BC Test Review
Evaluate each limit.
1) lim+ 4 x ln
x→0
1
x
5) lim+ 3tan x ln x
x→0
x
2) lim
x→0
6) lim
x→0
9) lim
x→1
5( x − 1)
ln x
e −1−x
1 − cos x
(
3
3
− x
x
e −1
10) lim
x→1
(
3) lim
x→ ∞
)
x
ln x
7) lim+ 5 ⋅ (tan x)
4
4
−
ln x
x−1
4) lim (2sec x − 2tan x)
x→
π
2
sin x
x→0
8) lim
x→0
2(e x − 1 − x)
x
2
)
Solve each related rates problem. Be sure to include units in your answers
1) A hypothetical square grows so that the length of its sides are increasing at a rate of 3 m/min.
How fast is the area of the square increasing when the sides are 12 m each?
2) A hypothetical square grows so that the length of its diagonals are increasing at a rate of 5
m/min. How fast is the area of the square increasing when the diagonals are 10 m each?
3) Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The area of
the spill increases at a rate of 16π m²/min. How fast is the radius of the spill increasing when
the radius is 14 m?
4) An observer stands 400 ft away from a launch pad to observe a rocket launch. The rocket
blasts off and maintains a velocity of 500 ft/sec. Assume the scenario can be modeled as a right
triangle. How fast is the observer to rocket distance changing when the rocket is 300 ft from
the ground?
32π
5) A spherical snowball is rolled in fresh snow, causing it grow at a rate of
in³/sec. How fast
3
is the radius of the snowball increasing when the radius is 8 in?
6) A hypothetical square grows at a rate of 25 m²/min. How fast are the sides of the square
increasing when the sides are 4 m each?
7) A spherical snowball is rolled in fresh snow, causing it to grow so that its radius increases at a
rate of 4 in/sec. How fast is the volume of the snowball increasing when the radius is 6 in?
8) A spherical balloon is inflated at a rate of 36π cm³/sec. How fast is the radius of the balloon
increasing when the radius is 3 cm?
9) A hypothetical cube grows at a rate of 64 m³/min. How fast are the sides of the cube increasing
when the sides are 2 m each?
10) A spherical balloon is inflated so that its radius increases at a rate of 3 cm/sec. How fast is the
volume of the balloon increasing when the radius is 3 cm?
Worksheet by Kuta Software LLC
[11] Water is dripping out of the bottom of a cone-shaped cup at 2 in3/sec. The cup is
15 inches deep and has a radius of 5 inches. [a] Find a formula for the volume of
water in the cup in terms of the radius only. [b] Find the rate at which the radius of
water is changing when the volume of water is 64π in3. [c] Find the rate at which the
height of water is changing when the volume of water is 64π in3. [d] Find the rate at
which the circular top surface area of water is changing when the volume of water is
64π in3
Theorems about Continuity & Derivatives
[1] Given a y-value of 2, find the value of c guaranteed by the intermediate value theorem for f  x   x3  3x on [0,3].
[2] Find the value of c guaranteed by Rolle’s theorem for f  x   x3  3x on 0, 3  .


[3] Find the value of c guaranteed by the mean value theorem for f  x   x3  3x on  0,3 .
 
.
 2 
[4] (calculator) Find the value of c guaranteed by the mean value theorem for f  x   sin x on 0,
[5] (calculator) Find the value of c guaranteed by the mean value theorem for f  x   e x on  1,1 .
[6 -9] Use the graph of c.
[6] Estimate to the nearest tenth the value of c guaranteed by Rolle’s theorem on the
interval 0,1.5].
[7] How many values of c satisfy the conclusion of Rolle’s theorem on [-1,2]?
[8] Estimate to the nearest tenth the values of c that satisfy the conclusion of the mean
value theorem on [-1,1.5].
[9] Estimate to the nearest tenth the values of c that satisfy the conclusion of the
intermediate value theorem on [-1,2] for y  1 .
x
on [0,2]?
x 1
x2 1
[11] Does the MVT apply to f  x  
on [0,2]?|
x 1
[10] Does the MVT apply to f  x  
4
[12] Does the MVT apply to f  x   x 5 on [-1,1]?
6
5
[13] Does the MVT apply to f  x   x on [-1,1]?

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