Math 1325 Final Exam Review. (Set it up, but do not simplify) lim

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1 . Given f( ), find Math 5 Final Eam Review f h f. h0 h a. If f ( ) 5 (Set it up, but do not simplify) If c. If f ( ) 5 f (Simplify) ( ) 7 f (Set it up, but do not simplify) ( ) 7 (Simplify) d. If f. Given f( ), represent f (), using a. f ( ) f ( ) 5 h0. Find the it for each of the following. a. 6 c. d. e. 8 f. 8 g. h. 9 i. h f. Find all values where the given functions are discontinuous and specify why they aren't continuous. a. k h. /7/06

2 if f 0 if if M 5 Final Eam Review page c. f if if 5. Find all values of k such that f is continuous. a. f f k if 5 k if 5 k if k if 6. Find the instantaneous rate of change for each function at the given value. f 7 at a. g 7 at 7. Find the slope of the line tangent to the graph at the given value. f at a. 9 f at 8. Find the derivative of the following. 5 a. pt t 6 t t y 5 7 c. f 7 d. y e. y 7 f. m t 6t5 t g. y ln h. rt t 8 5 t 5 Mar 06

3 i. y 8 j. y( )e k. y 5 7e l. m. y e y 5e ln( ) n. m t 5t t t o. y ln p. yln 9 q. y ln 5 7 M 5 Final Eam Review page 9. Given the function horizontal. f 5 8, find all values of where the tangent line is 0. For each of the following functions, find all critical values, state the interval(s) on which the function is increasing, state the interval(s) on which the function is decreasing and state all relative etrema. f a. f. A function f( ) has derivatives with the characteristics indicated in the table below. interval, 5 5,,,9 9, first derivative of f f( ) 0 f( ) 0 f( ) 0 f( ) 0 f( ) 0 second derivative of f f( ) 0 f( ) 0 f( ) 0 f( ) 0 f( ) 0 a. Over what intervals is the graph of f( ) increasing? Over what intervals is the graph of f( ) concave up? c. If f( ) is continuous on (, ), where does f( ) have a minimum value? (give the -value(s)) d. If f( ) is continuous on (, ), where does the graph of f( ) have an inflection point? (give the -value(s)). Given the derivatives f( ) and f ( ), find all values where f( ) has a relative minimum, and all values where f( ) has a relative maimum. a. and f ( ) 5 f ( ) 5 Mar 06

4 and ( ) f e 5 e f e e e ( ) 0 M 5 Final Eam Review page c. 8 f( ) and f( ). Find these from the given piecewise function graph. a. f 5 f c. f d. f e. f 6 f. Identify all discontinuities and eplain why the function is discontinuous at each point. g. Find all values of for which the function is not differentiable.. A demand equation, where p is the price in dollars and is the number of items demanded, is given by 6p p 9. Find the rate of change in the price with respect to the quantity, dp d. 5. Given that the function graphed below is f ( ), determine where f( ) is concave downward. 6. Given that the function graphed below is f, find the interval(s) on which f increasing and the interval(s) on which f is decreasing. is Mar 06

5 M 5 Final Eam Review page 5 R( ) The revenue, in hundreds of dollars, from selling tickets is given by Find the maimum revenue, and the number of tickets that must be sold to obtain that revenue. 8. A manufacturer of dress racks has determined that the cost in dollars to produce units per month is given by C0. 0, 000. Find the rate at which the cost per month is changing when production is increasing at the rate of units per month and the monthly production level is 80 units. 9. Given the following equations, find dy d by implicit differentiation. a. 9 y 7y y 0. Given the following equations and information, evaluate dy dt. a. given that d dt, 5 y 7y 7 y y 7 given that,, and y, y and d dt. Find the following. 5 8 a. 5 d 6 5 c. d. e d e 9 t t d dt t9 Mar 06

6 e. 5 5 d f. 9 8 d g. e d h. 9ln d i. ( )e 5 d. Evaluate each definite integral. M 5 Final Eam Review page 6 a a. 0 e d 9 d c. a d d. 5 e. 0 d ( e ) d 0,.. Find the area between f and g on. Find the area between the curves f and g. 5. Find the area completely enclosed by the graphs of f and g 6. Find f, y, f, y, f, y, f, y, f, y f, y y y a. y f, y e 8 y 7. A revenue equation is given by yy y for the following:. R 60, where is the number of units sold and revenue is in dollars. a. Find the marginal revenue when units are produced. Find the additional revenue brought in from the sale of the th unit. 8. A cost function is given by C 0 00, where is the number of units produced and cost is in dollars. a. Find the average cost per unit when 0 units are produced. Find an epression for marginal average cost. Mar 06

7 9. A profit function is given by M 5 Final Eam Review page 7 P , where is the number of units produced and sold, and profit is in dollars. Find the maimum profit than can be earned, and the number of units that must be produced and sold to earn that profit. 0. A company determines that the marginal cost for producing the th item can be determined by the function C. Find the cost function for producing items if the cost of producing 8 items is $50.. A company determines that the marginal cost for producing the th item can be determined by the function C 0.5e Find the cost function for producing items if fied costs are $00.. The percent of population living below the poverty level can be modeled with the function t P t e, where t is the number of years since 960. Find the rate of change in the percent of the population living below the poverty level in 965, the year after LBJ instituted the war on poverty.. The value, V, of the eports of U.S. goods, t years after 009, can be approimated by t V t.6e, 0.06 where V is given billions of dollars. Find the rate of change in the value of the eports in 0. Mar 06