[ Calculus I Archive ]
Review Questions:
Differentiation, relationship between and
,tangent lines, implicit differentiation,
related rates, higher derivatives
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Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
Question 7
Question 8
Question 9
Question 10
Question 11
The length m of a rectangle is decreasing at the rate of 2 cm/s and
the width w is increasing at the rate of 2 cm/s. When m = 12
cm and w = 5 cm, find the rates of change of (a) the area, (b) the perimeter,
and (c) the lengths of the diagonals of the rectangle. Which of these
quantities are decreasing and which are increasing?
Question 12
Let V be the volume and S the total surface area of a solid right circular
cylinder that is 5 ft high and has radius r ft. Find dV/dS when r
= 3.
Question 13
Sand falls onto a conical pile at the rate of 10 ft3/min. The
radius of the base of the pile is always equal to one half its altitude.
How fast is the altitude of the pile increasing when it is 5 ft deep?
Question 14
Suppose that a raindrop is a perfect sphere. Assume that, through condensaton,
the raindrop accumulates moisture at a rate proportional to its surface
area. Show that the radius increases at a constant rate.
Question 15
A boat is pulled in to a dock by a rope with one end attached to the bow
of the boat, the other end passing through a ring attached to the dock
at a point 4 ft higher than the bow of the boat. If the rope is pulled
in at the rate of 2 ft/s, how fast is the boat approaching the dock when
l0 ft of rope are out?
Question 16
A balloon is 200 ft off the ground and rising vertically at the constant
rate of 15 ft/s. An automobile
passes beneath it traveling along a straight road at the constant rate
of 55 ft/s. How fast is the distance between them changing one second
later?
Question 17
A light is at the top of a pole 50 ft high. A ball is dropped from the
same height from a point 30 ft
away from the light. How fast is the shadow of the ball moving
along the ground 1/2 second later? (Assume the ball falls a distance s
= 16t2 ft in t seconds.)
Question 18
Given a triangle ABC. Let D and E be points on the sides AB and AC, respectively,
such that DE is parallel to BC. Let the distance between BC and DE
equal x. Show that the derivative, with respect to x, of the area BCED
is equal to the length of DE.
Question 19
Water is being poured into an inverted conical tank (vertex down) at the
rate of 2 ft3/min. How fast is the water level rising when the
depth of the water is 5 ft? The radius of the base of the cone is 3 ft
and the altitude is 10 ft.
Question 20
Question 21
Question 22
Question 23
Question 24
Question 25
Question 26
Question 27
Question 28
Question 29
Question 30
Question 31
Last update: November 10, 1998.