The questions listed below are drawn from midterm and final exams from the last few years at OSU. As the text book and structure of the class have

Transcription

1 The questions listed below are drawn from midterm and final eams from the last few years at OSU. As the tet book and structure of the class have recently changed, it made more sense to list the questions in reference to where the material they cover comes from as opposed to presenting entire old eams. The eams that these were drawn from had 7 questions in an 80 minute format for midterms and had 0 questions in a 0 minute format for the final.

2 Questions from old mth eams listed by chapter. Chapter. Below is a table of revenue in thousands of dollars for the first 0 days a new product is on the market. What was the average rate of change of revenue from day to day 9? Day Revenue in thousands of $ s 0 dollars b) day 500 dollars c) day 500 dollars day 9 dollars e) day 8 dollars day. Recall that we defined the difference quotient for a function as: D. Q. = What is the difference quotient for the function = +? f ( + h) h + + h b) + h + h c) h + h + e) h +. Given the graph of a line, find the line s equation y = + 7 b) y = + 5 c) y = + 7 y = + 7 e) y =

3 . What is the equation of the line that goes through the point, 5 with a slope of? y = + b) y = + c) y = y = + e) y = Find the slope of the line with equation y 9 = 5 slope = b) slope = c) slope = slope = e) slope = 6. Find the equation of the line that goes through (-6, ) and is perpendicular to the line 8 y + 6 = y = + b) y = + 6 c) y = + y = + e) y = + 7. A car that sold for $6,800 in 999, was worth $, in 00. Assuming that its value is decreasing in a linear fashion, what should it be worth in 008? $, b) $, c) $, $, e) $, Find the equation for the straight line that goes through the points (, -) and (-, ). + 5y + = 0 b) 5y = 0 c) + 5y + 9 = 0 5y 9 = 0 e) none of them

4 9. Use the graph to find lim if = + 0 b) c) does not eist ( 5)( + 7) 0. Find the value(s) of at which f() is discontinuous if =. ( 5)( + ) {-7, 5/} b) {-7, -, 7/, 5} c) {-5, } {-, 5} e) f() is continuous over all reals. A business had an annual retail sales of $00,000 in 99 and $6,000 in 99. Assuming that the annual sales followed a linear pattern, what were the retail sales in 99? $8, b) $95, c) $8, $87, Find all the points at which the graph of = + has horizontal tangent lines. (0, -) and (, ) b) (0, -) c) (0, 0) and (, 0) (-,0)

5 . If = +, then which of the following will correctly calculate the derivative of f()? [( + ) + ] ( + ) b) [( + ) lim 0 + ] ( + ) c) ( lim ) ( + ) ( + + ) ( + ). Find lim 0 b) c) / does not eist 5. Find the intervals in which the function is continuous: = ( )( + ) (,) and (, ) b) (, ),(,),(,) and (, ) c) (, ),(,) and (, ) (, ),(, ),(,) and (, ) 6. The height s (in feet) of an object fired straight up from ground level with an initial velocity of 50 feet per second is given by s = 6t + 50t, where t is time in seconds. How fast is the object moving after 5 seconds? 50 ft/sec b) 50 ft/sec c) 0 ft/sec -0 ft/sec e) -50 ft/sec 7. A line has a slope of and goes through the point (, 7). 5 Which of the following points must the line also go through? ( 8, ) b) ( 6,) c) (,7) ( 6,) e) (,5)

6 8. If = 5 + then f ''( ) =? 7 b) c) 8 e) 9. The marketing research department for ACME Roadrunner Etermination Co. has determined that at a price of $5.79 per unit, monthly sales should be 000 units. They also determined that for every increase in price of $.00, monthly sales will decrease by 00 units. Determine the demand function for this product. Let P = price per unit and = number of units sold. P = b) P = c) P = P = If = then f ( + ) =? b) ( ) c) + + ( ) 5 + 5( ) + + ( ) + ( ) + 7. Evaluate lim =? + 0 b) c) 9 + e) the limit does not eist

7 . Determine the intervals on which 5 = is continuous.,,, + b) (, ), (,5), (5, + ), (, 5), ( 5,), (, + ) c) ( ),,,,5, ( 5, + ) e),,, + Refer to graph below for questions and. lim g ( ) =? 6 b) c) does not eist e) cannot be determined from the graph. lim g ( ) =? 0 b) c) 0 does not eist e) cannot be determined from the graph

8 5. The position equation for the movement of a particle is given by s = ( t ), where s is measured in feet and t is measured in seconds. Find the acceleration of the particle at time t = seconds. feet/sec b) 8 feet/sec c) 90 feet/sec 88 feet/sec 6. Give the sign of the second derivative of f at the indicated point on the graph. Positive b) Negative c) Zero The sign of the second derivative cannot be determined by looking at the graph of f. 7. Find the equation of the line that is tangent to the graph o y = + b) y = + c) y = y = 8 + f = at the point (, ). 8. If y = f () is a differentiable function and f ( 7) =. 0 and f '(7) = 0. 0, then a good estimate for f (9) is: 7. b) 0. 6 c)..7 e).

9 9. Which of the following will correctly calculate the derivative of the function w ( ) =, + using the definition of the derivative? c) ( + ) lim + 0 ( + ) lim b) lim ( + ) e) If y = f () is a differentiable function, then The slope of the graph of y = f () is 6 7 f ' = means: at the point where 8 6 = b) The slope of the graph of y = f () is at the point where = c) The point, 5 8 The point 7 6, 8 5 lies on the tangent line to the graph y = f () lies on the tangent line to the graph y = f (). The height in feet of a ball above ground at t seconds after being thrown straight upwards is given by the function h ( t) = 6t + 5t + 5. What is the velocity of the ball seconds after it was thrown? ft / sec b) ft / sec c) 0 ft / sec ft / sec e) ft / sec

10 . Given the graph of y = r(), we can see that r ( ) < 0 and r '( ) > 0 at the point labeled: A B E C D A b) B c) C D e) E. Find ( + ) lim 0 7 ( 7) 0 b) 6 c) -7 + e) Does not eist

11 Chapter. The demand function for a particular commodity is given by p = 600. Find the marginal revenue when = 0. $5,00.00 b) $0.00 c) $50.00 $.00. Find f '( ) if =. f '( ) = b) f '( ) = c) f '( ) = ( ) f '( ) =. Find the derivative of f() if = ( + 5)( + ). 6 b) c) 6. A manufacturer determines that the revenue derived from selling units of a certain item is given by R = Find the marginal revenue when = 0. $, b) $80.00 c) $,58.0 $ Find y if y =. 8 0 b) 8 c)

12 6. Which of the following statements is true of = 6 9? f is decreasing on (, ) b) f is increasing on (, ) c) f is increasing on (, ) f is decreasing on (, ) 7. The management of a large store has,600 feet of fencing to fence in a rectangular storage yard using the building as one side of the yard. If the fencing is used for the remaining three sides, find the area of the largest possible yard. Round answer to nearest whole square foot. 60,000 square feet b) 8, square feet c) 0,000 square feet 80,000 square feet 8. Find the values of that give relative etrema for the function 5 = 5. relative maimum at = 0 ; relative minimum at = 5 b) relative maimum at = ; relative minimum at = c) relative maimum at = ± ; relative minimum at = 0 relative maimum at = 0 ; relative minimum at = ± 9. Find the derivative of : = + 5 b) + 5 c) + 5

13 0. The marketing research department for a computer company used a large city to test market their new product. They found that the demand equation was p = If their cost equation is C = , find the number of units,, that will produce the maimum profit. 6 b) 7 c) Find all the critical numbers for =. 0 b),0, c),,. Marketing research has shown that at a price of $9.95 sales of the Fabulous Kitchen Gadget will be 5,000 units per month, and for every increase in price of $.00 the sales will decline by,000 units per month. Create a linear demand function based on this information, and from that create a revenue function. What is the revenue associated with sales of,500 units per month? $5.95 b) $7,85.00 c) $76,7.00 $750,09.95 e) $-9.8. Use the information from question, and the additional information that the cost function for the business is Cost ( ) = 5 +, 000, where is the number of units sold per month and Cost( ) is in US dollars. What is the marginal profit associated with sales of,500 units per month? $8.95/unit b) $9.95/unit c) $0.95/unit $.95/unit e) $.95/unit. If g ( t) = +, then g '( ) =? b) c) e)

14 5. Find the interval(s) over which the function = + is increasing. b), c), (, ) e) (,+ ) (, ) and, Find the critical number(s) for the function r ( ) = + 5 ± b) ± c) ± ± e) r() has no critical numbers 7. Locate the relative etrema for the function g ( ) = g() has a relative minimum at: = and NO relative maimum b) g() has NO relative minimum and a relative maimum at = c) g() has a relative minimum at: = and a relative maimum at = 0 g() has a relative minimum at: = 0 and a relative maimum at = e) g() has NO relative minimum and a relative maimum at = 0 8. Given the graph of y = f '( ) (the graph of the DERIVATIVE of f()); we can tell that the function f() increases on what interval(s)? (, ) and (,) b) (, ) c) (.,0) and (.,) (,.) and ( 0,.) e) it is impossible to answer the question with the given information

15 9. Find the open interval on which the function is concave downward: y = 6 There does not eist an interval for which y is concave downward. b) (,0) c) (0,) (, + ) e) ( 0, + ) 0. Given the graph of y = h' '( ), (the graph of the second derivative of the function h()) We can tell that the function h() is concave upward on the intervals: the function is only concave downward at = b) (, 5 ) and ( 8, ) c) ( 0, ) and ( 5, 8 ) and (,.5 ) (.75, 6 ) and ( 9.7,.5 ) e) ( 0,.75 ) and ( 6, 9.7 ). The graph of y = f () is given below. At which of the labeled points are both the first and second derivative of f() positive? A B C D point A b) point B c) point C point D e) the first and second derivative are not both positive at any of those labeled points

16 . A company s cost function is C = and its demand function is p = 90, where C and p are in dollars and is the number of units produced and sold. Find the price p that yields the maimum profit. $0.00 b) $60.00 c) $0. $.8 e) $ Find the equation of the line tangent to the graph of = + at the point (,) 5 7 y = + b) y = + c) y = + y = +. The graph of the function = + 5 has: two relative etrema and one inflection point b) one relative etremum and one inflection point c) one relative etremum and two inflection points two relative etrema and no inflection points 5. If the cost function of a business is C = and the demand function is p = 00 0, find the price p that maimizes profit. $8 b) $0 c) $0 $0

17 Chapter. Find the number of units,, that will minimize then average cost function if the total cost function is C = b) 5 c) Use the product rule to find the derivative of the function p ( ) = ( + 5)( + ) do not simplify your answer. ( )( + ) b) ( )( + ) c) ( )( + ) + ( + 5)( + ) ( )( + ) + ( + 5)( + ) e) ( + 5)( + ) + ( + 5)( + ). Use the quotient rule to find the derivative of the function do not simplify your answer. q( ) = + ( + ) b) ( + ) ( + )( ) ( + 5)( c) ( + ) + ) ( + )( ) + ( + 5)( ( + ) + ) ( + )( ) ( + 5)( e) ( + ) + ). The function h() is defined in terms of the differentiable function f (). Find an epression for h '( ) given that h( ) = ( ) ( ) f '( ) b) ( ) f '( ) c) ( ) + f '( ) ( ) + ( ) f '( ) e) ( ) f '( ) + ( )

18 5. The function h() is defined in terms of the differentiable function f (). Find an epression for h '( ) given that h( ) = f '( ) b) f '( ) ( ) ( ) c) ( f '( )) f '( ) ( ) ( ) ( ) f '( ) ( ) + ( ) e) ( ) f '( ) 6. If f () and h () are differentiable functions such that f ( ) =, f '() = 5, h ( ) = and h '() = 6 ; then find d d h( ) = b) - c) - e) If f () and h () are differentiable functions such that f ( ) =, f '() = 5, h ( ) = and d d ( f ( )) ( h ( )) = h '() = 6 ; then find [ ] b) 8 c) - e) The function h() is defined in terms of the differentiable function f (). Find an epression for h '( ) given that h ( ) = f ( ) f ( ) b) f ( ) c) f ( ) f ( ) (6) e) f ( ) ()

19 9. If f () and h () are differentiable functions such that f ( ) =, f '() = 5, f '() =, h ( ) =, h ( ) = 7, h' ( ) = and '() = 6 d d h( ) = h ; then find [ ] b) -5 c) 6 5 e) 0. When a company produces and sells thousand units per month, its total monthly profit is P 00 thousand dollars, where P = The production level at t months from the present is given by = + t How fast (with respect to time) will profits be changing months from now? ( when t = ) Round your answer to the nearest whole dollar. increasing at $8/mo. b) decreasing at $785/ mo. c) increasing at $765/mo. decreasing at $570/mo. e) problem cannot be solved with the given information. Find the absolute etrema on the interval [ 0, ] of =. + Maimum: (,) ; Minimum : (0,0) b) Maimum: (,) ; Minimum : (0,) c) Maimum: (, /5 ) ; Minimum : (-, -) Maimum: (, /5 ) ; Minimum : (0,0). Find all intervals for which the function is concave upward: =. + (, ) b) (, ) c) (, ) (, )

20 . If y is related to according to the formula respect to when = 0? 0 y =, at what rate is y changing with + 0 units of y per unit of b) 5 units of y per unit of c) 5 units of y per unit of 0 units of y per unit of e) -5 units of y per unit of

21 Chapter. Match the graph shown with the correct function. ( ) = e + b) ( ) = e ( + ) = e e) = e + f c) = e. Find the derivative of the function = ln( + ). + + b) + c) ( + )( + ) + e) e ( + ) z. Epand the following epression using logarithm identities ln 5 ln( z ) + ln(5) b) ln + ln z ln5 ln c) ln( z) ln(5) ln + ln z ln 5 ln e) ln( z 5). Find the slope of the tangent line to the graph of y ) = (ln e at the point where =. e b) + ) e (ln c) e e ( ln + ) e) ln( ) e

22 5. Find the equation of the line tangent to the graph = ln( ) + at the point (e,). e y = + b) y = e + e c) y = e + y = + e) y = e e 6. Find the derivative of y e =. e b) e c) e e e) e e 9 7. Choose the epression equivalent to: ln y ln(9 ) + ln(y) b) ln(9) ln(y) c) ln9 + ln ln ln y ln9 + ln ln + ln y 8. Find the critical number(s) of the function = ln. 0 b), 0, c) 9. If the point, ) is a point on the graph of the function ( 8 f = ( ) a. Then a =? b) 6 c) 6 e) 9 0. Write the equivalent logarithmic equation to the equation 7 = e 5. ln 7 = 5 b) ln 7 = e 5 c) e 5 ln 5 = ln = 5 e) ln 5 = 7

23 . Use logarithmic identities to write the following epression as a single logarithm. 7 log( y) + log( ) log 7 ( y) log 6 b) log 7 y ( ) c) 7 7 y log 7( y) ( ) log( 8 7y.5) e) log 8. Solve for : ln ln( + ) = ln6 +. e b) ± c) - ±. Solve for : = 0 {,} b) { 0, ln()} c) 5 0,ln ln() 0, ln(5). Evaluate d d e = e b) c) 0 e) e

24 0.t 5. The value of a computer t years after its purchase is v( t) = 00 e dollars. At what rate will the computer s value be changing years after its purchase? decreasing at $ 6.7 per year b) decreasing at $ 9.9 per year c) its value will not be changing after years increasing at $ 9.9 per year e) increasing at $ 6.7 per year 6. Solve for : ln( ) ln = 0 b) c) e e 7. Given y = ln( + ) ln( + ) ; evaluate dy d =0 ln() b) ln() c) e) 6

25 Chapter 5. The balance in an account triples in years. Assuming that the interest is compounded continuously, what is the annual percentage rate? 6.89% b) 8.5% c) 8.65% 5.%. A radio-active substance has a half-life of 80,000 years. You obtain some of this substance, how many years must pass for there to be 0% remaining? Carry 5 significant digits during all calculations, then round answer to the nearest whole year. 8,958 years b) 5,59 years c) 7,596 years 9,7 years e) not enough information is given to solve the problem. An account is opened with a principal of $600 paying.5% interest compounded monthly. How long will it take for the balance to reach $000? (Round your answer to the nearest tenth).0 years b) 0. years c).6 years.7 years e).6 years. A town currently, in 008, has a population of 0,000 people. The population is epected to grow eponentially, and will double in ten years, 08. Which of the following functions models this population growth, where t = 0 corresponds to the year 008, and Pop(t) is the population of the town in thousands? Pop b) Pop t) = 0 ln( t) ( t) = t + 0 ( c) Pop ( t) = 0 e t Pop t ( t) = 0 e e) Pop ( t) = 0 e 0.86 t

26 5. The decay constant for a radioactive element is.05 when time is measured in years. Find the half-life of this element. Round your answer to the nearest tenth of a year. 0.6 years b). years c). years 6. years e) 9.6 years 6. Radioactive cobalt has a half-life of 5. years. Find its decay constant. Round your answer to the nearest thousandth. 0. b) 0.68 c) e) The demand function for a product is q = p + Find E (), the elasticity of demand at price p = for this product. Round your answer to the nearest hundredth. -.0 b) 0 c).7.86 e) The demand function for a product is q = p + At a price of p =, this product is elastic b) inelastic c) both elastic and inelastic neither elastic nor inelastic e) the question cannot be answered with the given information 9. Given the function. = 0 0e where its domain is 0, we can tell that the graph will be increasing and concave up on its domain b) decreasing and concave up on its domain c) increasing and concave down on its domain decreasing and concave down on its domain

27 0. Given the function point indicated. g( t).0t = e at t = 0, find the percent rate of change of the function at the increasing at a rate of % b) decreasing at a rate of % c) increasing at a rate of 0% decreasing at a rate of 0%. A rumor spreads among 0,000 people in such a way that t days after the start of the rumor, the number of people who have heard the rumor is given by f (t), where 0000 f ( t) = 0. t + 50e How quickly will the rumor be spreading after 5 days? Round your answer to the nearest whole number. people/day b) 5 people/day c) 9 people/day 55 people/day e) 557 people/day

28 Chapter 6 dc. Find the cost function if the marginal cost function is = + 50 d $50.00., and the fied costs are C = b) C = c) C = C = e) C = dr. The marginal revenue for a certain product is given by: = d. Find the change in revenue when sales increase from 00 to 500 units. $0.00 b) $, c) $9, $, e) $7, dc. Find the cost function if the marginal cost function is = and fied costs are d $ C = b) C = c) 000 C = C = dr. The marginal revenue for a certain product is given by: = 0.. d Find the change in revenue when sales increase from 50 to 60 units. $90 b)$ c)$65 -$0

29 5 5. Evaluate: d C b) + C c) 5ln + C 5e + C 6. Find the value of k that makes the anti-differentiation formula true for: e d = ke + C k = b) k = c) k = k = 7. Find the function h() that has the following properties: i) h '( ) = + and ii) h ( 0) = ( ) = h + b) h ( ) = + c) h ( ) = + + h ( ) = Evaluate the following definite integral: ( + ) d The graph of = + is below. b) c) 5 6 e) 7