Reading Assignments

(Last modified: Tuesday, April 26, 2005, 3:10 PM )

I'll use Maple syntax for mathematical notation on this page.
All numbers indicate sections from *Calculus from Graphical, Numerical, and Symbolic Points of View, Second Edition* by Ostebee/Zorn.

** To read **: The Section *How to Use This Book: Notes for Students*
beginning on page xvii. All of Sections 1.1 and 1.2.

- How is the graph of y=f(x)+3 =x
^{2}+3 related to the graph of y=f(x)? Why? - How is the graph of y=f(x+3) =(x+3)
^{2}related to the graph of y=f(x)? Why? - Which of f(x), f(x)+2, and f(x+2) are even? odd?

** To read **: All of Appendix E, and through the section "Logarithm functions" of Section 1.3. Be sure to understand the definition of the logarithm function base b.

- How are the functions f(x)=3
^{x}and g(x)=log_{3}(x) related? - Solve for x in the equation: log
_{2}(x) + log_{2}(x^{3})=12

** To read **: All of Appendix F and finish Section 1.3.
Be sure to understand the definitions of sin(x) and cos(x) in terms of the unit circle.

- What are the domain and range of sin(x)?
- What is 120 degrees equal to in radians?
- What is the period of the cosine function? How can you tell?

** To read **: Through Example 4. Be sure understand the
section "Rates, amounts, and cars" beginning on page 36.

- Is the derivative of P positive or negative at t=5 ? Explain.
- Is the second derivative of P positive or negative at t=5 ? Explain.
- Give a value of t where the derivative of P is zero.

** To read **: All. Make sure to understand Examples 3 and 4.

- What does the term "locally linear" mean?
- Explain why the derivative of f(x)=|x| does not exist at x=0.

** To read **: All. Be sure to understand the definition of a stationary
point and the difference between local and global maxima and minima.

- Where does f have stationary points? Why?
- Where is f increasing? Why?
- Where is f concave up? Why?

** To read **: All. Think about why the Second Derivative Test makes sense.

- By looking at the graph of f '', how can you tell where f is concave up and concave down?
- By looking at the graph of f ', how can you tell where f is concave up and concave down?

** To read **: All. We'll talk about the formal defintion of the
derivative in detail during class.

- Let f(x)=x
^{3}. Find the slope of the secant line from x=-2 to x=4. - For a function f, what does the difference quotient ( f(a+h) - f(a) )/ h measure?
- Let f(x)=x
^{3}. What is the average rate of change of f from x=-2 to x=4?

** To read **: Through Theorem 4 on page 97. Be sure to understand Examples 1 and 2.

- What is the derivative of f(x)=x
^{3}? - Let f(x)=x
^{1/3}(i.e. the cube root of x). Use the graph of y=f(x) to explain why f'(x) does not exist at x=0.

** To read **: Through Theorem 6. Be sure to understand Example 4
and the defintions of left-hand and right-hand limits.

- Let g(x)=(x
^{2}- 9)/(x-3) as in Example 2.- Is g(x) defined at x=3? Why or why not?
- What is lim
_{x->3}g(x) ? Why?

- Is f(x)=|x| continuous at x=0? Why or why not?

** To read **: All. Be sure to understand the definition of an antiderivative
and Theorems 8, 9, and 10.

- Explain in your own words what an antiderivative of a function g(x) is.
- How many antiderivatives does f(x)=3x
^{2}have? Why?

** To read **: All. Be sure to understand Theorem 12 and the section "Proof by
picture" that follows.

- What is the 82nd derivative of f(x)=e
^{x}? - Do exponential functions model compound interest well? Explain.

** To read **: All. Be sure to understand the section "Differentiating the sine: an
analytic proof".

- What is lim
_{h->0}( cos(h) - 1) / h? - What is lim
_{h->0}sin(h) / h? - Why do we care about the limits in the first two questions?

** To read **: All. Be sure to understand Examples 3, 4 and 5.

- f(x)=x
^{2}sin(x). f'(x)=2x cos(x) - g(x)=sin(x) / (x
^{2}+ 1). g'(x) = cos(x) / (2x)

** To read **: Through Example 12. We'll consider evidence for why the Chain Rule is
true during class.

- f(x)= sin(x
^{2}). f'(x)=cos(2x) - g(x)=( sin(x) )
^{3}. g'(x)=3( cos(x) )^{2}

Reread the section, but no Reading Questions for today.

** To read **: All. Read Examples 2, 3, and 4 carefully.

- At which x-values can a continuous function f(x) achieve its maximum or minimum value on a closed interval [a,b]?
- What is the difference between an objective function and a constraint equation?

** To read **: All. Be sure to understand Examples 5 and 8.

- Why would you want to find the Taylor polynomial of a function?
- In your own words, briefly explain the idea of building the Taylor polynomial for a function f(x).

Reread the section, but no Reading Questions for today.

** To read **: All. Be sure to understand the statement of the Intermediate
Value Theorem.

- What are the
**hypotheses**of the Intermediate Value Theorem? - What is the
**conclusion**of the Intermediate Value Theorem?

** To read **: All. Be sure to understand the statement of the Mean Value Theorem
and the section "What the MVT says".

- What are the
**hypotheses**of the Mean Value Theorem? - What is the
**conclusion**of the Mean Value Theorem? - Explain the MVT using "car talk" (that is, using velocity).

** To read **: All. Be sure to understand the definition of the integral, Example 2,
and the section "Properties of the integral" beginning on page 306.

- What does the integral of a function f from x=a to x=b measure?
- Is the integral of f(x)=5x from x=-1 to x=3 positive or negative? Why?

** To read **: All. Be sure to understand the definition of the area function
and Examples 2, 3, and 4.

- Let f be any function. What does the area function A
_{f}(x) measure? - Let f(t)=t and let a=0. What is A
_{f}(1)?

** To read **: All, but you can skip the proof of the FTC in the section. We'll look
at a different approach in class.

- Find the area between the x-axis and the graph of f(x)=x
^{3}+ 4 from x=0 to x=3. - Does every continuous function have an antiderivative? Why or why not?

** To read **: Re-read the section for today.

- If f(x)=3x-5 and a=2, where is A
_{f}increasing? decreasing? Why? - How would your answer change if a=0?

** To read **: All. Be sure to understand Examples 8, 9, and 10.

- Explain the difference between a definite integral and an indefinite integral.
- What are the three steps in the process of substitution?
- Substitution attempts to undo one of the techniques of differentiation. Which one is it?

Reread the section, but no Reading Questions for today.

** To read **: All. Be sure to understand the definition of a Riemann sum
and Example 3.

- Explain, in your own words, the idea of Riemann sums for approximating integrals.
- If f(x) is decreasing on [a,b], will L
_{n}underestimate or overestimate the integral of f from a to b? How about R_{n}?

Reread the section, but no Reading Questions for today.

** To read **: All.

- Why would one want to study differential equations?
- Show that y(x)=x
^{(1/3)}is a solution to the differential equation y'(x)=1/(3y^{2}).

Reread the section, but no Reading Questions for today.

Maintained by Tommy Ratliff, ratliff_thomas@wheatoncollege.edu

Last modified: Tuesday, April 26, 2005, 3:10 PM