5A Reading Questions
1.2)
1: Tell me why Example 2 on p. 25 would be a terrible question to put on the ch. 1 test.
2: List all of the different types of functions discussed in this section. Pick 4 and tell me a realworld situation that those types of functions might be used to model.
3: Did you find anything difficult or confusing in the reading? Which parts? If you didn’t find
anything difficult or confusing, describe what you did find most interesting in the reading. Do you have
any questions?
1.3)
1: What is the difference between translating f(x) and stretching and reflecting f(x)?
2: State the 5 ways we can combine two old functions f and g to create new functions. Make up
your own functions f and g and create 5 new functions. Include the domains of the new functions. Make
them interesting! (i. e. no identity function!)
1.4)
1: In your own words, explain what it means for a line to be tangent to a curve.
2: In your own words, explain how we can use the slopes of secant lines to approximate the
slopes of a tangent line.
1.5)
1: In your own words, explain why the definition of a limit (p. 51) does not require that x = a.
2: This section suggests that we try to find limits of specific functions by taking values of x very
close to a and then looking for a pattern and guessing what the limit is. But then it warns us that this
method does not always work. Why?
1.6)
1: Explain how we can use the Product Law of Limits to obtain the Power Law.
2: Imagine you are a tutor in R125. A student comes in and asks for help with the following
problem.
𝑥 2 +𝑥−6
𝑥→2 𝑥−2
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 lim
What steps will you tell the student to take? What reasons will you give for each step?
1.8)
1: P. 83 lists three things that the definition of continuity implicitly requires. Come up with
three different functions and values of a so that each function is discontinuous at x = a in such a way
that each example fails one of the things listed, one for each requirement in the list.
2: Use the Intermediate Value Theorem to verify that f(x) = x3 + x + 1 has at least one real root.
2.1)
1: Section 2.1 gives two different definitions of the slope of the tangent line, one with x→a, and
one with h→0. Try using both to find the slope of the line tangent to f(x) = x3 at (1, 1). Which do you
prefer? Explain why.
2: Explain the difference between average velocity and instantaneous velocity.
2.2)
1: Example 1 on p. 118 shows two graphs, one of a function f and directly below it, the function
f’. Make up your own graph of a curve and call it g. Directly below, graph the function g’. Be sure to have
at least one point on g where g’ crosses the x-axis. Also, have at least one point on g where g’ does not
exist.
2: Explain in your own words why continuity does not guarantee differentiability.
2.3)
1: In your own words, explain why this section is so great! In other words, explain why it is
soooo much easier to find f’ using the techniques found in this section than by using the limit definition
of the derivative.
2: In your own words, explain why you should NOT use the Quotient Rule for finding the
derivative of every quotient. Make up your own example of a function that is a quotient where it would
be foolish to use the Quotient Rule.
2.4)
1: Rework example 2 on p. 148 by first multiplying the numerator and denominator of f(x) by
cos x and then applying the quotient rule to the transformed function. Extra credit for showing that your
derivative is equivalent to the book’s answer.
2: If f(x) = cos x, find f(25).
2.5)
1: What is the first step in finding f’(x) if 𝑓(𝑥) =
1
√cos 𝑥
?
2: Make up your own function g(x) such that finding g’(x) requires the use of the Chain Rule.
Explain why your example requires the Chain Rule.
2.6)
1: In your own words, explain why we must use implicit differentiation when we want to find
the slope of a line tangent to a given curve.
2: Make up your own example of a relation between x and y where implicit differentiation
would be the only way to find y’.
2.7)
1: This section gives many examples of rates of change: from physics, chemistry, biology, and
economics. Pick one specific example that was your favorite and explain why you liked it and what you
learned.
2. Read the quote by Joseph Fourier found on the bottom of p. 177. In your own words, explain
what a rate of change is and why this quote is important for this section.
2.8)
1) In your own words, explain why we had to learn the concepts in section 2.5 before we could
study related rates.
2) In your own words, explain why drawing a picture or diagram is so helpful in solving related
rates problems.
2.9)
1: Redo example 1 found on p. 188, only use 𝑓(𝑥) = √𝑥 + 4, 𝑎 = 0. Find √3.98 and √4.05. How
do your results compare with the books?
2: In your own words, explain the relationship between linear approximations and differentials.
3.1)
1: In your own words, describe the difference between a local maximum and an absolute
maximum of a function. Draw a graph of a function to illustrate.
2. T/F IF f has an absolute minimum value at c, then f’(c) = 0. If True, then justify your answer.
If False, then explain why it is false.
3.2)
1: In your own words, explain why we had to learn Rolle’s Theorem before we could learn the
Mean Value Theorem.
2: This section states that the main significance of the MVT is that it enables us to obtain
information about a function from information about its derivative. I disagree! Others have observed,
“The Mean Value Theorem is the midwife of calculus.” Google this quote to find out what it means, then
find a specific example of this in section 3.3.
3.3)
1: Let f be a continuous function. Suppose f is increasing but goes from being CD to CU – what
must occur? Suppose f goes from being increasing to decreasing but stays CD – what must occur?
Suppose f goes from being increasing to decreasing but stays CU – what must occur? Sketch an example
of each situation.
2. This section lists two tests that can be used to identify local maxima and minima. Which do
you think you will prefer? Explain why.
3: Extra Credit – Explain the math behind the following quote.
"In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This
was the first time a sitting president used the third derivative to advance his case for reelection. -- Hugo
Rossi"
3.4)
Skip Precise Definitions
1: In your own words, explain how to find the end behavior of a function.
2: In your own words, explain why knowing that lim
1
𝑥→∞ 𝑥 𝑟
= 0 is so helpful in finding the limits at
infinity of rational functions.
3.5)
Skip Slant Asymptotes
1: Explain how knowing the symmetry of a function can help you graph it more quickly.
2: This section lists 8 steps (A – H) to take in sketching the curve of y = f(x) by hand. Which step
do you think will be the most enjoyable for you to complete? Explain why.
3.7)
1: Explain why Step 1 (p. 258) in solving optimization problems is more like an English 1A
assignment than a Math 5A assignment.
2: In your own words, explain why, after finding Q expressed as a single variable function, it is
important to find the domain of that function.
3.8)
zero.
1: In your own words, explain what might happen if you choose x1 such that f’(x1) is close to
2: In example 1, we are told that Newton chose between 1, 2, and 3 for x1. Explain why x1 = 2
was the best choice of those three numbers.
3.9)
1: Up to now, our Ch. 3 problems have been in this pattern – Question: What is the derivative of
f(x)? Answer: f’(x). Explain how finding the antiderivatives of f(x) is like playing the game show Jeopardy.
(Google this game if you are not familiar with it.)
2: Explain how f(x) = x3 can have infinitely many antiderivatives.
4.1)
1: In your own words, explain how we can use rectangles for estimating the area under a curve.
2: In your own words, explain how the estimated area under a curve becomes more accurate as
one increases the number of subintervals.
4.2)
1: Section 4.2 discusses two ways in which the definite integral can be found
We can take xi# as any point in the ith subinterval.
We can use subintervals of unequal length.
Explain why, when we apply the lim 𝑡𝑜 ∑𝑛𝑖=1 𝑓(𝑥𝑖8 )∆𝑥, we don’t have to worry about the placement of
𝑛→∞
xi# in the ith subinterval or the length of each subinterval.
2: The most important sentence in this section is found at the bottom of p. 311. Discuss why this
is so. (Hint: Recall that we learned the limit definition of the derivative before we learned the easy way
of finding the derivative.)
4.3)
1: In your own words, explain how the FTC (Fundamental Theorem of Calculus) connects
differentiation and integration.
2: The FTC has two parts. Which one do you think you will use more frequently? Explain why you
chose as you did.
4.4)
1: In chapter 2, we learned that the derivative can be seen as the slope of the tangent line, a
rate of change, and a function in its own right. In this section, we learn that there are two
interpretations of an integral. What are they? How do you distinguish between them?
2: In your own words, explain the difference between displacement and total distance traveled.
4.5)
1: In your own words, explain the connection between U-Substitution (the Substitution Rule)
and the Chain rule.
2: On p. 344, the text integrates the same function with the same interval two different ways.
Explain the difference between these two methods. Which do you think you will prefer? Why?
5.1)
𝑏
1: In your own words, explain what ∫𝑎 𝑓(𝑥) − 𝑔(𝑥)] 𝑑𝑥 means. Assume f > g on [a, b].
𝑏
𝑑
2: Consider ∫𝑎 𝑓(𝑥)𝑑𝑥 and ∫𝑐 𝑔(𝑦)𝑑𝑦.
this context.
In your own words, explain what dx and dy means in
5.2)
1: In your own words, explain why, when finding the volume of a solid of revolution, it is sooo
helpful to sketch the region to be rotated to determine whether to use ∆x or ∆y. Why is this
determination important?
2: Let R be the region enclosed by y = x and y = x2. In your own words, explain why it makes a
difference whether you rotate R around the x-axis versus the y-axis.
5.3)
1: In your own words, explain why you would choose to use the method of cylindrical shells
instead of disks to find the volume of a solid.
2: In your own words, explain what (circumference)(height)(thickness) means.
5.4)
1: In your own words, explain how integration is used in computing work.
5.5)
1: In your own words, explain what 𝑤𝑖𝑑𝑡ℎ = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 means.
𝑎𝑟𝑒𝑎
2: How does the Mean Value Theorem for Integrals compare to the Mean Value Theorem? In
light of this, should we call the MVT the Mean Value Theorem for Derivatives?
1.7)
why.
1: Using the precise definition of a limit is often called the epsilon, delta (ε, δ) definition. Explain
2: We study this section at the end of the semester instead of when we were studying chapter
one concepts. Would you rather have learned it at the beginning of the semester or wait until now?
Explain why you feel this way.