e) Find the average revenue when 100 units are made and sold.

Transcription

1 Math 142 Week in Review Set of Problems Week 7 1) Find the derivative, y ', if a) y=x 5 x 3/2 e 4 b) y= 1 5 x 4 c) y=7x x 2 d) y=x x 10 x e) y= x7 5x 5 2 x 4 2) The price-demand function and the cost function for the production of DVD units is x = p and C(x) = 60, x. a) Find the average cost of making 100 units. b) Find the marginal cost of making 100 units. c) Find the marginal average cost when x = 100. d) Find the revenue when 100 units are made and sold. e) Find the average revenue when 100 units are made and sold. f) Find the revenue of making and selling 25units. g) Find the approximate revenue from the 25 th unit. h) Find the marginal average revenue function. i) Find the marginal average cost function.

2 3) Find the derivative of y= x 5 x 3/2 e 4 x 2 4)Find y when y=5x 2log 5 x 5). Find y ' when y=ln xe x 6) Find the equation of the tangent to the curve y=e x 2 at x = 0. 7) h x = 2x3 1 x 2 3, find h (x).. 8) Given f w = 5 3 w w 2 9) Find y when y= log x4 x 3 x 2, find f '(w).. Do not simplify. 10) Find the derivative of f x =8 x x 3 x 5 x 3. Do not simplify. 11) For f x = x 2 1 5, find f (x). 12) Find the values of x where the tangent line is horizontal for f x = x3 4 x 5. 13) For f x =e 5x4 2x 3 x, find f (x). 14) For f x = x 4 1 e x2 1, find f (x). 15) The cost function for Soccer League is modeled by C(x) = ln(3x 2 + 4x + 10). Find C (20) and interpret. x represents the number of players in the league, and C(x) represents the cost in hundreds of dollars. 16. Given R(x) = 4x 2 0.5x + 8, find the marginal average revenue. 17. Given C(x) = 250 5x, find the marginal average cost for x = 8.

3 18. Ecology Inc., creates sidewalk lanterns which run on solar energy. They have determined a price demand function for their product of x = p. They have fixed costs of $504, and variable costs of $12 per item. a) Find the domain of the price-demand function, p(x). b) Find the cost function, C(x). c) Find R(x), the revenue function in terms of the quantity produced and find its domain. d) Find the quantity they should produce and sell to maximize profit. Find the derivatives of each of the following functions. 19) f x =2x x log 4 x 20) f x =ln x e x 5x 21) f x =e x x ) f x = ex x ) f x =ln ln x ) f x =log x 2 Find the second derivative of each of the following: 25) f x =x 2 e x 26) f x =ln x 2 1

4 The cost function for Car Rentals is given by C(x) = x 2, where x is the number of units rented in hundreds, and C(x) is measured in dollars. 27) What is the average change in cost when the rentals increase from 200 to 350? 28) Find the marginal average cost for 1, 000 rentals. 29) Approximate the revenue from the sale of the 5 th item when the price-demand function is given by 5x + 2p = ) Find the derivative of f x =ln[ x x 10]. Do Not Simplify 31) Find the derivative of f x = e2x e 2x e 2x e 2x 32) Find y ' if y = 8 u and u = ln x at x = e 4. 33) a. Given the price p of an item is given by p= 1 5 x 20, find the elasticity of demand at p = $4. b. Should the price be raised, lowered, or stay the same? c. If the price changes by 50%, how will the demand change? 34) Find the interval(s) where f(x) is increasing when f x =e x x 2 6x 1 ln x 1 35) Find the equation of the tangent to y= e x 1 at x = 0.

5 36. Given the graph below of f (x), find the value of x where f(x) has a local minimum. 37) S ketch the graph of a function f that satisfies the following: Domain: (, 2) (2, ) Vertical asymptotes: x = 2 Horizontal asymptote: y = 2 x-intercepts: (4, 0) y-intercept: (0, 3) 38) Find the absolute extrema (locations and values) of f(x) = x 2 e 0.1x on [ 5, 30] 39)Find all local extrema of each of the given functions on its domain. Use the Second Derivative Test when it applies. a. f(x) = x 4 4x 3 80x b. g(x) = 0.2x + ln (5x 20) c. h(x) = 2x 5 15x 4 90x

6 40) Find the global extrema for the following functions on the given intervals. a. f x =20 4x 250 x 2 on the interval (0, ) b. f x = x4 e x on the interval (0, ) c. f x =4x ln x 7x on the interval (0, ) 41) Find all the asymptotes for each of the following functions a. f x = 2x2 5x 10 x 1 b. f x = 7x2 6x 10 2x 2 1 c. f x =5x 2 7x x 2 20 d. f x = 4x3 8x 2 10x 20 3x 2 7x 1 42) A 300-room hotel in Salt Lake City is filled to capacity every night at $80 a room. For each $1 increase in rent, 3 fewer rooms are rented. If each rented room cost $10 to service per day, how much should the management charge for each room to maximize gross profit? What is the maximum gross profit?

7 43) A fence is to be built to enclose a rectangular area of 800 square feet. The fence along three sides is to be made of material that costs $6 per foot. The material for the fourth side costs $ 18 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be built. 44) A pharmacy has a uniform annual demand for 200 bottles of a certain antibiotic. It costs $10 to store on bottle for one year and $40 to place an order. How many times during the year should the pharmacy order the antibiotic in order to minimize the total storage and reorder costs?