91Ó°ÊÓ

If \(f(x)\) is a linear function, \(f(1)=0,\) and \(f(2)=1,\) what is \(f(3) ?\)

Write the equation of the tangent line to the curve \(y=x^{3}+3 x-8\) at \((2,6).\)

Find the indicated derivative. $$\frac{d}{d x}\left(x^{-3}\right)$$

In industry, the relationship between wages and the quit ratio of employees is defined to be the percentage of employees that quit within 1 year of employment. The quit ratio of a large restaurant chain that paid its employees the minimum hourly wage (\$7.25 per hour) was .2 or 20 employees per 100. When the company raised the hourly wage to \(\$ 8,\) the quit ratio dropped to \(.18,\) or 18 employees per 100 (a) Assuming a linear relationship between the quit ratio \(Q(x)\) and the hourly wage \(x,\) find an expression for \(Q(x)\) (b) What should the hourly wage be for the quit ratio to drop to 10 employees per \(100 ?\)

A company manufactures and sells fishing rods. The company has a fixed cost of \(\$ 1500\) per day and a total cost of \(\$ 2200\) per day when the production is set at 100 rods per day. Assume that the total cost \(C(x)\) is linearly related to the daily production level \(x\). (a) Express the total cost as a function of the daily production level. (b) What is the marginal cost at production level \(x=100 ?\) (c) What is the additional cost of raising the daily production level from 100 to 101 rods? Answer this question in two different ways: ( 1 ) by using the marginal cost and ( 2 ) by computing \(C(101)-C(100).\)

A salesperson's weekly pay depends on the volume of sales. If she sells \(x\) units of goods, her pay is \(y=5 x+60\) dollars. Give an interpretation of the slope and the \(y\) -intercept of this straight line.

A T-shirt shop owner has a fixed cost of \(\$ 230\) and a marginal cost of \(\$ 7\) per T-shirt to manufacture \(x\) T-shirts per day. Let \(C(x)\) denote the cost to manufacture \(x\) T-shirts per day. (a) Find \(C(x).\) (b) If the shop owner decides to sell the T-shirts at \(\$ 12\) each, find \(R(x),\) the total revenue from selling \(x\) T-shirts per day.

Apply the three-step method to compute the derivative of the given function. $$f(x)=3 x^{2}-2$$

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. $$f^{\prime}(0), \text { where } f(x)=10^{1+x}$$