Math 116 Review A ball is thrown upward from the top of a 200-foot cliff. The initial velocity of the ball is 125 feet per

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1 Math 6 Review You may only use a calculator if the problem is labeled calc.. Find the equation of the tangent line that is tangent to the graph of f and parallel to the given line. Page of 5 f x x, line is x y 5. A ball is thrown upward from the top of a 00-foot cliff. The initial velocity of the ball is 5 feet per second, which implies that the position function is s t 6t 5t 00, where the time t is measured in seconds. Find the height, the velocity, and the acceleration of the ball when t 3 seconds 3. Find the derivative of the given function. f x 3x x x. Find the equation of the tangent line to the graph of the function f at the given point. 3 ; at the point 0, f x x x 5. When the admission price for a baseball game was $6 per ticket, 36,000 tickets were sold. When the price was raised to $7, only 33,000 tickets were sold. Assume that the demand function is linear. The fixed costs for the park are $85,000 and the cost per ticket is $0.0. a. Find the profit P as a function of x, where x is the number of tickets sold. b. Find the marginal profit when 8,000 tickets are sold. 6. Find the equation of the tangent line to the graph of the function f at the given point. f x ; at the point,3 x 3 x

2 7. The value V of a machine t years after it is purchased is given by V t 0, 000 t. Find the rate of Page of 5 depreciation when t Find the second derivative of the given function. f x x x 9. The total cost of producing x units of a particular commodity Cx and the unit price of that commodity px are given by the functions C x x x 57 and px x a. Use marginal cost to estimate the cost of the st unit. 5 8 respectively. b. Use marginal revenue to estimate the revenue derived from the sale of the st unit. 0. When the price of a certain commodity is p dollars per unit, the manufacturer is willing to supply x hundred units, where 3p x. How fast is the supply changing with respect to time when the price is $ per unit and is increasing at the rate of 87 cents per month?. When the price of a certain commodity is p dollars per unit, consumers demand x hundred units of the commodity, where p 3px x 79. How fast is the demand changing with respect to time when the price is $5 per unit and is decreasing at the rate of 30 cents per month?

3 . A manufacturer s total cost is 3 Page 3 of 5 C q q 6q 00 dollars when q units are produced. The current 6 level of production is units. Estimate the amount by which the manufacturer should decrease production to reduce the total cost by $ Find the equation of the tangent line to the graph of the given equation at the given point. 3 x xy y, at 0,. The profit for a product is increasing at a rate of $5600 per week. The demand and the cost functions for the product are given by p x and C 00x 500. Find the rate of change of sales with respect to time when the weekly sales are x units. 5. A manufacturer s total cost is 3 C q 0.00q 0.05q 0q 000 dollars, where q is the number of units produced. a. Use marginal analysis to estimate the cost of producing the 5 st unit. b. Compute the actual cost of producing the 5 st unit. c. Sketch a quick graph of the cost and the marginal cost of the 5 st unit as it related to the problem. Use a window of [9,5]. Why do you think the marginal cost was so close to the actual cost of producing the 5 st unit? 6. The demand x for a web camera is 30,000 units per month when the price is $5 and 0,000 units when the price is $0. The initial investment cost is $75,000 and the cost per unit is $7. Assume that the demand is a linear function of the price. a. Find the profit P as a function of x. b. Approximate the change in profit for a one-unit increase in sales when x, For the function 3 f x x 6 x (a) Find the intervals of increase or decrease and the local extrema values. (b) Find the intervals of concavity and the inflection points. (c) Use part (a) and part (b) to sketch the graph of the function

4 Answers:. y x. a) s3 3feet b) v3 9 ft/s Page of 5 c) a3 3 ft/s 3. f ' x. y x x P x x 7.8x 85, a) b) ' 8,000 P $5.8 / tkt 6. y x 7 7. V ' 3 65 dollars/year 8. f " x 9. a) x 3 C x x Cost of producing the 5 ' b) 0..7 dt. dt. q ' st unit R x x Revenue derived from the sale of the C' 0 $ the supply is increasing at a rate of 7 units/month st unit R' 0 $ the demand is increasing at a rate of 7. 7 units/month 70 y x. dt production should be decreased by 0. units the rate of change of sales with respect to time is units/week

5 5. a) Cost of 5 st unit C' 50 $0.50 b) Cost of 5 st unitc C , , $03.0 Page 5 of 5 c) $ $6500 $0.50 unit 50 5 Answers may vary. 6. a) 7. (a) P x x 3x 75,000 b) dp $, profit increases by dollars, 3 is a local minimum (b) CU..:,,, C. D.:, IP:,, f 0,