numbers

Derivative Formulas That We Know So Far Show/Hide Formulas

Each student will solve two problems.

Round 1 Show/Hide Round 1

Round 2 Show/Hide Round 2

Exam X2 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• The Exam will last the full duration of the class period.
• No books, notes, calculators, or phones
• Eight problems, 25 points each, printed on front & back of three sheets of paper. All problems are based on suggested exercises.
  • Four problems about finding derivatives using various methods that we have studied (in sections 2.3, 2.4, 2.5, 2.6, 3.3)
  • Four problems about using derivatives to find things.
    • Related rates (Section 2.7)
    • Exponential Growth in Biology or Exponential Decay of Radioactive Substance (Section 3.4)
    • Velocity & Acceleration (Problems about this appear in Sections 2.3, 2.4, 2.5.)
    • Slope or Equation of the Tangent Line and/or Normal Line. (Problems about this appear in Sections 2.3, 2.4, 2.5, 3.3.)

Students Solving Problems and Discussing Their Solutions

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

Scoring: If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their R07 score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their R07 score will be 0/5.

Students find their Student Number in the lists below. The problems to be solved are listed farther down the page.

Student Numbers for Tue Oct 24 Recitation Meetings Show/Hide Student Numbers

Recitation Part 1: Finding Critical Numbers of Functions Show/Hide Part 1

If there is time remaining: Recitation Part 2 Show/Hide Part 2

Quiz Q6 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• Two Problems, 15 points each, about Maximum and Minimum Values, based on Suggested Exercises from Section 4.1, printed on front & back of one sheet of paper.

Students Solving Problems and Discussing Their Solutions

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

Scoring: If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their R07 score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their R07 score will be 0/5.

Students find their Student Number in the lists below. The problems to be solved are listed farther down the page.

Student Numbers for Tue Oct 31 Recitation Meetings Show/Hide Student Numbers

Rolle's Theorem: If a function \(f\) satisfies the following three requirements (the hypotheses)

1. \(f\) is continuous on the closed interval \([a,b]\).
2. \(f\) is differentiable on the open interval \((a,b)\).
3. \(f(a)=f(b)\).

There is at least one number \(x=c\) with \(a \lt c \lt b\) such that $$f'(c)=0$$ In other words, $$\text{the slope of the tangent line at }x=c\text{ is }m=f’(c)=0$$

Remark: The theorem does not give you the value of \(c\). If \(c\) exists, you'll have to figure out its value.

Students 1,2: Consider the function \(f(x)=x^3-3x+5\) on the interval \(\left[-\sqrt{3},\sqrt{3}\right]\).

1. Verify that the function and the interval satisfy the three hypotheses of Rolle's Theorem. Explain clearly.
2. Find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem. Show all details clearly.
3. Draw a graph of \(f(x)\) on the interval and draw a tangent line to illustrate your result.

Students 3,4: Consider the function \(\cos{(x)}\) on the interval \([\frac{\pi}{6},\frac{13\pi}{6}]\).

1. Verify that the function and the interval satisfy the three hypotheses of Rolle's Theorem. Explain clearly.
2. Find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem. Show all details clearly.
3. Draw a graph of \(f(x)\) on the interval and draw a tangent line to illustrate your result.

Students 5,6: Consider the function \(f(x)=x+\frac{1}{x}\) on the interval \([\frac{1}{3},3]\).

1. Verify that the function and the interval satisfy the three hypotheses of Rolle's Theorem. Explain clearly.
2. Find all numbers \(c\) that satisfy the conclusion of Rolle's Theorem. Show all details clearly.
3. Draw a graph of \(f(x)\) on the interval and draw a tangent line to illustrate your result.

The Mean Value Theorem: If a function \(f\) satisfies the following three requirements (the hypotheses)

1. \(f\) is continuous on the closed interval \([a,b]\).
2. \(f\) is differentiable on the open interval \((a,b)\).

There is at least one number \(x=c\) with \(a \lt c \lt b\) such that $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ In other words, $$\text{slope of the tangent line at }c \ \text{ is equal to the slope of the secant line from }a\text{ to }b$$

Remark: The theorem does not give you the value of \(c\). If \(c\) exists, you'll have to figure out its value.

Students 7,8: Consider the function \(f(x)=x^3-3x+2\) on the interval \([-2,2]\).

1. Verify that the function and the interval satisfy the two hypotheses of the Mean Value Theorem. Explain clearly.
2. Find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. Show all details clearly.
3. Draw a graph of \(f(x)\) on the interval and draw a tangent line and a secant line to illustrate your result.

Students 9,10: Consider the function \(f(x)=\ln{(x)}\) on the interval \([1,4]\).

1. Verify that the function and the interval satisfy the two hypotheses of the Mean Value Theorem. Explain clearly.
2. Find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. Show all details clearly.
3. Draw a graph of \(f(x)\) on the interval and draw a tangent line and a secant line to illustrate your result.

Students 11,12: Consider the function \(f(x)=\frac{1}{x}\) on the interval \([1,3]\).

1. Verify that the function and the interval satisfy the two hypotheses of the Mean Value Theorem. Explain clearly.
2. Find all numbers \(c\) that satisfy the conclusion of the Mean Value Theorem. Show all details clearly.
3. Draw a graph of \(f(x)\) on the interval and draw a tangent line and a secant line to illustrate your result.

Correspondence between sign behavior of \(f'(x)\) on an interval \( (a,b) \) and increasing/decreasing behavior of the graph of \( f(x) \) on the interval \( (a,b) \)

• If \(f'(x)\) is positive on an interval \( (a,b) \) then \(f(x)\) is increasing on the interval \( (a,b) \).
• If \(f'(x)\) is negative on an interval \( (a,b) \) then \(f(x)\) is decreasing on the interval \( (a,b) \).
• If \(f'(x)\) is zero on a whole interval \( (a,b) \) then \(f(x)\) is constant on the interval \( (a,b) \).

The First Derivative Test for Local Extrema

• Test 1: \(f'(c)=0\) or \(f'(c) DNE\). (If the number \(c\) passes Test 1, then \(c\) is called a partition number for \(f'(x)\).)
• Test 2: \(f(c)\) exists. (If the number \(c\) passes both Test 1 and Test 2, then \(c\) is called a critical number for \(f(x)\).)
• Test 3: \(f(x)\) is continuous at \(c\).
• Test 4: \(f'(x)\) changes sign at \(c\).(If the number \(c\) passes Tests 1,2,3,4, then \(x=c\) is the location of a local max or local min of \(f(x)\). The corresponding \(y\) value, \(f(c)\), is called the local max value or local max value.)

Correspondence between sign behavior of \(f''(x)\) on an interval \( (a,b) \) and concavity behavior of the graph of \( f(x) \) on the interval \( (a,b) \)

• If \(f''(x)\) is positive on an interval \( (a,b) \) then \(f'(x)\) is increasing on the interval \( (a,b) \), which in turn means that \(f(x)\) is concave up on the interval \((a,b)\).
• If \(f''(x)\) is negative on an interval \( (a,b) \) then \(f'(x)\) is decreasing on the interval \( (a,b) \), which in turn means that \(f(x)\) is concave cown on the interval \((a,b)\).

Related terminology: An inflection point is a point on the graph of a function where the function is continuous and the concavity changes (from up to down or from down to up).

Students 13,14: Consider the function \(f(x)=\sin{(x)}-\cos{(x)}\) and the interval \([-2,2]\).

1. Find the intervals on which \(f\) is increasing or decreasing.
2. Find the local maximum values and local minimum values of \(f\).
3. Find the intervals on which \(f\) is concave up or concave down.
4. Find the \((x,y)\) coordinates of all inflection points of \(f\).

Students 15,16: Consider the function \(f(x)=xe^{(-x)}\).

1. Find the intervals on which \(f\) is increasing or decreasing.
2. Find the local maximum values and local minimum values of \(f\).
3. Find the intervals on which \(f\) is concave up or concave down.
4. Find the \((x,y)\) coordinates of all inflection points of \(f\).

Students 17,18: Consider the function \(f(x)=x^4-2x^2+3\).

1. Find the intervals on which \(f\) is increasing or decreasing.
2. Find the local maximum values and local minimum values of \(f\).
3. Find the intervals on which \(f\) is concave up or concave down.
4. Find the \((x,y)\) coordinates of all inflection points of \(f\).

Quiz Q7 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• Three Problems, 10 points each, printed on front & back of one sheet of paper.
  • One Mean Value Theorem problem based on Suggested Exercises from Section 4.2.
  • One problem about Derivatives and the Shapes of Graphs, based on Suggested Exercises from Section 4.3.
  • One problem about Curve Sketching, based on Suggested Exercises from Section 4.4.

Students Solving Problems and Discussing Their Solutions

A pair of students will work together to write the solution of a Suggested Homework Problem on the blackboard or whiteboard. The emphasis should be on writing a very clear solution, with key steps explained. Write large and clear!

Multiple pairs of students should be able to work simultaneously. There may be two or three rounds. The Recitation Instructor will decide how best to choreograph the rounds. After one round of students has finished writing their solutions on the white boards, they will sit down and the Instructor will lead a discussion of the solutions on the board. The Instructor and the students in the class will talk about what is good and not so good in each solution. Then the boards will be erased and the next round of students will come to the board and write the solution to their problems.

Scoring: If a student (either alone or as part of a team of two students) presents a solution to their assigned problem on the white board, their R07 score will be in the range 1/5 to 5/5 (depending on whether they have prepared ahead of time). If they do not present a solution, their R07 score will be 0/5.

Students find their Student Number in the lists below. The problems to be solved are listed farther down the page.

Student Numbers for Tue Nov 7 Recitation Meetings Show/Hide Student Numbers

Meeting Part 1: Optimization Problems (Section 4.5)

Students 1,2: (Suggested Exercise 4.5#2) Find two numbers whose difference is 100 and whose product is a minimum. (You must use calculus and show all details clearly. No credit for just guessing values.)

Students 3,4: (Suggested Exercise 4.5#7) Find the dimensions of a rectangle with perimeter 100m whose area is as large as possible. (You must use calculus and show all details clearly. No credit for just guessing values.)

Students 5,6: (Suggested Exercise 4.5#11) If 1200 cm² of material is available to make a box with a square base and an open top, find the largest possible volume of the box. (You must use calculus and show all details clearly. No credit for just guessing values.)

Students 7,8: (Suggested Exercise 4.5#15) Find the point on the line \(y=2x+3\) that is closest to the origin. (You must use calculus and show all details clearly. No credit for just guessing values.)

Students 9,10: (Suggested Exercise 4.5#17) Find the points on the ellipse \(4x^2+y^2=4\) that are farthest away from the point \((1,0)\) (You must use calculus and show all details clearly. No credit for just guessing values.)

Students 11,12: (Suggested Exercise 4.5#22)Find the area of the largest rectange that can be inscribed in a right triangle with legs of lengths 3cm and 2cm if two sides of the rectangle lie along the legs. (You must use calculus and show all details clearly. No credit for just guessing values.)

Students 13,14: (Suggested Exercise 4.5#25) A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30 ft, find the dimensions of the window that has the greatest area. (You must use calculus and show all details clearly. No credit for just guessing values.)

Students 15,16: (Suggested Exercise 4.5#30) A cone-shaped paper drinking cup is to be made to hold 27 cm³ of water. Find the height and radius of the cup that will use the smallest amount of paper. (You must use calculus and show all details clearly. No credit for just guessing values.)

Students 17,18: (Suggested Exercise 4.5#39) Find an equation of the line through the point \((3,5)\) that cuts off the least area from the first quadrant. (You must use calculus and show all details clearly. No credit for just guessing values.)

Meeting Part 2: Newton's Method (Section 4.6)

Students work in pairs on this (Class Drill on Using Newton's Method)

Exam X3 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• The Exam will last the full duration of the class period.
• No books, notes, calculators, or phones
• Five problems, printed on front & back of three sheets of paper. All problems are based on suggested exercises.
  1. A problem about Maximum and Minimum Values (Section 4.1)
  2. A problem about Derivatives and the Shapes of Graphs and/or Curve Sketching (Sections 4.3 and 4.4)
  3. A problem about Optimization (Section 4.5)
  4. A problem about Newton's Method (Section 4.6)
     • Note: I will not give you the formula to use for Newton's Method. The formula is presented in the Book, and it was presented in Lecture and in Recitation. You should learn the formula by doing exercises that require you to use the formula. There are exercises of this type in the Homework List, and you did Class Drills in Class on Mon Nov 6 and in Recitation on Tue Nov 7. You can see those Class Drills in the calendar entries for those days.
  5. A problem about Antiderivatives (Section 4.7)
     • Note: I will not give you the give you the Basic Antidifferentiation Formulas. The formulas are presented in the Book, and they were presented in Lecture. You should learn those formulas by doing exercises that require you to use those formulas. There are exercises of this type in the Homework List.

Reminders about Studying

• The most important issue is, Can you successfully write down the solution to a problem? Therefore, the centerpiece of your studying should be, Practicing writing down the solutions to problems.
• Mathematical concepts get presented to you in the Book and in Lecture. Examples are also presented in both places. I generally try to not present a particular type of example in class if a similar example is already presented well in the book. Rather, I try to present examples in class that are different from the examples that are presented in the book. Therefore, for your studying, you should be sure to study not just the examples that I do in class, but also the examples that are presented in the book!
• The book and my lectures are not supposed to present examples similar to all of the kinds of problems that you need to know how to solve. The idea is that the book and my lectures teach you the concepts and show you some examples, and from there, you need to be able to generalize and solve different problems.
• In writing my Quizzes and Exams, I aim to include a mixture of
  • Problems that are based on problems from the Homework List and that are similar to a class example.
  • Problems that are based on problems from the Homework List and that are not similar to a class example, but that are similar to a book example.
  • Problems that are based on problems from the Homework List but that are not similar to any class or book example.
  Therefore, your studying and practice problems should include studying and practicing of all types of problems.

Students work in pairs on common problems.

All students in the room work on problem [1] for about 5 minutes, then the Instructor discusses that problem. Then all students work on problem [2] for 5 minutes, and the Instructor discusses that problem, and so on. Some problems will need more time.

Meeting Part 1: Antiderivatives Satisfying an Extra Condition (Section 4.7)

[1] (Suggested Exercise 4.7 #15) Let $$f(x)=7x-3x^5$$

1. Find the General Antiderivative, \(F(x)\).
2. Find the Particular Antiderivative that satisfies \(F(1)=5\).

[2] (not like a book exercise) Let $$f(t)=3e^t-4$$

1. Find the General Antiderivative, \(F(t)\).
2. Find the Particular Antiderivative that satisfies \(F(0)=8\).

[3] (review of prerequisites) Draw the first quadrant of the unit circle, with important famous angles \(\theta=0,\pi/6,\pi/4,\pi/3,\pi/2 \) shown, along with the \((x,y)\) coordinates of the points where the rays of those angles intersect the circle.

[4] (4.7#27) Find \(f(t)\) such that $$f'(t)=10\cos t - \sec^2 t \ \text{ for } \ -\pi/2 \lt t \lt \pi/2 \ \text{ and that } \ f(\pi/3)=13$$

[5] (4.7#20) Suppose that $$f''(x)=30x-\sin x$$ Find \(f(x)\).

Problems about Position, Velocity, and Acceleration

Remember that for an object moving in one dimension, the velocity, \(v(t)\), is the derivative of the position, \(s(t)\). That is, $$s'(t) = v(t)$$

Therefore, position, \(s(t)\), is an antiderivative of the velocity, \(v(t)\).

Also remember that the acceleration, \(a(t)\), is the derivative of the velocity, \(v(t)\).

Therefore, velocity, \(v(t)\), is an antiderivative of the acceleration, \(a(t)\).

Furthermore, recall that when an object falls freely under the influence of gravity, it is known that the object will have constant acceleration with a value $$a=-32 \ \text{ft/s}^2$$ The negative sign may be confusing. The reason for the negative sign is that the positive position direction is up. Since gravity makes objects fall down, it is acclerating them in the negative position direction. Hence, the acceleration gets a negative sign.

[6] (based on 4.7#40, similar to 4.7#47) Suppose that an object is moving in one dimension with velocity $$v(t)=9\sqrt{t} \ \text{ ft/s}$$

1. Find the general form of the position function, \(s(t)\). That is, find the General Antiderivative of \(v(t)\), but instead of calling it \(V(t)\), call it \(s(t)\).
2. Suppose that it is also known that the initial position is \(s(0)=13 \ \text{ft}\). Find the position function. That is, find the Particular Antiderivative that satisfies \(s(0)=13\).

[7] (based on 4.7#43) Suppose that an stone is dropped off a tower that is 400 feet tall and falls freely. Let position be defined to be zero at ground level, and remember that the positive position direction is up.

1. What is the value of the initial position of the stone, \(s(0)\)?
2. What is the value of the initial velocity of the stone, \(v(0)\)?
3. The acceleration of the stone is constant. What is the value of the the acceleration, \(a\)?
4. Given that the velocity, \(v(t)\), is an antiderivative of the acceleration, find the general form of the velocity function, \(v(t)\). That is, find the General Antiderivative of \(a(t)\), but instead of calling it \(A(t)\), call it \(v(t)\).
5. Knowing what you know about the initial velocity, \(v(0)\), find the particular form of the velocity function, \(v(t)\). That is, find the Particular Antiderivative of \(a(t)\) that satisfies $$v(0) = \text{ initial velocity that you identified earlier}$$
6. Given that the position, \(s(t)\), is an antiderivative of the velocity, find the general form of the position function, \(s(t)\). That is, find the General Antiderivative of \(v(t)\), but instead of calling it \(V(t)\), call it \(s(t)\).
7. Knowing what you know about the initial position, \(s(0)\), find the particular form of the position function, \(s(t)\). That is, find the Particular Antiderivative of \(v(t)\) that satisfies $$s(0) = \text{ initial position that you identified earlier}$$ The formula that you have found for the position function, \(s(t)\) gives the position of the stone above ground level at time \(t\).
8. What is the time when the stone reaches ground level?
9. What is the speed of the stone when it strike the ground?

Quiz Q8 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• Two Problems, 15 points each, printed on front & back of one sheet of paper.
  1. A Problem about Areas and Distances, based on Suggested Exercises from Section 5.1.
  2. A problem about Definite Integrals, based on Suggested Exercises from Section 5.2.

Students work in pairs on common problems.

All students in the room work on problem [1] for about 10 minutes, then the Instructor discusses that problem. Then all students work on problem [2] for 10 minutes, and the Instructor discusses that problem, and so on. Some problems will need more time.

[1]: Students work in pairs on this Class Drill: Definite Integrals for a Graph Made Up of Geometric Shapes

[2]: Students work in pairs on this Class Drill: Computing Definite Integrals Using Geometry

Recall The Evaluation Theorem (ET)

(the relationship between definite integrals and antiderivatives)

If \(f(x)\) is continuous on the interval \([a,b]\), then $$\int_a^bf(x)dx\underset{\text{ET}}{=}F(b)-F(a)$$ where \(F(x)\) is any antiderivative of \(f(x)\).

Use the Evaluation Theorem to evaluate the integrals. Show all details clearly and use correct notation.

Basic Problems

[3]: (5.3#3)$$\int_{-2}^{0}\left(\frac{1}{2}t^4+\frac{1}{4}t^3-t\right)dt$$

[4]: (5.3#13)$$\int_{1}^{2}\left(\frac{x}{2}-\frac{2}{x}\right)dx$$

[5]: (5.3#7)$$\int_{0}^{\pi}\left(5e^x+3\sin x\right)dx$$

Harder Problems that Involve Rewriting the Integrand Before Integrating

[6]: (5.3#18)(A lot of rewriting on this one, but it results in a very simple integral!) $$\int_{0}^{\pi/3}\left(\frac{\sin \theta +\sin \theta \tan^2 \theta}{\sec^2 \theta}\right)d\theta$$

[7]: (5.3#9) $$\int_{1}^{4}\left(\frac{4+6u}{\sqrt{u}}\right)du$$

[8]: (5.3#23)$$\int_{1}^{e}\left(\frac{x^2+x+1}{x}\right)dx$$

Students work in pairs on common problems.

All students in the room work on problem [1] for about 10 minutes, then the Instructor discusses that problem. Then all students work on problem [2] for 10 minutes, and the Instructor discusses that problem, and so on. Some problems will need more time.

Recitation Part 1: Using the Evaluation Theorem

Recall The Evaluation Theorem (ET) as presented in the book and in lecture on Mon Nov 20, using the terminology of antiderivatives:

(the relationship between definite integrals and antiderivatives)

If \(f(x)\) is continuous on the interval \([a,b]\), then $$\int_a^bf(x)dx\underset{\text{ET}}{=}F(b)-F(a)$$ where \(F(x)\) is any antiderivative of \(f(x)\).

And recall The Evaluation Theorem (ET) re-cast using the notation of indefinite integrals, as presented in the book and in lecture on Mon Nov 27:

(the relationship between definite integrals and indefinite integrals)

If \(f(x)\) is continuous on the interval \([a,b]\), then

Use the Evaluation Theorem to evaluate these three basic definite integrals. Show all details clearly and use correct notation.

[1]: (5.3#11) $$\int_{0}^{1}x\left(\sqrt[3]{x}+\sqrt[4]{x}\right)dx$$ Give an exact answer and a decimal approximation, rounded to 3 decimal places.
Hint: You’ll have to rewrite the integrand as a sum of power functions before integrating.

[2]: (5.3#15) $$\int_{0}^{1}\left(x^{10}+10^x\right)dx$$ Give an exact answer and a decimal approximation, rounded to 3 decimal places.
Hint: You’ll have to do some sleuth work to figure out one of the antiderivatives. Try checking your book in Section 5.3.

[3]: (5.3#29) $$\int_{-1}^{2}\left(x-2|x|\right)dx$$ Give an exact answer.
Hint: Remember that the function \(|x|\) is a piecewise-defined function. That is, the formula for \(|x|\) depends on which piece of the domain that you are in. That will mean that you will need to break up this definite integral on the interval \([-1,2]\) into two definite integrals, each on a smaller interval.

Recitation Part 2: Using the Net Change Theorem

Recall The Net Change Theorem (NCT) as presented in the book and in lecture on Mon Nov 27, using the terminology of antiderivatives:

(the integral of a rate of change of a quantity is the net change of that quantity)

If \(F(x)\) is differentiable on the interval \([a,b]\), then $$\int_a^bF'(x)dx\underset{\text{NCT}}{=}F(b)-F(a)$$

[4]: (5.3#51) If \(w'(t)\) is the rate of growth of a child in pounds per year, what does the integral below represent? $$\int_{5}^{10}w'(t)dt$$

[5]: (5.3#52) If oil leaks from a tank at a rate of \(r(t)\) gallons per minute at time \(t\), what does the integral below represent? $$\int_{0}^{120}r(t)dt$$

[6]: (5.3#59) An object moves along a line with velocity $$v(t)=3t-5 \ \text{ for } \ 0\leq t \leq 3$$

1. Find the displacement of the object during the time interval. Give an exact answer.
2. Illustrate your result for (a) using a graph of the velocity \(v(t)\).
3. Find the distance traveled by the object during the time interval. Give an exact answer and a decimal approximation, rounded to 3 decimal places.

[7]: (5.3#60) An object moves along a line with velocity $$v(t)=t^2-2t-3\ \text{ for } \ 0\leq t \leq 6$$

1. Find the displacement of the object during the time interval \([2,5]\). Give an exact answer.
2. Illustrate your result for (a) using a graph of the velocity \(v(t)\).
3. Find the distance traveled by the object during the time interval \([2,5]\). Give an exact answer and a decimal approximation, rounded to 3 decimal places.

Quiz Q9 Information

• Section 100: Sit in Alternate Seats and rows in Morton 235
• Section 110: Sit in Alternate Seats and rows in Morton 237
• 20 Minutes at the end of class
• No books, notes, calculators, or phones
• Three Problems, 10 points each, printed on front & back of one sheet of paper.
  1. A Problem about Evaluating Definite Integrals, based on Suggested Exercises from Section 5.3.
  2. A Problem about Evaluating Definite Integrals, based on Suggested Exercises from Section 5.3.
  3. A problem about the Fundamental Theorem of Calculus, based on Suggested Exercises from Section 5.4.

Recitation Part 1: Using the Fundamental Theorem of Calculus, Part 1

Recall the Fundamental Theorem of Calculus, Part 1

If \(f\) is continuous on the interval \([a,b]\), then $$\frac{d}{dx}\left(\int_a^xf(t)dt\right)\underset{\text{FTC1}}{=}f(x) \text{ for } \ a \lt x \lt b$$

[1]: (5.4#6) The function \(g(x)\) is defined by the integral: $$g(x)=\int_{3}^{x}e^{t^2-t} \ dt$$ Find \(g'(x)\).

[2]: (5.4#10) The function \(g(x)\) is defined by the integral: $$g(x)=\int_{0}^{x}\sqrt{1+\sqrt{t}} \ dt$$ Find \(g'(x)\).

[3]: (5.4#10) The function \(h(x)\) is defined by the integral: $$h(x)=\int_{0}^{\tan x}\sqrt{1+\sqrt{t}} \ dt$$ (Hint: You will need the Chain Rule.)

Recitation Part 2: Computing the Average Value of a Function on an Interval

Recall the Definition of the Average Value of a Function on an Interval

If \(f(x)\) is continuous on the interval \([a,b]\), then the Average Value of \(f(x)\) on the interval \([a,b]\) is defined to be the number

[4]: Find the average value of the function \(f(x)= \frac{1}{x}\) on the interval \([1,4]\). Simplify your answer.

[5]: Find the average value of the function \(f(x)= \sin (x) \) on the interval \([0,\pi]\). Simplify your answer.

[6]: Find the average value of the function \(f(x)= \sec^2(\theta)\) on the interval \([0,\pi/4]\). Simplify your answer.

Recitation Part 3: The Substitution Method

Recall The Substitution Method as Introduced in Monday’s Lecture

[7]: (5.5#3) Find the Indefinite Integral by using the Substitution Method. $$\int x^4 \sqrt{1+x^5} \ dx$$

[8]: (5.5#13) Find the Indefinite Integral by using the Substitution Method. $$\int \frac{1}{7-5x} \ dx$$

[9]: (5.5#19) Find the Indefinite Integral by using the Substitution Method. $$\int e^x \sqrt{1+e^x} \ dx$$

[10]: (5.5#20) Find the Indefinite Integral by using the Substitution Method. $$\int \cos^4 \theta \sin \theta \ d\theta$$

[11]: (5.5#27)

1. (Review) Use the Chain Rule or the Quotient Rule to find the derivative of \(\sec x\). (Either rule can be used. One is simpler.)
2. Find the Indefinite Integral by using the Substitution Method. $$\int \sec^3 x \tan x \ dx$$ (Hint: Rewrite the integrand as \(\sec^2 x \sec x \tan x \).)