Applications of Derivatives Practice Problems for AP Calculus
Review the following concepts if needed:
- Maximum and Minimum Problems for AP Calculus
- Inverted Cone (Water Tank) Problem for AP Calculus
- Shadow Problem for AP Calculus
- Angle of Elevation Problem for AP Calculus
- Distance Problem for AP Calculus
- Area and Volume Problem for AP Calculus
- Profit, Revenue and Cost Problems for AP Calculus
Part A The use of a calculator is not allowed.
- A spherical balloon is being inflated. Find the volume of the balloon at the instant when the rate of increase of the surface area is eight times the rate of increase of the radius of the sphere.
- A 13-foot ladder is leaning against a wall. If the top of the ladder is sliding down the wall at 2 ft/sec, how fast is the bottom of the ladder moving away from the wall when the top of the ladder is 5 feet from the ground? (See Figure 8.4-1.)
- Air is being pumped into a spherical balloon at the rate of 100 cm3/sec. How fast is the diameter increasing when the radius is 5 cm?
- A woman 5 feet tall is walking away from a streetlight hung 20 feet from the ground at the rate of 6 ft/sec. How fast is her shadow lengthening?
- A water tank in the shape of an inverted cone has an altitude of 18 feet and a base radius of 12 feet. If the tank is full and the water is drained at the rate of 4 ft3/min, how fast is the water level dropping when the water level is 6 feet high?
- Two cars leave an intersection at the same time. The first car is going due east at the rate of 40 mph and the second is going due south at the rate of 30 mph. How fast is the distance between the two cars increasing when the first car is 120 miles from the intersection?
- If the perimeter of an isosceles triangle is 18 cm, find the maximum area of the triangle.
- Find a number in the interval (0, 2) such that the sum of the number and its reciprocal is the absolute minimum.
- An open box is to be made using a piece of cardboard 8 cm by 15 cm by cutting a square from each corner and folding the sides up. Find the length of a side of the square being cut so that the box will have a maximum volume.
- What is the shortest distance between the point and the parabola y = –x2?
- If the cost function is C(x) = 3x2 + 5x +12, find the value of x such that the average cost is a minimum.
- A man with 200 meters of fence plans to enclose a rectangular piece of land using a river on one side and a fence on the other three sides. Find the maximum area that the man can obtain.
- A trough is 10 meters long and 4 meters wide. (See Figure 8.4-2.) The two sides of the trough are equilateral triangles. Water is pumped into the trough at 1 m3/min. How fast is the water level rising when the water is 2 meters high?
- A rocket is sent vertically up in the air with the position function s =100t2 where s is measured in meters and t in seconds. A camera 3000 m away is recording the rocket. Find the rate of change of the angle of elevation of the camera 5 sec after the rocket went up.
- A plane lifts off from a runway at an angle of 20°. If the speed of the plane is 300 mph, how fast is the plane gaining altitude?
- Two water containers are being used. (See Figure 8.4-3.)
- The wall of a building has a parallel fence that is 6 feet high and 8 feet from the wall. What is the length of the shortest ladder that passes over the fence and leans on the wall? (See Figure 8.4-4.)
- Given the cost function C(x) = 2500 + 0.02x + 0.004x2, find the product level such that the average cost per unit is a minimum.
- Find the maximum area of a rectangle inscribed in an ellipse whose equation is 4x2 + 25y2 = 100.
- A right triangle is in the first quadrant with a vertex at the origin and the other two vertices on the x- and y-axes. If the hypotenuse passes through the point (0.5, 4), find the vertices of the triangle so that the length of the hypotenuse is the shortest possible length.
- If y = sin2(cos(6x – 1)), find .
- The graph of f ' is shown in Figure 8.5-1. Find where the function f: (a) has its relative extrema or absolute extrema; (b) is increasing or decreasing; (c) has its point(s) of inflection; (d) is concave upward or downward; and (e) if f (3)= –2, draw a possible sketch of f. (See Figure 8.5-1.)
- (Calculator) At what value(s) of x does the tangent to the curve x2 + y2 = 36 have a slope of –1.
- (Calculator) Find the shortest distance between the point (1, 0) and the curve y = x3.
Part B Calculators are allowed.
One container is in the form of an inverted right circular cone with a height of 10 feet and a radius at the base of 4 feet. The other container is a right circular cylinder with a radius of 6 feet and a height of 8 feet. If water is being drained from the conical container into the cylindrical container at the rate of 15 ft3/min, how fast is the water level falling in the conical tank when the water level in the conical tank is 5 feet high? How fast is the water level rising in the cylindrical container?
(Calculator) indicates that calculators are permitted.
Solutions for these practice problems can be found at: Solutions to Applications of Derivatives Practice Problems for AP Calculus
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