Exam 2 Review 1190

Exam 2 Review -1190 1. Find y ⅆ x ⅆ for y = 5 x 3 ⅇ x A. y ⅆ x ⅆ = 15 x 2 ⅇ x B. y ⅆ x ⅆ = 15 x 2 ⅇ x + 5 x 3 ⅇ x C. y ⅆ x ⅆ = 15 x 2 e D. y ⅆ x ⅆ = 5 x 3 ⅇ x E. y ⅆ x ⅆ = 15 x 2 ⅇ x − 5 x 3 ⅇ x 2. Find ⅆ x ⅆ ln ( x 2 + 6 x ) A. ln ( 2 x + 6 ) B. x 2 + 6 2 x + 6 C. 1 2 x + 6 D. 2 x + 6 x 2 + 6 E. ( 2 x + 6 ) ln ¿ ) 3. If D ( x )= g ( f ( x )) , x f ( x ) f ' ( x ) g ( x ) g ' ( x ) 3 5 -2 -4 9 1 -4 10 -10 12 -4 1 0 3 -13 Find D ' ( 1 ) .

A. -148 B. -13 C. -130 D. -100 E. 3 4. Suppose f ' ( x ) = 3 x 2 ( x + 2 ) ( x − 7 ) 2 , determine the values of x where the tangent line to y = f ( x ) is horizontal. A. x = 0 and x =− 2 only B. x = 7 only C. x =− 2 and x = 7 only D. x = 2 and x =− 7 only E. x = 0 and x = 7 only 5. Find the x-values of the point of inflection of f ( x ) = x 3 − 12 x 2 Hint: (Point of inflection is where function changes from concave up to concave down or vice versa) A. x = 0,8 B. x = 0 , − 8 C. x = 4 D. x = 0,12 E. None of the above 6. Find the Relative rate of change for x = 1 , if f ( x ) = 10 − 3 ⅇ x Hint: R.R.O.C= f ' ( x ) f ( x )

9. For the demand function x=D(p)=100 ⅇ − 0.25 p Find the value of p that will maximize the revenue. (Hint: Revenue is maximum when E ( p )= − p D ' ( p ) D ( P ) = 1 ¿ 10. Given the graph of f(x) below,what is the open intervals over which f’ (x) is negative

11. Suppose a company is able to determine functions that predict its revenue and costs as R(x) = x(30 - x) and C(x) = 6x + 44. A) Find the marginal profit function. B) Find the range of x values for which the profit is positive 12. Find the absolute extrema of g(x) = 5 ln(x) - x over the interval (0,  ). 13. Owners of a bike rental company that charges customers between $5 and $50 per day have determined that the number of bikes rented per day n can be modeled by the linear function n(p) = 300 - 6p, where p is the daily rental charge. How much should the company charge each customer per day to maximize revenue?

17. Using the first derivative test, what are the local extrema for the function s ( x ) = − 2 3 x 3 + 10 x 2 − 48 x − 1 18. A company projects their revenue and cost functions will be R ( x ) = 3 x 2 + 10 x and C ( x ) = 1 3 x 3 − 4 x 2 − 5 x where xis the number of units manufactured (in thousands) and R(x) and C(x) are in thousands of dollars. How many units should be manufactured to maximize profit. 19. A gas station company finds that with annual upkeep and certification, the maintenance cost(in hundreds of dollars) after x years for each fuel pump is C ( x )= 8 − 1 . 2 x + 0 . 02 x 2 . After how many years is the average cost of the fuel pumps minimized?

20. Recall the price elasticity of demand for a price function x = D(p), where x is the quantity and p is the price, is given by the following: E ( p ) =− pD ' ( p )/ D ( p ) a) Compute the elasticity of demand for x=D(p) =60000-100p =200 b) Explain using complete sentences what this means and what should be done to maximize revenue c) Find the price p that should be charged to maximize revenue.