Derivatives

Review Questions:

Differentiation, relationship between $f$ and $f$ ^',tangent lines, implicit differentiation, related rates, higher derivatives

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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.