Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
Math 131 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 6.1, 6.5, and 6.7)
Note: This collection of questions is intended to be a brief overview of the exam material. When studying, you
should also look at your notes, the suggested homework problems from the textbook, as well as the other week-inreviews for this material.
1. The following table shows the number of heartbeats after t minutes of a patient after a surgery.
t (min)
Heartbeats
36
2530
38
2556
40
2610
42
2720
44
2920
a) Find the average heart rate between 38 and 42 minutes. Interpret your answer.
b) Estimate the patient’s heart rate (i.e. instantaneous heart rate) after 42 minutes. Interpret your answer.
c) Find an appropriate model, h(t), for the data. (Round to four decimal places, if necessary).
d) Use your rounded model to find the patient’s heart rate after 42 minutes. How does this compare with
your estimate above?
e) Use your rounded model to find the average heart rate between 38 and 42 minutes. How does this compare
with your answer above?
1
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
2. Find lim +
x→−7
x2 + 4x − 21
.
|x + 7|
3. Consider the graph of a function shown below.
4
3
2
1
−4
−3
−2
−1
1
2
3
4
x
−1
−2
−3
−4
a) If the graph above is f (x), determine where any local and/or absolute extrema of f (x) occur.
b) If the graph above is f (x), determine where f ′ (x) is negative.
c) If the graph above is f ′ (x), determine where f (x) is decreasing.
d) If the graph above is f ′′ (x), determine where f ′ (x) is decreasing.
2
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
4. Find the domain of the function f (x) =
5. Consider the function g(x) =
√
8
x2 − 5x + 6
.
loga 32 x − 9
x2 + 1
.
x−3
a) Find an expression for the slope of the secant line through (x, g(x)) and (x + h, g(x + h)).
b) Find an expression for the slope of the tangent line at the point (x, g(x)).
3
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
6. Find the inverse of the function f (x) =
2x − 3
, if it exists.
x+3
7. Where is the function f (x) continuous?

sin(x + π ) − 3
x<0






 p
13
x5 − x3 − 3ex 0 ≤ x ≤ 9
f (x) =






 ln(x − 7)
x>9
x2 − 7x − 30
8. Find the derivative of f (x) = log3
q
7
(x3 − π x) csc(5x2 )
4
.
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
9. Sketch the graph of a function that satisfies all of the following conditions:
• f ′ (−3) = f ′ (−5) = 0
• f ′ (x) < 0 if −5 < x < −3
• f ′ (x) > 0 if x < −5, −3 < x < 1, or x > 1
• f ′′ (x) < 0 if x < −4 or x > 1
• f ′′ (x) > 0 if −4 < x < 1
• lim+ f (x) = −∞
x→1
• lim− f (x) = ∞
x→1
• lim f (x) = 2
x→∞
x
, write the function f (x) which results from shifting g(x) to the left
2−x
3 units, then compressing horizontally by a factor of 2, then stretching vertically by a factor of 5, then
reflecting about the y axis, and then shifting down by 18 units.
10. Starting with the function g(x) =
sin x
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
x→−1 x2 + x
11. Find lim
5
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
12. On Io, one of Jupiter’s many moons, a volcanic eruption pushes a rock downward off a 175-meter cliff with
an initial speed of 10 meters per second. If the acceleration due to gravity is -1.765 m/s2 , find an equation
s(t) that gives the rock’s position above the ground t seconds after it was pushed off the cliff.
13. Mark each of the following curves as f (x), f ′ (x), or f ′′ (x).
14. If g(3) = −2, g ′ (3) = 8, g ′ (−2) = 1, f (3) = 2, f ′ (3) = −4, f ′ (−2) = 7, and h(x) = 12 f (g(x))+ f (x)·g(x),
find h ′ (3).
6
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
7x + π
.
15. Find lim √
x→−∞ 16x2 − 3x + e
16. Find the linearization of the function f (x) = (x + 1)2 at a = 3 and use it to approximate (4.12)2 .
7
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
17. A fence is built to enclose a rectangular area of 800 square feet. The fence along 3 sides costs $6 per foot,
and the fourth side costs $18 per foot. Find the dimensions of such a rectangle that will minimize the cost.
18. Find
Z b √
5 4
x − (ln 19)x2
1
−x8
+ 3 − πe
x
x
dx.
8
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
19. Consider the function f (x) =
ex
.
x
a) Use calculus to find any critical numbers of f (x), as well as the intervals where the function is increasing/decreasing, the values of any local extrema, the intervals where the function is concave upward/downward,
and any inflection points.
b) Find the absolute minimum of f (x) on the interval [0.5, 3], if it exists.
c) Find the absolute minimum of f (x) on the interval [−1, 3], if it exists.
9
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
20. The growth rate of a population of bacterial cells in a petri dish is given by f (t) = 0.2e0.5t + 10 cells per day,
where t is the number of days since the start of the experiment, 0 ≤ t ≤ 20. After 3 days, there were 300
bacteria cells.
a) Find the increase in the number of cells between days 8 and 15.
b) How many cells were in the dish after 7 days?
c) What is the average number of bacteria cells in the population from day 10 to day 15?
21. Find
g ′ (θ )
if g(θ ) =
Z 2eθ √
3
1 + u2
csc θ
u
du.
10
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
22. Find the area bounded by the curves y = 0.6x + 3 and y = x2 − 3x + 1 on the interval [−3, 2].
sec a
23. Find the derivative of g(a) = e a2 +5a ln 2sin(3a) .
24. Polonium-210 has a half-life of 140 days. If a sample has a mass of 180 mg, how long will it take for the
mass to decay to 8 mg?
11
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
25. Find each of the following integrals:
sin 4x
a)
Z
b)
Z
π
dt
(3 − e−t )et
c)
Z
1
dx
8x(ln x)7
d)
Z
p
3
(1 + cos 4x)2
3q2 − 4
dx
(3q3 − 12q + 18)2
dq
12
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
26. Solve the following equation for x:
logb 2x =
3
log 9 + logb 6 − logb 18
2 b
27. Find the equation of the tangent line to the curve y = sec x + cos 2x at x = π /3.
√
Hint: sin(π /3) = 23 , cos(π /3) = 21 , and tan(π /3) =
√
Also, sin(2π /3) = 3/2 and cos(2π /3) = −1/2.
13
√
3, and sec x =
1
cos x ,
csc x =
1
sin x ,
and cot x =
1
tan x .
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
28. Consider the function g(t) =
f is [−4, 9].
Z t
−4
f (x) dx, where the graph of f (x) is shown below. Note that the domain of
a) Find g(7).
b) Where does g have an absolute max on [−4, 7]?
c) Where is g increasing?
14
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
29. When using a Riemann sum with left endpoints to approximate the area under the curve f (x) = x2 − 5x − 3
on the interval from [6, 10] using 16 rectangles, what is the height of the 4th rectangle?
x2 − 2x − 8
, if
x4 − x3 − x2 − 5x − 30
any exist. If there are vertical asymptotes, describe the behavior near each asymptote.
30. Use calculus to find the horizontal and vertical asymptotes of the function f (x) =
Hint: x4 − x3 − x2 − 5x − 30 = (x − 3)(x2 + 5)(x + 2).
15
Math 131 WIR, copyright Angie Allen, Fall 2011
(Problem 12 is courtesy of Sherry Scarborough and Heather Ramsey,
and the figure for Problem 28 is courtesy of Joe Kahlig)
31. Calculate the rate of flow in a small human artery where we take η = 0.03, R = 0.006 cm, l = 3 cm, and
P = 4000 dynes/cm2 . (Give your answer to three significant figures.)
32. Find the area bounded by the curves y2 =
x
and y2 = x − 4.
2
33. The dye dilution method is used to measure cardiac output with 9 mg of dye. The dye concentrations, in
mg/L, are modeled by f (t) = 51 t(16−t), 0 ≤ t ≤ 16, where t is measured in seconds. Find the cardiac output.
16