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Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
Math 131 Week-in-Review #5 (Sections 2.6, 2.7, 2.8, and 3.1)
1. The position of an object moving in a straight line is given by s = f (t) = t −1 − 2t, where s is measured in meters and
t in seconds. Find the velocity and speed of the object after four seconds using the definition of the derivative at a
number (form 2).
2. The following limit represents the derivative of some function f at some number a. State such an f and a:
lim
x→π /4
tan x − 1
x − π /4
Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
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3. A pie has just been taken out of the oven and is cooling before being eaten. Suppose that T (m) is the temperature of
the pie, in degrees Fahrenheit, after it has been out of the oven m minutes.
a) Interpret T (5) = 250.
b) Interpret
T (3) − T (1)
= −10.
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c) What are the units of T ′ (m), and what is the sign of T ′ (m)?
d) Interpret T ′ (5) = −8.
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Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
4. Use the definition of the derivative to find f ′ (x) if f (x) =
the line tangent to f (x) at the point (2, 3).
√
5x − 1, and then use your formula to find the equation of
5. Create a chart showing the graphical relationships between f , f ′ , and f ′′ .
Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
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6. Consider the graph of f (x) below.
a) Arrange the derivatives at the given points from smallest to largest.
b) At what points do f ′ (x) and f ′′ (x) have the same sign?
7. Sketch the graph of a function f that is continuous everywhere, but it is not differentiable at x = −4 because there is
no tangent line nor at x = 3 because the tangent line is vertical.
Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
8. Sketch the graph of the derivative of each of the following functions.
a)
b)
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Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
9. a) Sketch a function whose slope is positive and decreasing.
b) Sketch a function whose slope is getting less negative.
c) Sketch a function that is increasing whose slope is also increasing.
10. Sketch the graph of a function that satisfies all of the following conditions:
• f ′ (1) = f ′ (−1) = 0
• f ′ (x) < 0 if |x| < 1
• f ′ (x) > 0 if 1 < |x| < 2
• f ′ (x) = −1 if |x| > 2
• f ′′ (x) < 0 if −2 < x < 0
• inflection point at (0, 1)
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Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
11. Let F be an antiderivative of the function f whose graph is shown below.
a) Where is F increasing?
b) Where is F decreasing?
c) Where is F concave up?
d) Where is F concave down?
e) Use the above information to sketch a graph of F if F(0) = 0.
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Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
12. The table below gives values of P(t), the population of a small city in Texas in year t. (Midyear estimates are given.)
t
P(t)
1994
29, 036
1996
29, 672
1998
32, 300
2000
36, 205
2002
38, 260
a) Find the average rate of growth from 1996 to 2000, and interpret your answer. (Round your final answer to the
nearest integer, if necessary.)
b) Estimate and interpret P ′ (2000). (Round your final answer to the nearest integer, if necessary).
Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
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13. Find the derivative of each of the following:
a) h(x) = (x − 4)(2x + 3)
b) y =
√
x(x − π )
c) g(u) =
√
√
2u + 5u
d) y = aek +
c
b
+
k k2
14. Given F(x) is an antiderivative of f (x), create a chart showing the graphical relationships between the antiderivative
F(x) and its first and second derivatives.
Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
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15. Find where f (x) = (1 + 2x)2 has a horizontal tangent line.
16. Find the equation of the normal line to the curve y = 2x − 3x2 at the point (1, −1) using the definition of the derivative
at a number (form 2).
Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
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17. Sketch the graph of a function f (x) which has a point of most rapid increase and the graph of a function g(x) which
has a point of least rapid decrease.
18. Find a second-degree polynomial P such that P(2) = 7, P ′ (2) = 6, and P ′′ (2) = 10.
19. If j(x) =
2ex √
d2 j
8
− x3 + π x − 1.83, find 2 .
−5
dx
20. The equation of motion of a particle is s = 3t 3 − 6t 2 + 3t + 1, where s is in meters and t is in seconds.
Find the velocity and acceleration functions of the particle, as well as the acceleration after 1 second.
Math 131 WIR, copyright Angie Allen (Problems 6, 8, and 11 are courtesy of Joe Kahlig)
d
21.
dx
12
!
√
4
x−8 + 3x2 − x5
√
3
8 x2
22. Use calculus to determine on what intervals the function f (x) = x3 − 5x2 + x is increasing/decreasing and concave
upward/downward.