4. Applications of Derivatives

Consider the following cost functions.

a. Find the average cost and marginal cost functions.

C(x) = 1000+0.1x, 0≤x≤5000, a=2000

Consider the following cost functions.

c. Interpret the values obtained in part (b).

C(x) = 1000+0.1x, 0≤x≤5000, a=2000

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = 500+0.02x, 0≤x≤2000, a=1000

The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground. Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].

Consider the following cost functions.

c. Interpret the values obtained in part (b).

C(x) = 500+0.02x, 0≤x≤2000, a=1000

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

Complete the following statement. If dy/dx is small, then small changes in x will result in relatively ______ changes in the value of y.

Suppose w(t) is the weight (in pounds) of a golden retriever puppy t weeks after it is born. Interpret the meaning of w'(15) = 1.75.

Define the acceleration of an object moving in a straight line.

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 

a. Determine the average and marginal costs for x = 3000 lawn mowers.

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 

b. Interpret the meaning of your results in part (a).

Explain Rolle’s Theorem with a sketch.

The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.

a. How is dS/dt related to dr/dt if h is constant?

The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation S = πr√(r² + h²). 

a. How is dS/dt related to dr/dt if h is constant?

Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation

______

S = πr √ r² + h².

c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?