Discussion Questions and Problems

Discussion Questions

M6-1 Explain how to find the slope of a straight line.
M6-2 Explain how to find the slope of a nonlinear function at a particular point, using limits.
M6-3 What is the procedure for finding the maximum or minimum of a function? How is the second derivative used in this process?
M6-4 What are critical points?

Problems

M6-5 Find the derivatives of the following functions.
1. $Y = 2 X^{3} - 3 X^{2} + 6$ $Y = 2 X^{3} - 3 X^{2} + 6$
2. $Y = 4 X^{5} + 2 X^{3} - 12 X + 1$ $Y = 4 X^{5} + 2 X^{3} - 12 X + 1$
3. $Y = 1 / X^{2}$ $Y = 1 / X^{2}$
4. $Y = 25 / X^{4}$ $Y = 25 / X^{4}$
M6-6 Find the second derivative for each of the functions in Problem M6-5.
M6-7 Find the derivatives of the following functions.
1. $Y = X^{6} - 0.5 X^{2} - 16$ $Y = X^{6} - 0.5 X^{2} - 16$
2. $Y = 5 X^{4} + 12 X^{2} + 10 X + 21$ $Y = 5 X^{4} + 12 X^{2} + 10 X + 21$
3. $Y = 2 / X^{3}$ $Y = 2 / X^{3}$
4. $Y = 25 / 2 X^{4}$ $Y = 25 / 2 X^{4}$
M6-8 Find the second derivative for each of the functions in Problem M6-7.
M6-9 Find the critical point for the function $Y = 6 X^{2} - 5 X + 4$ $Y = 6 X^{2} - 5 X + 4$ Is this a maximum, minimum, or point of inflection?
M6-10 Find the critical points for the function $Y = (\frac{1}{3}) X^{3} - 5 X^{2} + 25 X + 12.$ $Y = (\frac{1}{3}) X^{3} - 5 X^{2} + 25 X + 12.$ Identify each critical point as a maximum or a minimum or a point of inflection.
M6-11 Find the critical point for the function $Y = X^{3} - 20.$ $Y = X^{3} - 20.$ Is this a maximum, minimum, or point of inflection?
M6-12 The total revenue function for a particular product is $T R = 1, 200 Q - 0.25 Q^{2} .$ $T R = 1, 200 Q - 0.25 Q^{2} .$ Find the quantity that will maximize total revenue. What is the total revenue for this quantity?
M6-13 The daily demand function for the AutoBright car washing service has been estimated to be $Q = 75 - 2 P,$ $Q = 75 - 2 P,$ where Q is the quantity of cars and P is the price. Determine what price would maximize total revenue. How many cars would be washed at this price?
M6-14 The total revenue function for AutoBright car washing service (see Problem M6-13) is believed to be incorrect due to a changing demand pattern. Based on new information, the demand function is now estimated to be $Q = 180 - 2 P^{2} .$ $Q = 180 - 2 P^{2} .$ Find the price that would maximize total revenue.
M6-15 The total cost function for the EOQ model is

$T C = \frac{D}{Q} C_{o} + \frac{Q}{2} C_{h} + D C$ $T C = \frac{D}{Q} C_{o} + \frac{Q}{2} C_{h} + D C$

In a particular inventory situation, annual demand (D) is 2,000, ordering cost per order $(C_{o})$ $(C_{o})$ is $25, holding cost per unit per year $(C_{h})$ $(C_{h})$ is $10, and purchase (material) cost per unit (C) is $40. Write the total cost function with these values. Take the derivative and find the quantity that minimizes cost.
M6-16 Find the second derivative of the total cost function in Problem M6-15, and verify that the value at the critical point is a minimum.

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