Unit #3 - Differentiability, Computing Derivatives, Trig Review
Some problems and solutions selected or adapted from Hughes-Hallett Calculus.
Derivative Interpretation and Existence
1. The cost, C (in dollars), to produce q quarts of ice
cream is C = f (q). In each of the following statements,
identify the units of the input and output quantities.
(a) f (200) = 600
(b) f 0 (200) = 2
(a) Express the relationship f (60) = 40 in a sentence,
including units.
(b) Express the relationship f 0 (60) = 6 in a sentence,
including units.
(c) Use the information in parts (a) and (b) to estimate the dosage for a 65 kg child.
2. The time for a chemical reaction T (in minutes), is
a function of the amount of catalyst present, a (in
milliliters), so T = f (a). State the units of the input
and the output for the following expressions.
(a) f (5) = 18
(b) f 0 (5) = −3
7. Sketch a smooth graph that satisfies each description
below (each part is a separate graph).
(a) The derivative is everywhere positive and gradually
increasing.
(b) The derivative is everywhere negative and gradually increasing in value (smaller negatives, approaching zero).
3. Suppose C(r) is the total cost of paying off a car loan
borrowed at an annual interest rate of r%.
(a) What are the units of C(r)?
(b) What are the units of C 0 (r)?
(b) What is the sign of C 0 (r)?
(c) The derivative is everywhere positive and gradually
decreasing.
8. Consider the function graphed below.
4. A magnetic field, B, is given as a function of the distance, r, from the center of the wire as follows:
B=
r
B0
r0
when r ≤ r0 , and
r0
B0
r
when r > r0 . Here r0 and B0 are positive constants.
B=
(a) Sketch a graph of B against r.
Use your graph to answer the following questions.
(As you do so, think about what the meaning of
the constant B0 is.)
(a) At what x-values does the function appear to not
be continuous?
(b) At what x-values does the function appear to not
be differentiable?
(b) Is B a continuous function of r?
9. Identify any x-values at which the absolute value function f (x) = 8|x + 6|, is
(c) Is B a differentiable function of r?
5. Consider the function
(
1 + cos(πx/2)
g(x) =
0
(a) not continuous, and then
for − 2 ≤ x ≤ 2
for x < 2 or x > 2
(b) not differentiable.
10. Let f (t) be the number of centimeters of rainfall that
has fallen since midnight, where t is the time in hours.
Write a one sentence interpretation of each of the following mathematical statements.
Sketch the graph, and use your sketch to answer the
following questions.
(a) Is the function continuous at x = 2 and x = −2?
(b) Is the function differentiable at x = 2 and x = −2?
(a) f (8) = 2.5
6. The dosage (D, in mg) of some drugs, like children’s
Tylenol, depends on the mass of the patient, M in
kg. We would express the relationship as a function,
D = f (M ).
(b) f −1 (7) = 11
(c) f 0 (8) = 0.3
(d) (f −1 )0 (7) = 4
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The Derivative as a Function
11. Consider the function f (x) shown in the graph below.
15. For the function f (x) shown in the graph below, sketch
a graph of the derivative.
Carefully sketch the derivative of the given function
(you will want to estimate values on the derivative function at different x values as you do this). Use your
derivative function graph to estimate the following values on the derivative function.
at x =
the derivative is
-3
-1
1
3
12. Below is the graph of f (x). Sketch the graph of f 0 (x).
16. A child fills a pail by using a water hose. After finishing, the child plays in a sandbox for a while before
tipping the pail over to empty it. If V (t) gives the volume of the water in the pail at time t, then the figure
below shows V 0 (t) as a function of t.
13. Below is the graph of f (x). Sketch the graph of f 0 (x).
At what time does the child:
(a) Begin to fill the pail?
(b) Finish filling the pail?
0
14. Below is the graph of f (x). Sketch the graph of f (x).
(c) Tip the pail over?
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Computing Derivatives
Below are a small sample of problems involving the computation of derivatives. They are not enough to properly
learn and memorize how to apply all the derivative rules. You should practice with as many problems as you need to
become proficient at computing derivatives.
Further practice problems can be found in any calculus textbook.
From Hughes-Hallett 5th edition,
Section 3.1 - 7-47 (odd)†
Section 3.2 - 1-33 (odd)
Section 3.3 - 3-29 (odd)
Section 3.4 - 1-49 (odd)
Section 3.5 - 3-39 (odd)
Section 3.6 - 1-33 (odd)
From Hughes-Hallett 6th edition,
Section 3.1 - 7-49 (odd)†
Section 3.2 - 1-25 (odd)
Section 3.3 - 3-29 (odd)
Section 3.4 - 1-55 (odd)
Section 3.5 - 3-47 (odd)
Section 3.6 - 1-41 (odd)
4x2 tan x
.
sec x
(a) Find f 0 (x).
17. Let f (x) = 4ex − 9x2 + 5. Compute f 0 (x).
30. f (x) =
√
−5
18. Let f (x) = 2x6 x + 3 √ . Compute f 0 (x).
x x
7x2 + 7x + 5
√
19. Let f (x) =
.
x
(a) Compute f 0 (x).
31. f (x) = 7 sin x + 12 cos x
(a)Compute f 0 (x).
(b) Find f 0 (3).
(a) Compute f (t).
(b) Find f 0 (1).
32. Let f (x) = cos x − 2 tan x. Compute f 0 (x).
20. Let f (t) = 7t−7 .
0
(b) Find f 0 (3).
5 sin x
3 + cos x
(a) Compute f 0 (x).
33. f (x) =
0
(b) Find f (3).
21. Let f (x) = 4ex + e1 . Compute f 0 (x).
(b) Find f 0 (2).
34. f (x) = 7x(sin x + cos x)
22. Let f (x) = 4ex + 4x. Compute f 0 (x).
(a) Compute f 0 (x).
(a) Find f 0 (3).
2
23. f (x) = (3x − 2)(6x + 3).
35. Let f (x) = cos(sin(x2 )). Compute f 0 (x).
(a) Compute f 0 (x).
(b) Find f 0 (4).
√
x−4
. Compute f 0 (9).
24. Let f (x) = √
x+4
4x + 3
.
3x + 2
(a) Compute f 0 (x).
36. Let f (x) = 2 sin3 x. Compute f 0 (x).
25. Consider f (x) =
7 − x2
7 + x2
0
(a) Compute f (x).
38. Let f (x) = −3 ln[sin(x)]. Compute f 0 (x).
(b) Find f 0 (5).
39. Let f (x) = 2 ln(4 + x). Compute f 0 (x).
40. Let f (x) = arcsin(x).
26. Consider f (x) =
(a) Compute f 0 (x)
(b) Find f 0 (1).
27. Let f (x) = −2x(x − 3).
(a) Compute f 0 (x).
4x3 − 3
28. f (x) =
x4
(a) Compute f 0 (x).
41. Let f (x) =
(b) Find f 0 (−5).
† For
(b) Find f 0 (0.4).
arccos(14x)
. Compute f 0 (x).
arcsin(14x)
42. Let
P =
(b) Find f 0 (2).
V 2R
2.
(R + r)
dP
Calculate
, assuming that r is variable and R and
dr
V are constant.
ex
29. g(x) =
. Compute g 0 (x).
5 + 4x
† For
dy
.
dx
37. Let y = (8 + cos2 x)6 . Compute
this section, simplify products and fractions before you differentiate, rather than using the product rule and quotient rule.
this section, simplify products and fractions before you differentiate, rather than using the product rule and quotient rule.
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Trigonometric Functions
43. A population of animals oscillates sinusoidally between
a low of 100 on January 1 and a high of 1100 on July 1.
Graph the population against time and use your graph
to find a formula for the population P as a function of
time t, in months since the start of the year. Assume
that the period of P is one year.
51. Consider the trig graph below.
44. Find the period and amplitude of r = 0.4 sin(πt) + 1
√
45. Find one solution to the equation 1 = 2 cos(2x).
46. Find one solution to the equation 3 = 2 sin(3x+1)+2.
47. Find sin θ and tan θ if cos θ =
0 ≤ θ < π/2.
8
17 ,
48. Find sin θ, sec θ, and cot θ if tan θ =
assuming that
11
60 .
Find both a cosine and a sine based formula for this
graph.
49. Find a formula for the graph of the function y = f (x)
given in the figure below.
52. Consider the trig graph below.
50. Find a formula for the graph of the function f (x)given
in the graph below.
Find both a cosine and a sine based formula for this
graph.
53. Sketch the graph of y = 2x cos(2πx) on the interval
x = [−4, 4].
√
54. Sketch the graph of y = x cos(2πx) on the interval
x = [0, 9].
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