Unit 4 Practice Exam
1. Approximate the root using a linearization centered at an appropriate nearby
number
a. √50
3
b. √65
i. 50 is roughly 49 and the square root of 49=7
The tangent line at (49,7) has the slope
f ’(49)=1/2(49)-1/2
1
1
=2 ×
√49
=1/14
√50 ≈ 𝐿(50) = 7 +
1
(50 − 49) ≈ 7.071
14
ii. 43 =64 which is roughly 65
The tangent line at (64,4) has the slope
2
1
f ‘ (64)=3 64− ⁄3
1
=3 ×
1
3
( √64)2
3
1
= 48
√65 ≈ 𝐿(65) = 4 +
≈ 4.02
1
(65 − 64)
48
2. If y=tan(x), x=π, and dx=.5, what is dy?
a. -.25
b. -.5
c. 0
d. .5
e. .25
i. (tanπ)(dx)
=(tan π) (.5)
=(0)(.5)
=(0)
The answer is then C
3. Use the linearization (1+x)k at x=0 is L(x)=1+kx to approximate
a. (1.002)100
3
b. √1.009
i. 1+0.002(100)
=1.2
ii. 1+0.009(1/3)
=1.003
4. For the function with the properties f(x)=x3-x, a=1, dx=.01, find
a. True change ∆𝑓 = 𝑓(𝑎 + 𝑑𝑥) − 𝑓(𝑎)
b. Estimated Change 𝑑𝑓 = 𝑓 ′ (𝑎)𝑑𝑥
c. The approximation error |∆𝑓 − 𝑑𝑓|
(Express part C in scientific notation e.g. <1.0x10-n)
i. True change
∆𝑓 = 𝑓(1.01) − 𝑓(1)
=(0.030301)-(0)
=0.030301
ii. Estimated change
3(1)2 dx
=3(.01)
0.03
iii. Approximation error
|0.030301-0.03|
=0.000301
is < 1.0x10-3
5. When a circular plate of metal is heated in an oven, it’s radius increases at a
rate of 0.01cm/sec. at what rate is the plates area increasing when the radius
is 50 cm?
i. A=πr2
dA/dt=2πr dr/dt
dA/dt=100π(0.01)
dA/dt=π
6. Inge flies a kite at a height of 300 ft, the win is carrying the kite horizontally
away at a rate of 25 ft/sec. How fast mush she let out the string when the Kite
is 500 ft away from her?
i.
7. True or False. If radius is increasing at a constant rate, the circumference is
as well.
i. False
8. The radius r and the surface area (S) are related to the equation S=4πr2.
Write an equation that relates dS/dt to dr/dt.
i.
14. A trucker handed in a ticket at a toll booth showing that in 2 h she had covered
159 mi on a toll road with a speed limit of 65 mph. The trucker was cited for
speeding. Why?
15. It took 20 sec for the temperature to rise from 0 F to 212 F when a thermometer
was taken from a freezer and placed in boiling water. Explain why at some moment
in that interval the mercury was rising at exactly 10.6 degrees F/sec.
16. Find the local extrema, the intervals on which the function is increasing and
decreasing in the function g(x)=x1/3(x+8)
17. On the moon, the acceleration due to gravity is 1.6 m/sec2.
a. If a rock is dropped into a crevasse, how fast will it be going before it
hits the bottom 30 sec later?
b. How far below the point of release is the bottom of the crevasse?
c. If instead of being released from rest, the rock is thrown into the
crevasse from the same point with a downward velocity of 4 m/sec,
when will it hit the bottom and how fast will it be going when it does?