Math 125
Carter
Test 3 Fall 2013
General Instructions: I have tried to give three or four problems of each type that I
envision for the exam. When I write the exam, I will copy this file, edit it down, and (of
course) change some (maybe all) of the problems. Write yourself 3-5 tests, and work each
one in 70 minutes. Study the problems that you have missed, and work the next test.
1. Compute the derivatives for each of the following expressions. Do not simplify your
results!
(a)
i. y = x3 + 2x2 − 10x + 4.
ii. y = x4 + 3x3 + 5x2 − 3x − 10.
iii. y = 3x2 − 2x + 8
iv. y = (x − 2)(x2 + x)
(b)
i. y = (x2 + 2x − 5)(x − 4).
ii. y = (x − 2)(3x − 2)(x2 − 4).
iii. y = ex cos (x).
iv. y = (ln (x))ex .
(c)
ii.
iii.
iv.
(d)
ex
.
sin (x)
(3x−2)
.
(2x+5)
ln (x)
.
x
x2 −2x+2
.
x3 −1
i. y =
i. y = tan (x2 + 1).
√
ii. y = x2 + 1.
2
iii. y = cos ex +4 .
√
iv. y = sin x2 + 1.
(e)
i. y = cos (ex ).
ii. y = ln
3x+5
x2 +8
.
iii. y = arcsin x/4.
iv. y = arctan ex .
(f)
i. f (t) = ln (sin (t)).
√
ii. f (x) = x2 + 1 + arctan x.
iii. f (x) = ln ex + 4.
iv. f (x) = sin (ln (x)).
(g)
i. f (θ) =
ii. f (x) =
3
cos (θ)
.
1+sin (θ)
√
x3 +8·(x2 −4x)5
.
(x3 +x2 +3x+4)(cos (x))
1
2
iii. f (x) = 8x +3x+2 .
iv. f (x) = log6 (x + 8)4 (sin (x2 )).
(h)
i.
ii.
iii.
iv.
y
y
y
y
= ln (arcsin (x)).
= arcsin (ln (x)).
= arctan (ex ).
= earctan (x) .
2. (10 points) Compute the equation of line tangent to the curve:
(a) x2 + y 2 = 25 at the point (−3, 4).
(b) x3 + y 3 = 117 at the point (5, −2).
(c) x2/3 + y 2/3 = 25 at the point (−27, 64).
3. (a) (10 points) There are two points at which the tangent to the curve f (x) = x3 − x
is horizontal. Determine the coordinates of each.
(b) (10 points) A cartoon coyote plans to shoot an anvil upwards off the edge of a 600
meter tall cliff at the rate of 40 meters per second. Assume that the acceleration
due to cartoon gravity is −10 meters per square second.
i.
ii.
iii.
iv.
When does the anvil reach its acme?
How high above the ground does the anvil travel?
How many seconds does it take to reach the ground?
What is the anvil’s terminal velocity?
(c) (10 points) Determine the equation of the line tangent to the curve y = sin (x) at
the point x = π6 .
4. (10 points)
(a) A large coffee cone 10 centimeters in radius and 15 centimeters tall drips coffee
at a rate of 2 cubic centimeters per second. How fast is the height of the coffee
dropping when the height is 3 centimeters? The volume of a cone is V = π3 r2 h
where r denotes the radius and h denotes the height.
(b) The radius of a spherical ball of ice is decreasing at a rate of 2 centimeters per
minute. How fast is the volume decreasing when the radius is 10 centimeters?
The volume of a sphere is given by V = 4π
r3 where r denotes the radius.
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(c) A girl flies a kite at a height of 300 feet, the wind carrying the kite horizontally
away from her at a rate of 25 feet/second. How fast must she let out the string
when the kite is 500 feet away from her?
(d) Water runs into a conical tank at the rate of 9 ft.3 /min. The tank stands vertex
down and has a height of 10 feet and a base radius of 5 feet. How fast is the water
level rising when the water is 6 feet deep?
2
(e) A hot air balloon rising straight up from a level field is tracked by a range finder
150 meters from the liftoff point. At the moment that the range finder’s elevation
angle is π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the
balloon rising at that moment?
(f) All homework problems and worked examples from section 3.10.
5. (10 points)
(a) Determine the Maximal and minimal values for the function f (x) = 25 − x2 over
the interval −6 ≤ x ≤ 2. At what x-coordinates do these occur?
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(b) Determine the Maximal and minimal values of the function f (x) = e−x over the
interval −1 ≤ x ≤ 3. At what x-coordinates do these occur?
(c) Determine the Maximal and minimal values of the function y = sin (x) over the
interval π/6 ≤ x ≤ 2π/3. At what x-coordinates do these occur?
(d) Determine the Maximal and minimal values of the function y = x3 − x for over
the interval −2 ≤ x ≤ 5. At what x-coordinates do these occur?
6. (10 points) Use the idea of local linearization to give approximate values for the expressions indicated. Choose your point x = a nearby the indicated points judiciously.
√
(a) 26
(b) (0.0013)1/3
(c) 50.022
7. I found these problems at the bottom of the file for the previous test 2. They too are
interesting problems that are worth your mastery.
(a) Compute the equation of the line tangent to the function f (x) = x tan (x) at the
point x = π4 .
(b) Compute
dy
dx
where sin (x + y) = x cos (y).
(c) Compute the third derivative of y =
1
.
1+x
(d) Compute f 0 (x) for f (x) = ln [(x + 1)(2x + 9)].
(e) Determine the maximal and minimal values for the function f (θ) = cos (θ)+sin (θ)
over the interval [0, 2π]. For which angles θ do these optima occur?
(f) Determine the maximal and minimal values for the function f (x) = −x2 +10x+43
over the interval [3, 8].
(g) The radius of a sphere is increasing at a rate of 14 inches per minute. Determine
the rate at which the volume is increasing when the radius is 8 inches.
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(h) A tank in the shape of a right circular cone of radius 300 centimeters and height
500 centimeters leaks water from its vertex (which points down) at a rate of 10
cubic centimeters per minute. Find the rate at which the water level is falling
when the level is 200 centimeters.
(i) Compute the equation of the line tangent to the curve at the particular point,
(a, f (a)) where y = 4 ln (9x + 2) at a = 2.
(j) Sand falls into the shape of a conical pile at the rate of 2 m3 per minute. The
proportion of the radius to the height is always 1/3. At what rate is the height
increasing when the height is 5 meters? The volume of a cone is V = π3 r2 h where
r denotes the radius and h denotes the height.
(k) A boat is pulled into a dock by a rope through a windlass that is 3 feet above the
boat. The rope is being hauled in at 1 foot per second. At what rate is the boat
approaching the dock when it is 4 feet from the dock?
(l) Determine the Maximum and minimum values of the functions on the given intervals.
i.
f (x) = −x2 + 10x + 8 on [3, 6].
ii. f (θ) = cos (θ) + sin (θ) on [0, 2π].
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