Supplemental Review Problems for Unit Test : 1 Marginal Analysis (Sec 7) Be prepared to calculate total revenue given the price - demand function; to calculate total profit given total revenue and total cost; to calculate marginal average cost (or revenue or profit) by calculating average cost (or revenue or profit) FIRST, and then calculating the derivative of that function; Be able to interpret mathematical quantities such as: If C() is the total cost (in dollars) of producing items of a product, then C(1) - C(11) is the EXACT cost of producing the 1th item C (11) is the APPROXIMATE cost of producing the 1th item C (1) represents the instantaneous rate of change of total cost at a production level of 1 items C (1)=15 means: at a production level of 1 items, a unit increase in production will increase total cost by approimately $15; or equivalently, the marginal cost at a production level of 1 items is $15 per additional item produced; or equivalently, the approimate cost of producing the 1th item is $15 If P() represents the total profit (in dollars) from the production and sale of units of a product, then P(1) - P(11) is the EXACT profit from the production of the 1th item P (11) is the APPROXIMATE profit from the production of the 1th item P (1) represents the instantaneous rate of change of total profit at a production level of 1 items P (1)= - 15 means: at a production level of 1 items, a unit increase in production will decrease total profit by approimately $15; or equivalently, the marginal profit at a production level of 1 items is - 15 dollars per additional unit produced; or equivalently, the production of the 1th unit results in a DECREASE in total profit of approimately $15 [7a] If the cost function for producing items is given by C ( ) = 500 + 00 0 0, find the marginal cost function m [7b] For a particular product, the price-demand equation is p= 0 0 01, where p is the price and is the quantity, and the cost function is C ( )= 5000 + What is the revenue function r? [7c] For a particular product, the cost function is C ( )= 5000 + And the revenue function is r ( ) = 0 01 What is the profit function w? 000 [7d] If the profit function for producing items is given by P ( ) = 01 + 10, find the marginal profit function y [7e] If the cost function is given by C ( )= 5000 + 00, find the average cost function 0 a 1 [7g] If the total revenue function for a commodity is r ( )= 0, find the marginal 10 revenue at = 0 [7h] If the profit function is given by P ( )= 0 00, find the marginal average profit function n [7i] The price p (in dollars) and the demand for a particular product are related by the equation = 1 - p Find the revenue function r
[7k] Given the revenue function R(), using marginal analysis, what is the mathematical interpretation of R (0) = - $0? [7k] Given the profit function P(), what is the mathematical interpretation of P(1) - P(0) = $ 150? [7l] Given the cost function C(), using marginal analysis, what mathematical epression is used to find the approimate cost from producing the 17th unit? Continuity and Graphs (sec 41) - Be able to determine where a polynomial, rational or radical function is continuous; Know how to recognize infinite limits, that is, limits where substitution of the target number for yields a non-zero number divided by 0; [41a] Where is f() = 5 - continuous? [41b] Where is f ( )= 4 4 + continuous? [41c] Find the vertical asymptotes of f( )= + 4 + 5+ 4 [41d] Solve ( 5 - )( + 1)( - ) < 0 [41e] Solve ( + 1 )( ) < 0 ( + ) [41f] Where is f( )= 5 continuous? [41g] Where is f( ) continuous? 4 [41h] Where is f( )= 5 continuous?, [41j] Where is f() continuous if f( ) = 5,, [41m] lim = + > 5 = 5 < 5 Derivatives and graphs (Sec 4,4, and 44) Be able to use f to determine intervals on which f is increasing or decreasing and the graph of f is rising or falling; Be able to us f to determine intervals on which the graph of f is concave up or concave down; Be able to construct sign charts for f, f and f ; Be able to find critical values ( in domain f where f is zero or not defined); Be able to use the first derivative test and the second derivative test to find local etrema; Be able to find inflection points; Be able to calculate limits at infinity; Be able to find horizontal and vertical asymptotes; Be able to use information about f, f and limit information to sketch or recognize graphs of functions [4i] Find all intervals where f() = + - 6 is increasing [4j] Find all intervals where f() = + - 6 is decreasing [4k] Given f() = - - +, select the choice that gives the correct sequence of true (T) and false (F), respectively, for I, II I f() has a local minimum at = 1 II f() has a local maimum at = - [4l] Find all intervals where f() = + - 1 is increasing [4m] Find all intervals where f() = + - 1 is decreasing [4n] Given f() = + - 1, select the choice that gives the correct sequence of true (T) and false (F), respectively, for I, II I f() has a local minimum at = 1 II f() has a local maimum at = -
[4q] Given the graph of f (), select a formula for f() a) f() = - + + b) f() = - + + c) f() = - d) f() = + [4i] Given the graph below, select the correct statement a) f () > 0 and f () > 0 on (-, ) b) f () > 0 and f () < 0 on (-, ) c) f () < 0 and f () > 0 on (-, ) d) f () < 0 and f () > 0 on (-, ) 9 [4j] Find f () if f( ) = ( + 4 ) [4k] Find where all local maima occur for f( )= + 7 [4l] Find where all local minima occur for f( )= + 7 [4m] Find where all local maima occur for f( )= + 4 [4n] Find where all local minima occur for f( )= + 4 [4o] Find all intervals where the graph of f( )= 5 + is concave upward [4p] Find all intervals where the graph of f( )= 5 + is concave downward [4q] Find where all inflection points of f( )= 5 + occur [4r] Choose the correct statement for f() if f() is continuous on (-, ), and: f () = 0 and f () = 0; f () = 0 and f () < 0; f () = 0 and f () = ; f () = and f () = a) f() is a local minimum b) f() is a local maimum c) f() is neither a local maimum nor a local minimum d) There is insufficient information to determine whether f() is a local maimum, a local minimum or neither
[4s] Assume f() is continuous on (-, ) and determine whether the following statements are True or False I If f (-) = 0 and f (-) > 0, then f(-) is a local minimum II If f (-) = 0 and f (-) = 0, then f(-) is neither a local maimum nor a local minimum III If f (-) = 0 and f (-) = -, then f(-) is a local maimum IV If f (-) = - and f (-) = -, then f(-) is a local maimum [4v] Given f( )= + 4, select the graph of f() Note: + 4= ( ) ( + 1) [4w] Given the graph below of the derivative f (), select a possible graph of f() [44a-j] Identify the intervals where f () > 0 Identify the intervals where f () < 0 Identify the intervals where f () > 0 Identify the intervals where f () < 0 Identify the intervals where f () is increasing Identify the intervals where the graph of f() is concave down Identify where inflection points occur
[44k] Find lim( 7 + 5 ) [44l] Find lim ( 7 + 5 ) 5 [44m] Find lim + 5 [44n] Find lim + 5 + [44o] Find lim 4 + 5 [44p] Find the vertical asymptote(s) of f( )= + 1 ; of f + ( )= + 1 ; of f + ( )= 1 5 [44q] Find the horizontal asymptote of f( )= ; of f 5 5 ( )= ; of f( )= [44s] Find all the horizontal asymptotes of f( )= + 1 [44u] Given f( ) f f ( ), ( ) ( ), ( ) 8 = = =, find all local etrema + + ( + ) 4 [44v] Given f( ) f f ( ), ( ) ( ), ( ) 8 = = =, find all the inflection points of + + ( + ) 4 the graph of f() [44z] Given the information below, select the graph of y = f() lim f ( ) =, lim f ( ) =, lim f ( ) =, lim f ( ) =, lim f ( ) = 1 1 1 + 1 1 + ± f(-) =, f(0) = 0, f() =, f (0) = 0 f () > 0 over (-, -1) U (-1,0) f () < 0 over (0,1) U (1, ) f () < 0 over (-1,1) f () > 0 over (-,-1) U (1, )
4 Absolute maima and minima - two methods (section 45) Method 1 - use to find absolute maima and minima of a continuous function f on [ a, b] A Find Critical Values ( values where f () = 0 or f () is not defined) in [ a, b ] B Calculate f at endpoints and critical values found in A C Largest value calculated in step B is absolute maimum value; Smallest value calculated in step B is absolute minimum value Method - to find absolute maimum or absolute minimum value of a continuous function f in an interval I (not necessarily closed) with only one critical value A Find Critical Value ( value where f () = 0 or f () is not defined) in interval I - verify that there is only one C V in I B Use the second derivative to test critical value (f > 0 => absolute minimum at CV; f < 0 => absolute maimum at CV) C Return to the problem to review eact information requested -- if ma or min value was requested, you must substitute the critical value into the original function f [45a] Given the following graph, find the absolute maimum and minimum of f() over the interval [ 1, 4] [45b] Find where the absolute maimum and minimum of f( )= 6+ 8 occur over the interval [0, 4] [45c] Find the absolute maimum and minimum of f( )= 6+ 8 over the interval [0, 4] 16 [45d] The absolute maimum of f( )= 5+ +, over the interval ( -, 0 ) is: [45f] Find the absolute maimum value of f( )= 10 + 7 over the interval [0, 4] [45g] Find the absolute minimum value of f( )= over the interval [0,16] [45h] A company manufactures and sells hundred televisions per month If the cost and 7 price-demand equations are C ( )= + 5+ 100 and p= 15, determine the level 8 8 of production that will maimize profit
[45k] A farmer has 500 feet of fence and wishes to build three rectangular enclosures (see figure) Find the total area equation A() to be maimized < > [45l] A fence is to be built to enclose a rectangular area of 600 square feet The fence along sides is to be made of material that costs $6 per foot The material for the fourth side costs $15 per foot Let be the length in feet of the epensive side Find the total cost equation C() to be minimized epensive side < > [45m] A fence costing $ 1000 is to be built to enclose a rectangular area The fence along sides is to be made of material that costs $5 per foot The material for the fourth side costs $1 per foot Let be the length in feet of the epensive side Find the total area equation A() to be maimized epensive side < > [45n] A rectangular piece of cardboard that measures 6 by 8 inches is to be formed into a rectangular candy bo by cutting squares with length from each corner and folding up the sides Find the volume equation V() to be maimized [45o] A package shipping company will accept a package if its length plus its girth (the distance all the way around) does not eceed 18 inches Find the volume equation V(), of the largest package with a square end that can be shipped
Supplemental Review Problems for Unit Test - KEY 1 Marginal Analysis (Sec 7) [7a] m () = 00-04 ; [7b] r () = 0-01 ; [7c] w() = 17-01 - 5000; [7d] y() = - 0 + 000/ 5000 1 [7e] a ( )= + 00 ; [7g] 16 ; 0 [7h] n ( )= 1+ 00 ; [7i] r ( )= 4 1 ; [7k] At a production level of 0 units, a unit increase in production will decrease total revenue approimately $0, or equivalently, at a production level of 0 units, total revenue is decreasing at a rate of approimately $0 per unit [7k] The eact profit from the production of the 1st item [7l] C (16) Continuity and Graphs (sec 41) [41a] ( -, ) ; [41b] ( -, - 4) U ( - 4, ) [41c] = -1 ; [41d] ( - 1, ) U ( 5, ) [41e] ( -, - 1 ) U (, ) [41f] [ /5, ) [41g] ( -, 4/ ) [41h] ( -, ) [41j] all real ecept = 5 ; [41m] - Derivatives and graphs (Sec 4,4, and 44) [4i] 1, [4j] 1, [4k] F, F [4l] ( -, - ) U ( 1, ) [4m] ( -, 1 ) [4n] T, T [4q] b) f() = - + + [4i] a) f () > 0 and f () > 0 on (-, ) ( ) + ( + ) 7 8 [4j] f ( ) = 88 + 4 18 4 [4k] ma at = - 1 [4l] min at = 1 ; [4m] ma at = - [4n] min at = [4o] ( 0, ) [4p] ( -, 0 ) [4q] = 0 [4r] f () = 0 and f () = 0 implies (d); f () = 0 and f () < 0 implies (b); f () = 0 and f () = implies (a); f () = and f () = implies (c) [4s] I is true; II is false; III is true; IV is false; [4v] [44a-j] f () > 0 on (b,d)u(d,0)u(h, ); f () < 0 on (-,b)u(0,e)u(e,h) f () > 0 on (a,d)u(e,k); f () < 0 on (-,a)u(d,e)u(k, ) f () is increasing on (a,d)u(e,k); f() is concave down on (-,a)u(d,e)u(k, ) Inflection points occur at = a and = k [44k] - [44l] [44m] [44n] [44o] 0
[44p] = - 1; none; = 1 [44q] y = -5/; none; y = 0 [44s] y = 1 [44u] local ma at = ; [44v] 4, 1 9 [44z] 4 Absolute maima and minima - two methods (section 45) [45a] absolute maimum is 0 (at = and at = 4); absolute minimum is - ( at = 1) [45b] absolute ma is at = 0; absolute minimum is at = ; [45c] absolute ma is 8; absolute minimum is - 1; [45d] absolute maimum is - [45f] absolute maimum is 64; [45g] absolute minimum is - 1/4 [45h] = 4 1 [45k] A ( )= 65 700 ; [45l] C ( )= 1+ ; 17 [45m] A ( )= 100 ; [45n] V ( )= 48 8 + 4 10 [45o] V ( )= 18 4