4. Applications of Derivatives

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?

Resistors connected in parallel If two resistors of R₁ and R₂ ohms are connected in parallel in an electric circuit to make an R-ohm resistor, the value of R can be found from the equation

1/R = 1/R₁ + 1/R₂

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If R₁ is decreasing at the rate of 1ohm/sec and R₂ is increasing at the rate of 0.5 ohm/sec, at what rate is R changing when R₁ = 75 ohms and R₂ = 50 ohms?

Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft³/min.

a. What is the relation between the variables h and r in the figure?

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Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.

b. How many revolutions per minute is 0.6 rad/sec?

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Right circular cylinder 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.

b. How is dS/dt related to dh/dt if r is constant?

Right circular cylinder 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.

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

Right circular cylinder 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.

d. How is dr/dt related to dh/dt if S 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.

b. How is dS/dt related to dh/dt if r is constant?

Economics

Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².

a. Find the average cost per machine of producing the first 100 washing machines.