Mat 210 Business Calculus Final Exam Review Spring Final on April 28 in COOR HALL 199 at 7:30 AM

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1 f ( Mat Business Calculus Final Eam Review Spring Final on April 8 in COOR HALL 99 at 7: AM. A: Find the limit (if it eists) as indicated. Justify your answer. 8 a) lim (Ans: 6) b) lim (Ans: -) c) lim (Ans: DNE) d) lim (ans:/) e) lim (Ans: DNE) 9 f) lim (Ans: ) g) lim (Ans: 6) h) lim (Ans ½) I) 7 lim (Ans: ) j) lim (Ans: ) k) lim (Ans: ) e. B: Prove the Limits lim 6. 9 lim =6 9. lim =. 6. lim e lim lim lim 9 =6 9 =DNE. i) Let f ( ) if if a. Find lim f ( ). If the limit does not eist clearly eplain why. Ans: - b. Find lim f ( ). If the limit does not eist clearly eplain why. Ans: DNE, because left hand limit and right limits are different. c. Use the definition of continuity to determine if f () is continuous at. Ans: continuous d. Use the definition of continuity to determine if () Ans: no f ) is continuous at.

2 if if ii) f( ) if if a) Find lim f( ), lim f( ), lim f ( ) eist clearly eplain why. b) Find f ( ), f, f, f, f 6 DNE. If the limit does not c) Use the definition of continuity to determine if f () is continuous at,,. Ans: at continuous; at discontinuous. At discontinuous. Consider the function f ( ). a. Find the average rate of change of f () between the values and. Ans: 7 b. Using the limit definition, find the instantaneous rate of change of f () at. Ans: c. Using your answer to part (b), find the equation of the tangent line at the point (,7). Ans: y 7 ( ). Find the derivatives of the following functions. 6 7 a) f ( ) b) g( ) 6 Ans : Ans : c) h( ) d. y, y Ans : ( ) e. y ( )( ) ( )( ) ( )( ) f y ) ( ) g) y ( ) y, y ( )

3 b) h) f ( ) 6 e f ( ) 6 e i) g( ) 7 ln j) h( ) ln( ) g ( ) 7 9 h ( ) k) g( ) 8 7 ln g ( ) /. a) Let f ( ) 7, find derivative of f ( 9) b) Let f ( ). Does f() has inverse function. For what interval it has inverse function? Find the derivative of f () /. Answer: (, ), f () / c) 6. a) Use idea of linear approimation to approimate 6 6. b) Prove that for close to, and illustrate this approimation by drawing the graphs of y and y on the same screen. We consider f ( ) f '( ) f '(), as a Now using linear approimation formula we find f ( ) f ( a) f '( a)( a) f ( ) f () f '()( ) c) Prove that ( ) m m for close to and use this approimation to find approimation of / ( 6.9) d) Use the linear approimation of f ( ) to approimate the value of / ( 6.9). Solution. From f ( ) we have f '( ) /

4 7. Derivative Practice Problems: Answers. y =. y =,,. y = +. y = y = + 6. y = y = / 6 / 6 8. y = 6/ + / / / 9. y = sqrt() /( ). y =. y = e..e.. y = e 8e. y = e e. y = e + 9 e. y = ln() / 6. y = ln(/) -/ 7. y = ln( ) / 8. y = ( + )(e + ) ( )e ( e ) 9. y = ( + )(ln ) ln (6 ). y = ( ) ( 6 ) ( 6). y = ( - 6) 9 9( 6) ( 6). y = ( - + ) ( )( ). y = e + e. y = ln ( + + ) 8. Integration Practice Problems Answer:. 99 d. 98 c d. d. c. ln c

5 t. dt t. ( e ln ) d. t t c. e ln c 9. Integration by substitution Answer. ( )( ) d 6. 6 ( ) c;. ( )( 7) d. 8 ( 7) c;. 6 d /. ( 6) c;. ( ) e e d ( ) ln d e e d d. (e ) c;. c; ( ) 6. (ln ) c; 7. e c.. Logarithmic Integral: Answer:.... d d ln e d e d 6. ln( ) c;.ln(ln ) c;. ln(e ) c;. ln( 6) c

6 . The following all involve the "e" rule.. e d. e c;..e. d.... e c;. e d. e c;. e d. e c.. Integration by parts Answer.. ln d. ln d. ( ). e d d e d. e e c. ln c. ln c;. ( ) ( ) c. 6. e e e c. Integration by substitution: Answer:. ( )( )... d 8 e ( ) d e ln( ) d. ( ) 6. d 6 e e d e d. 6 ( ) c; /. ( 8) c;.ln( e ) c;.(ln( )) c; 7. (e ) c; 7. ln d / 6. 9 ( e) c; / 7. (ln ) c.

7 . Definite Integral: Answer. ( 6 ) d =. e d= ( e e ). e d e =. ln d =. Area: Find the area of the following:. Bounded by and y ais and y Answer: /. Enclosed by y and y on [-,] Answer: 9/ Solution : ( ( )) d ( ( )) d () () () 9. Application:. The daily profits in dollars of a firm is given by P ( ) ( 8 ), where is the number of items sold. Find instantaneous rate of change where =. Interpret your answer. P ( ) 8 P () () 8 instantaneous rate of change= where =. That means marginal profit is. Algebraically find the following for the function f ( ) a. All critical values f ( ) 6 ( ) or So, critical values are -,, b. the intervals where the function is increasing Critical values -

8 Test values - - Sign of f () f () (, ) (,) (, ) So, function increases on (, ) (, ) c. the interval where the function is decreasing Decreases at (,) d. the interval where the function is concave up Inflection values -.. Test values - - Sign of f () f () (,.) (., ) (,.) (., ) concave up (.,) and (., ) e. the interval where the function is concave down concave down (,.) and (,.) f. all the inflection points f ( ) 6 6( 6 ) or.. Elasticity: a) Consider the demand function p. d Then p dp Find the value of for which the elasticity E =. p d E dp By plug in value of p, E= and derivative we get

9 ( ) b) The weekly sales of Honolulu Red Oranges is given by q = 96 8p. Calculate the price elasticity of demand when the price is $ per orange. Also, calculate the price that gives a maimum weekly revenue. Answer.6, $6 c) Suppose the likelihood that a child will attend a live musical performance can be modeled by q =.( ). ( ) Here, q is the fraction of children with annual household income thousand dollars who will attend a live musical performance during the year. Compute the income elasticity of demand E at an income level of $,. (Round your answer to two decimal places.) Answer:.8 d) The relation between traffic volume and ependiture of building roads are given.6 as T ( K).K, K is ependiture. Find the elasticity of T with respect to K. Find also the consequences if ependiture increases by %. K K.6 Solution: E T ( )..6K.6.6 T( K).K The increase in % of ependiture would lead to.6% increase in traffic volume. The % increase in price would lead to a.6% increase in traffic volume.. Related Rates:. A car is traveling at mph due south at a point / mile north of an intersection. A police car is traveling at mph due west at a point / mile east of the same intersection. At that instant, the radar in the police car measurers the rate at which the distance between the cars is changing. What does the radar gun register?. A -foot ladder leans against the side of a building. If the top of the ladder begins to slide down the wall at the rate of ft/s, how fast is the bottom of the ladder sliding away from the wall when the top of the ladder is 8 ft off the ground?. The radius of a circular puddle is growing at a rate of cm/s. How fast is its area growing at the instant when the radius is cm? (Round your answer to the nearest integer.). Answer: 7 sq. cm/s

10 How fast is the area growing at the instant when it equals cm? (Round your answer to the nearest integer.) Answer: sq. cm/s. A rather flimsy spherical balloon is designed to pop at the instant its radius has reached 6 centimeters. Assuming the balloon is filled with helium at a rate of 7 cubic centimeters per second, calculate how fast the radius is growing at the instant it pops. Round your answer to two decimal places.) Answer:. cm/s. A circular conical vessel is being filled with ink at a rate of cm /s. How fast is the level rising after cm have been poured in? The cone has a height of 6 cm and a radius cm at its brim. (The volume of a cone of height h and crosssectional radius r at its brim is given by V = / πr h) Answer:.77 cm/s. Consumers and producers surplus: a) The demand and supply curves are D ( q) q and S ( q). Find q 6 consumer s surplus and producer s surplus. Solution. For equilibrium D ( q) S( q) q q q and q 6 p D q ) S( ) ( q Now PS = And CS = q pq S( q) dq ( q ) dq 6 q D( q) dq p q q 6 dq ln