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# ELEM CALC & ITS APPLICS MA 123

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Date Created: 10/23/15

Limits and continuity Chapter 3 Practicereview problems The collection of problems listed below contains questions taken from previous MA123 exams Limits and onesided limits 1 Suppose Ht t2 5t 1 Find the limit a 15 b 1 c 9 d 6 8 2t 5 t2 7 4 2 Find the limit lim 7 m2 2 7 2 a 2 b 4 c 6 d 8 e The limit does not exist 3 Find the limit lim 75 z 7 5 1 1 1 1 77 b 77 0 d 7 a 10 lt gt 5 c lt gt 5 e 10 4 C t 1 m2 7 7m 12 ompu e a 0 b 1 c 71 d 2 e The limit does not exist 2 7 3 2 5 Find lim 771 T 7 1 a 1 b 0 c 71 d 2 e The limit does not exist 2 7 6 Find the limit or state that it does not exist lini1 m7 7 a 8 b 720 c 715 d 9 13 Does Not Exist 2 7 7 7 Compute lim 710 13 13 a 5 b 4 C 3 d 2 e 1 i h 42 7 16 8 Compute in a 4 b 5 C 6 d 7 e 8 3 9 Find the limit lim t7gt0 W a 0 b 1 c 2 d 3 e The limit does not exist 10 11 12 13 14 15 16 17 18 1951 Find the limit as x tends to 0 from the left lim 27 x70 x a 13 b 12 C 0 d 12 e 13 471 Find the limit lim u 70 h Hint Evaluate the quotient for some negative values of h close to 0 a 0 b 2 C 2 d 4 e 4 Compute lim x73 x 7 3 a 4 b 74 C 0 1 Doesn t exist 3 Cannot be determined i i i i 7 1 x2 if x lt 2 Find the limit of x as x tends to 2 from the left if x 7 3 if m 2 2 a 5 b 6 C 7 d 8 e 9 i i i i x3 7 2 if x Z 2 Find the limit of x as x tends to 2 from the left if x 7 1 2 if m lt 2 a 5 b 6 C 7 d 8 13 Does not exist i 4x2 7 1 if x lt 1 For the function x 7 3m 2 if m 2 1 Find mlinlfr a 5 b 3 C 1 d 0 e The limit does not exist 2 8 15 if lt 2 Let Hz 3 M T 4x 7 if x gt 2 F d l In 221 W a 15 b 20 C 30 d 35 e The limit does not exist 75x 7 if x lt 3 L t 8 C35 2716 3523 F d l In 313 W a 6 b 76 C 77 d 78 e The limit does not exist 7t if t lt 1 suppose 752 if t Z 1 Find the limit t a 71 b 1 C 0 d 2 e The limit does not exist 19 20 21 22 23 24 25 7 it if t lt 1 suppose i t3 if t Z 1 Find the limit ft a 72 b 71 c 1 d 2 e The limit does not exist Suppose the total cost7 Cq7 of producing a quantity q of a product equals a xed cost of 1000 plus 3 times the quantity produced So total cost in dollars is Cq1000 3g The average cost per unit quantity7 Aq equals the total cost7 Cq7 divided by the quantity produced7 q Find the limiting value of the average cost per unit as q tends to 0 from the right In other words nd l A 1351 q a 0 b 3 c 1000 d 1003 e The limit does not exist i i i i 3 Find the limit lim 7 taco 1 t2 a 0 b 1 c 2 d 3 e The limit does not exist m2 m 1 Find the limit lim 4100 395 2 a 1 b 13 c 0 d 19 13 The limit does not exist 4 2 Find the limit lim w 9400 33 83 l 9 a 0 b 1 c 2 d 3 e The limit does not exist 2 2 Find the limit lim 3 Hm at 23 a 0 b 1 c 2 d 3 e The limit does not exist Suppose the total cost7 Cq7 of producing a quantity q of a product is given by the equation Cq 5000 5q The average cost per unit quantity7 Aq equals the total cost7 Cq7 divided by the quantity produced7 q Find the limiting value of the average cost per unit as q tends to 00 In other words nd lim Aq qaoo a 5 b 6 c 5000 d 5006 e The limit does not exist 26 27 28 29 30 31 Continuity and differentiability Bt iftg W 5 iftgt3 Find a value of B such that the function t is continuous for all t a 35 b 45 C 53 d 54 Suppose e 52 ifxlt2 ifx22 A x Suppose that x 7 1 2 Find a value of A such that the function x is continuous at the point x 2 a A8 b A1 c A2 d A3 e AO t ift 3 suppose t A3 if tgt3 2 Find a value of A such that the function t is continuous for all t a 12 b 1 c 32 d 2 13 52 2 2 3 if lt3 Consider the function x m m7 3xB 1fxgt3 Find a value of B such that x is continuous at x 3 a 6 b 9 c 12 d 15 13 There is no such value of B 2 1 Find all values of a such that the function x ill 2m Z is continuous everywhere a a71only a71anda1 b a72anda2 a 72 only C 13 all real numbers d Which of the following is true for the function x given by 2x 7 1 if x lt 71 x x21 if 71 x 1 x 1 if x gt 1 a f is continuous everywhere b 1 is continuous everywhere except at x 71 and x 1 c f is continuous everywhere except at x 71 d f is continuous everywhere except at x 1 None of the above 3 V 32 Which of the following is true for the function a b C d e V f 90 90 7 1 7 f is differentiable at z 1 and z 2 f is differentiable at z 17 but not at z 2 f is differentiable at z 27 but not at z 1 f is not differentiable at either z 1 or z 2 None of the above Rates of change and derivatives Chapter 2 Practicereview problems The collection of problems listed below contains questions taken from previous MA123 exams Average rates of change Word Problems 1 A train travels from A to B to C The distance from A to B is 10 miles and the distance from B to C is 40 miles The average velocity from A to B was 20 miles per hour and the average velocity from B to C was 40 miles per hour What was the average velocity from A to C in miles per hour a 1805 b 903 c 1003 d 1803 e 1005 2 A train travels from city A to city B It leaves city A at 1030 am and arrives at city B at 130 pm The distance between the cities is 150 miles What was the average velocity of the train in miles per hour a 60 b 150 c 50 d 75 e 130 3 A train travels from city A to city B to city C The distance from A to B is 20 miles The distance from B to C is 45 miles The train took 1 hour for the trip from A to B7 stopped at city B for 30 minutes7 and then went from B to C at an average velocity of 30 miles per hour What was the average velocity of the train for the entire trip in miles per hour 65 65 a 65 b 25 c 7 d 50 e g 4 A train travels from A to B to C The distance from A to B is 30 miles and the distance from B to C is 80 miles The train leaves A at 1000 AM and arrives at C at 300 PM The average speed from A to B was 30 miles per hour What was the average speed from B to C in miles per hour a 20 b 25 c 30 d 35 e 40 5 A train travels from city A to city B The cities are 600 miles apart The distance from city A at 25 hours after the train leaves A is given by dt 502 252 What is the average velocity of the train in miles per hour during the trip from A to B Hint First nd how long it takes for the train to get from A to B a 50 b 55 c 60 d 65 e 70 6 John leaves at 900 am and drives from Lexington to Ashland arriving at 1100 am He stops for two hours since his girlfriend Mary is not yet ready Then they drive together from Ashland to Columbus arriving at Columbus after a three hour drive The distance from Lexington to Ashland is 110 miles and the distance from Ashland to Columbus is 130 miles Find the average velocity of John s car in miles per hour for the entire trip including the two hour stop correct to two decimal places a 3381 b 3342 c 3500 d 3429 e 3447 20 7 8 10 11 12 13 14 15 16 Average rates of change lfgz m 7 12 what is the average rate of change of gz with respect to x as x changes from 73 to 3 a 4 b 2 C 0 d 2 e 4 2 Suppose that ht Find the average rate of change of ht from t 5 to t 10 a 705 b 704 c 05 d 04 e 02 Find the average rate of change of the function Rt x 2t 1 7 as if changes from 1 to 9 a g b g c i d 4 e 2 If gz lz 7 7 what is the average rate of change of gz with respect to x as x changes from 73 to 3 a 2 b 1 C 0 d 1 e 2 Find the average rate of change of the function Ct 252 7 1 as if changes from 71 to 2 a 0 b 1 C 2 d 3 e 4 Let 95 52 7 35 1 Find a value A 2 0 such that the average rate of change ofgs from 0 to A equals 8 a 0 b 8 c 11 d 15 e 22 Suppose ft t3 1 Find a value A greater than 0 such that the average rate of change of ft from 0 to A equals 2 a 1 0 v2 0 v3 d 2 e v5 Compute w where fz 3x2 1 a 12 b 12 h c 12 1 2h 1 12 1 3h 13 None of the above What is the average rate of change of 95 52 7 4 as 3 changes from 1 to 1 h a 63h b 2h c 42h d 2 e h Let fz 2x2 7 3m Find the average rate of change of fz from x 3 to z 3 h a 97h b 90 c 9 d 9721 8 92h 17 18 19 20 21 22 23 24 Let 9t t 7 52 1 What is the average rate of change of 9t as if changes from 4 to 4 h a 7127211 b h2 c h22h d 1272 e 1 lfft3t24then flt1hgt7 f1 h a 43h b 34h c 63h d 83h e 84h lf ft 1t then ft h 7 m h a 1012 b 1tth c 71tth d 1tt7h 8 71050540 Instantaneous rates of change Consider a triangle with base z and height 2x Find the instantaneous rate of change of the area of the triangle with respect to z when x 5 a 1 b 2 c 5 d 10 e 20 Find the instantaneous rate of change of the function Ht t3 at t 2 a 2 b 3 c 8 d 12 e 27 In what follows7 you may use the following formula for the derivative of a quadratic function If pm sz Bx 07 then pm 214m B lf 95 352 5 7 2 what is the value of 95 when the instantaneous rate of change of 95 with respect to 5 equals 1 a 72 b 71 c 0 d 1 e 2 lf95 352 25 7 2 what is the value of 5 for which the instantaneous rate of change of 95 with respect to 5 equals 8 a 2 b 1 C 0 d 1 e 2 Suppose the price of a good is given by the quadratic function Pt 258 1425 01t2 What is the instantaneous rate of change in the price when t 3 a 18 b 20 c 22 d 24 e 26 pm pm nu pamp pm mm mm mm Let gz 2 4x 5 Find a value of 0 between 1 and 10 such that the average rate of change of gz from x 1 to z 10 is equal to the instantaneous rate of gz at z c 4 w5o 5 m55 5 Find a nonnegative number A such that the average rate of change of Ft t2 7 2t 1 1 from t 1 to t A equals the instantaneous rate of change of Ft at t 2 AO m A2 A3 m A4 A5 Suppose the cost Cq in dollars of producing a quantity q of a product equals 1 2 Cq 500 2q gq The marginal cost M C q equals the instantaneous rate of change of the total cost Find the marginal cost when a quantity of 10 items are being produced 2 m 6 10 m m 5m Tangent lines Find the slope of the tangent line to the graph of x 3x2 7 7x 4 at z 2 a 5 b 6 C 7 d 8 e 9 Find the equation of a line tangent to the curve y 2x2 z 1 at z 2 a y911x72 b y119m72 d y1322x73 e y74x1 C y 22 13z73 Suppose Cm 2 z 7 2 For what value of z is the tangent line to the graph of y Cm parallel to the m axis a m71 b x0 C x2 d z12 e x712 Suppose 95 52 45 1 Find a point of the graph oft 95 such that the tangent line to the graph is parallel to the s axis a 279 b 474 C 473 d i478 393 i471 What is the value of x such that the slope of the tangent line to the graph of x 2 7 10x 1 14 is 6 a 6 b 7 c 8 d 9 e Thereis no suchz 33 34 35 36 Suppose 95 52 1 Find a point on the graph oft 95 such that the tangent line to the graph is parallel to the line with equation 25 s a 071 W 11541 C 172 d 327134 e 25 Suppose ht represents the height of an object above the ground at time 157 where the height is measured in feet and the time t is measured in seconds If W i 716t2 482 144 what is the velocity of the object at time t 0 b d 64 feet per second a 48 feet per second 144 miles per hour c 32 furlongs per fortnight e 96 feet per second lf ht represents the height of an object above ground level at time t and ht is given by 1125 7162 96t1 nd the height of the object at the time when the velocity is zero a 144 b 145 c 148 d 150 e 160 Suppose the position Pt of an object at time t is given by t2 1 Find a value of t at which the instantaneous velocity of the object equals the average velocity on the interval 07 1 a 12 b 1 C 32 d 2 e 52 Limits and continuity Chapter 3 Practicereview problems The collection of problems listed below contains questions taken from previous MA123 exams 6 Find the limit or state that it does not exist lini1 m4 Limits and onesided limits 1 Suppose Ht t2 5t 1 Find the limit 15 b 1 c 9 d 6 2 Find the limit gig a 2 4 c 6 d 8 3 Find the limit 3315 a 7 b a c 0 d g i 277m12 4 Compute a 0 b 1 c 71 d 2 5 Find iii a 1 b 0 C 1 d 2 74 a 8 b 20 C 715 g i QmZ 73 4 574 7 Compute 33 f T t a 5 b 4 c 3 d 2 8 Compute a 4 b 5 C 6 d 7 x2m720 13 2t 5 e The limit does not exist e e The limit does not exist 13 The limit does not exist 13 Does Not Exist e 1 9 10 11 12 13 14 15 16 xiT3 F d th 1 t l 7 in e 1m1 tits 0 b 1 c 2 d 3 e The limit does not exist Find the limit as x tends to 0 from the left lim x7vO 2x a 13 b 12 C 0 d 712 e 713 Find the limit lim h70 Hint Evaluate the quotient for some negative values of h close to 0 a 0 b 2 C 2 d 4 e 4 Compute lim x7 x 7 3 a 4 b 74 C 0 1 Doesn t exist e Cannot be determined i i i i 7 1 x2 if x lt 2 Find the limit of x as x tends to 2 from the left if x 7 3 if m 2 2 5 b 6 c 7 d 8 e 9 i i i i x3 7 2 if x Z 2 Find the limit of x as x tends to 2 from the left if x 7 1 2 if m lt 2 5 b 6 C 7 d 8 e Does not exist i 4x2 7 1 if x lt 1 For the function x 7 3m 2 if m 2 1 F d l In with W 5 b 3 C 1 d 0 e The limit does not exist 2 8 15 if lt 2 Let Hz g M l 4x 7 If x gt 2 Find mler 15 b 20 c 30 d 35 e The limit does not exist 17 18 19 20 21 22 23 75m 7 if m lt 3 L t 8 C35 252716 iszB F d l m 122 W a 6 b 76 c 77 d 78 e The limit does not exist it if tlt 1 suppose 7 t2 t Z 1 Find the limit ft a 71 b 1 c 0 d 2 e The limit does not exist SH 088 m 7 it if t lt 1 pp 7 t3 if t 2 1 Find the limit ft a 72 b 71 1 d 2 e The limit does not exist Suppose the total cost7 Cq7 of producing a quantity q of a product equals a xed cost of 1000 plus 3 times the quantity produced So total cost in dollars is Cq1000 3q The average cost per unit quantity7 Aq equals the total cost7 Cq7 divided by the quantity produced7 q Find the limiting value of the average cost per unit as q tends to 0 from the right In other words nd 1 A 1351 q a 0 b 3 c 1000 d 1003 e The limit does not exist i i i i 3 Find the limit llm 7 taco 1 t2 0 b 1 c 2 d 3 e The limit does not exist m2 m 1 Find the limit lim mace 3m 22 a 1 b 13 c 0 d 19 13 The limit does not exist 4 2 Find the limit lim 9400 33 83 9 a 0 b 1 c 2 d 3 e The limit does not exist 24 25 26 27 28 29 30 i i i i 2x2 Find the limit lin1 7 xaoo z 23 0 b 1 c 2 d 3 e The limit does not exist Suppose the total cost7 Cq7 of producing a quantity q of a product is given by the equation Cq 5000 5q The average cost per unit quantity7 Aq equals the total cost7 Cq7 divided by the quantity produced7 q Find the limiting value of the average cost per unit as q tends to 00 In other words nd 330140 a 5 b 6 c 5000 d 5006 e The limit does not exist Continuity and differentiability Bt if t g 3 Suppose W 5 if t gt 3 Find a value of B such that the function t is continuous for all t a 35 b 45 C 53 d 54 e 52 7 A x if x lt 2 Suppose that x 7 1 2 if m 2 2 Find a value of A such that the function x is continuous at the point x 2 a A8 b A1 c A2 d A3 e A0 t if t g 3 suppose t 14 if tgt3 Find a value of A such that the function t is continuous for all t a 12 b 1 C 32 d 2 e 52 2 2 3 f lt 3 Consider the function x m m 7 3x B If x gt 3 Find a value of B such that x is continuous at x 3 a 6 b 9 12 d 15 13 There is no such value of B i i x2 2x if x lt a i i Find all values of a such that the function x 71 if m gt a is continuous everywhere a a 71 only b a 72 only c a 71 and a 1 d a 72 and a 2 13 all real numbers 31 32 Which of the following is true for the function x given by 2m 7 1 if m lt 71 m21 if 71 x 1 m 1 if m gt 1 a f is continuous everywhere b 1 is continuous everywhere except at z 71 and z 1 f is continuous everywhere except at z 71 d f is continuous everywhere except at z 1 13 None of the above Which of the following is true for the function x lz 7 1 a f is differentiable at z 1 and z 2 b 1 is differentiable at z 17 but not at z 2 f is differentiable at z 27 but not at z 1 d f is not differentiable at either z 1 or z 2 13 None of the above The idea of the integral Chapter 8 Practicereview problems The collection of problems listed below comprises questions taken from previous MA123 exams 1 Estimate the area under the graph of fz 2 2 on the interval 07 2 by dividing the interval into four equal parts Use the right endpoint of each interval as a sample point a 1175 square units b 571875 square units c 95 square units 1 575 square units 775 square units 24 2 Estimate the area under the graph of y 2 2x 3 for z between 72 and 2 Use a partition that consists of 4 equal subintervals of 727 2 and use the right endpoint of each subinterval as a sample point 22 b 23 c 24 d 25 e 26 VVVVVV 35233 1010 v of 0 A2 of 39 2929 99 0 039 VII 3 29 0 O o to of 3 A O O O 0 O 10 w 9 1vo V O Q 2029 9 O of O O 0102 vow Q Q 020 3 O Q 0 1 8 0 Q 102 V Q Q to o O 3 to O 2 m H 3 A train starts from rest velocity equal to 0 miles per hour at 1200 noon The velocity increases at a constant rate until 1215 when the velocity equals 64 miles per hour How far does the train travel from 1200 to 1215 a 7 b s c g d 10 e 11 25 4 Use a calculator to estimate the integral 21 dz 1 Use three 3 subintervals and the left endpoint of each subinterval to determine the height of the rectangles used in the approximation The approximate value of the integral is a 166 b 168 c 172 d 174 e 178 5 225 Use a calculator to estimate the integral logz dz Use ve 5 subintervals and the left endpoint of each subinterval to determine the height of the rectangles used in the approximation The approximate value of the integral is a 131 b 128 c 113 d 104 08 116 6 7 8 9 10 Use a calculator to estimate the integral 12 1112 dz Use four subintervals and the right endpoint of each subinterval to determine the height of the rectangles used in the approximation The approximate value of the integral is a 0218 b 0297 c 0352 0470 e 0521 9 Estimate the area under the graph of y 2x2 for z between 1 and 5 Use a partition that consists of 4 equal subintervals of 17 5 and use the right endpoint of each subinterval as a sample point a 92 b 94 c 96 d 102 L 0 1 2 3 4 5 95 Suppose you estimate the integral 20 1 m2 dm 10 by the sum of the areas of 10 rectangles of equal base length Use the right endpoint of each base to determine the height What is the area of the rst left most rectangle 144 Suppose you estimate the area under the graph of x 3 from x 5 to z 25 by adding the areas of b 244 c 341 d 441 e 541 rectangles as follows partition the interval into 20 equal subintervals and use the right endpoint of each interval to determine the height of the rectangle What is the area of the 11th rectangle a b 4096 30 1 7dm as the sum of areas of rectangles You break the interval 1000 1331 c 2744 d 3375 You want to estimate the integral 107 30 into 20 subintervals of equal length If you use the left endpoint of each subinterval to determine the height of each rectangle7 which estimate is correct Hint Draw a picture 3013gt111 1 3013lt111 1 a 10353510 11 12 29 10353510 11 12 29 301 1 1 1 1 301 1 1 1 1 7d lti i i i d d gti i i i C1021112 29130 1021112 29130 301dlt111 11 e 10 m a 10 12 14 2830 117 11 12 13 14 15 16 17 20 Suppose you estimate the integral mzdz 10 length Use the left endpoint of each base to determine the height What is the area of the rst leftmost rectangle by the sum of the areas of 50 rectangles of equal base 20 b 30 c 40 d 50 e 60 10 Evaluate the sum 21 k k4 56 b 60 c 63 d 73 e 74 Write the sum 7 9 11 13 15 17 19 21 in summation notation as N ZA 2k k2 What are the values of A and N a A1N10 b A2N10 c A1N9 d A2N9 e A3N9 4 Evaluate the sum Zlt5k 1 k2 a 112 b 66 c 29 d 64 48 6 Evaluate the sum Zltk2 7 1 k3 a 35 b 60 c 82 d 98 e 122 6 Evaluate the sum Zltk2 7 k2 a 60 b 63 c 67 70 e 72 Suppose you estimate the integral 6 dm 2 by evaluating the sum 7L imam km k1 If you use Am 27 what value should you use for n a 25 b 10 c 30 20 e 15 118 18 19 20 21 You make two estimates using rectangles for the integral 0117m2dm The rst estimate uses 50 equal length subintervals and the left endpoint of each subinterval The second estimate uses 50 equal length subintervals and the right endpoint of each subinterval What is the difference between the two estimates rst minus second 8 4 1 i b i i d i i a 50 M 50 C 50 M 50 50 The integral 6 x3dx 1 is computed as i A A 3 mg wz What is the value of A a 8 5 c 7 d 6 e 4 Suppose you estimate the integral A6 35 dz by adding the areas of n rectangles of equal base length7 and you use the right end point of each subinterval to determine the height of each rectangle If the sum you evaluate is written as A f B 7k 71 b A4B4 i 4 101 n what are A and B a A2B4 C 13 None of the above A1B2 m AzB2 Suppose that you estimate the integral A8 35 dz Zmflt2kmy 101 by evaluating a sum If you use 12 intervals of equal length7 what value should you use for Am a 01 b 02 c 03 d 04 05 119 22 23 24 11 N Suppose that the integral fm dm is estimated by the sum 2 fa kAm Am The terms 1 k1 in the sum equal areas of rectangles obtained by using right endpoints of the subintervals of length Am as sample points If N 207 then what is Am a 05 b 1 c 5 d 1 3 Cannot be determined 52 N Suppose that the integral fm dm is estimated by the sum 2 fa kAm Am The terms 2 k1 in the sum equal areas of rectangles obtained by using right endpoints of the subintervals of length Am as sample points If fm 7 and N 507 then nd the area of the second rectangle 1 a 116 b 19 c 18 d 14 3 12 12 N Suppose that the integral mdm is estimated by the sum Z a kAm Am7 where 6 k1 Am 2 and N 30 The terms in the sum equal areas of rectangles obtained by using right endpoints 6 of the subintervals of length Am as sample points What is a a 2 b 3 C 4 d 5 120 Formulas for integrals integrals antiderivatives and the Fundamental Theorem of Calculus Chapter 10 Practicereview problems The collection of problems listed below comprises questions taken from previous MA123 exams 1 Pl 4 5l 6l 7 m Let 55 1t23dt Find h m 4 a ltz23gt4 2z b 1 e W on e 252 Let Fz m2t2 7 375 1 dt Find F 3 1 a 7 b 8 c 9 10 e 11 If m Fa 5244mm 2 ndF 3 49 64 a 3 21 c i d 27 e 36 1 Let Am t2 t4 156 dt Find the value of x on 17 50 where Am takes its minimum value 0 1 b 12 14 16 c 25 d 502 504 506 e 50 Find 12 323s lds 0 a 181 804 c 132 d 55 e 1 2 Find x2 2x 1 dz 1 a 4 b 193 c 9 d 3 e 293 Evaluate the limit i 1 3135 2 WW 151 where x m2 Hint Draw a picture and relate the limit to an integral a 15 b 14 C 13 d 12 e 1 147 8 9 10 11 12 13 14 Let x g i Evaluate the integral 03 x dx a 72 b 92 c 112 d 132 3 152 1 1 1 Suppose x dx 7 4 and gx dx 5 What is the value of 2 x dx 7 0 0 0 a 19 b 17 c 15 13 e 11 6 Use the Fundamental Theorem of Calculus to compute xx 3dx 1 a 373 m 383 a 393 d 403 13 413 Find the general antiderivative x 52 dx a 3x52C b x2 1 0 c 72x2 3C d 7x2 1C e x530 Find 3 x 1 m dx 4 x3 x4 7 x a if 1 C b 2 2 t C C 433 1 C x 7 2 x4 x d 2 C e 353 C What is the average of the function ht t2 1 on the interval 17 4 Recall that the average of t on an interval 11 equals the constant value A such that the area under the graph of the constant function A equals the area under the graph of t for the interval 11 In other words7 b tdt bAdt a a 8 b 10 c 12 d 14 e 16 What is the average of the function ht t3 1 on the interval 17 4 7 247 257 267 277 279 a E b E E d E 393 E 148 Computing some integrals Chapter 9 Practicereview problems The collection of problems listed below comprises questions taken from previous MA123 exams 1 Evaluate the sum 3691230 a 55 b 550 c 110 d 275 165 2 Evaluate the sum 6 8 10 12 100 The sum equals a 2518 b 2530 c 2538 d 2540 2544 3 Use the summation formula below to evaluate 33 43 403 ika 71201 12 151 4 31 608391 b 608400 5 653479 d 672400 672391 4 Suppose the equation 11121131141 1111 1 2 2 2 2 39 39 39 2 7 A holds for all non negative integer values of 71 What is the value of the constant A a 1 b 2 C 3 d 4 e 5 5 Evaluate the sum 10 15 20 25 30 1000 Note that each term is a multi le of 5 The sum p equals 31 100460 b 100485 100495 d 100500 13 100525 6 Evaluate the sum 8101214600 The sum equals 90288 b 90294 c 90300 d 90306 13 90312 20 7 Evaluate the sum 26102 7 2k k1 a 15200 b 15600 a 16000 d 16400 16800 130 8 9l 10 11 12 13 14 Evaluatethesum 6912151821300 15147 b 15150 a 15143 d 15157 If n kzik nn1nil7 A ndA a 1 b 3 C 5 d 8 Given the summation formula loelow7 determine the correct value of A Z k5 i 7122712 2n 7 An 12 k1 7 12 a 1 b 0 d 2 1 The formula for the sum of the sixth power of k is 7 A717 1 21716 7 7713 21715 n k1 42 What is the value of A a 7 b 21 C 15 d 6 Evaluate the limit 1 n2 n 1 1m Hoe 3n 2 a 112 b 111 C 110 d 19 Evaluate the limit as 71 tends to in nity 8713 7n 5 1m Waco 2713 8712 a 2 b 3 C 4 d 6 Evaluate the limit 1 2713 n 23 a 0 b 1 C 2 d 3 131 13 15160 e 10 e 3 e 3 e 18 e 8 e 4 15 Evaluate the limit 1 24682n lim 2 naoo 77 1 b 2 c 3 d 4 e 5 16 Evaluate the limit 1 2712 3 n 42 a 0 b 1 C 2 d 3 13 Does not exist 17 Evaluate the limit 4m 12 1m 7 4200 4352 3 a 1 b 2 C 4 d 8 13 Does not exist 18 Find 7L 1 4k 13 Z 9 k1 a 0 b 2 C 4 d 8 13 Does not exist 19 Find 1 123456n lim mace 2712 a 0 b l l d l 8 Does not exist 8 4 2 20 Evaluate the limit 1 1 n 3520 E 2 2k k1 a 0 b 12 1 d 32 13 2 21 Evaluate the limit 1 1 5 k 21330 g B t k1 3 5 7 9 11 a 5 b g C 5 d g 3 3 22 Evaluate the limit as 71 tends to in nity 7L 1 4k 3 13 Z t k1 a 6 b 5 C 3 d 4 e 7 132 23 The limit n 4 15 3k 331 Z 2 lt2 t I k1 is obtained by applying the de nition of the integral to 5 dx 2 What is x a 1 32 M4 b 1 4904 C 1 2954 3 4 d M 5254 e m 2 2 7 24 Let 71 if x lt 0 x 2 if 0 xlt2 3 if 2 g x Evaluate the integral 3 72 a 1 b 2 C 3 d 4 e 5 2 5 The integral is computed as n 3 12 3k 333 Z 2 lt2 t I k1 What is x a 6x2 b 8x2 c 12x 4353 e x21 0247 235W Hint Think of the de nite integral as an area 26 Evaluate the integral a 0 b 1 c 2 d 3 e 4 27 Given that the area of the ellipse 4x2 y2 4 is 271397 evaluate the integral 1 V 4 7 4x2 dx 0 Hint Think of the de nite integral as an area a g b E c 7139 d 27139 e 47139 133 28 29 30 31 32 Evaluate 6 V 62 7 m2 dm 0 Hint Think of the de nite integral as an area a 47139 b 167139 C 97139 d 67139 e 257139 Suppose you are given the following data points for a function x 1 2 3 4 fz 2 4 8 12 If f is a linear function on each interval between the given points7 nd 4 1 Hint Draw a picture a 13 19 c 20 d 26 e 40 Evaluate the integral 9 4 18 18 7 3x dm 0 a 54 48 c 60 d 24 e 36 0 8 Compute Suggestion Draw agraph 74 40 b 41 c 42 d 43 e 44 7n277n5 Compute a 53 b 59 c 79 1 1 e Thelimit does not exist 134 Word Problems Chapter 7 Practicereview problems The collection of problems listed below contains questions taken from previous MA123 exams Maxmin problems 1 A eld has the shape of a rectangle with two semicircles attached at opposite sides Find the radius of 3 4 6 the semicircles if the eld is to have maximum area7 the perimeter of the eld equals 1007 and the width of the eld twice the radius of the semicircles is at most 18 Caution Be sure your answer satis es all conditions The radius equals a 6 b 7 c s d g e 10 Find the area of the largest rectangle with one corner at the origin and the opposite corner in the rst quadrant and on the line y 10 7 2m Assume the sides of the rectangle are parallel with the axes a 732 b 672 5 552 d 492 3 252 If you sell an item at price 97 your revenue will equal the price p times the number sold7 71 Suppose price is linearly related to the number sold by the equation 71 100 7105 710 How should you set the price to maximize revenue The price should equal 10 b A rectangle in the rst quadrant has one corner at 07 0 and the opposite corner on the curve y 2 7 2 15 c 20 d 25 e 30 What is the largest possible area of this rectangle 84 42 24 C 3 mg 393 3 3 20 ltbgt Find the length of the shortest line segment that connects the point 47 0 in the z y plane to the line 3421 8 5 5 Find the area of the triangle of minimum area with base equal to the unit interval 0 g x g 1 on the z 10 b 7W 16 x 12 EVE c d e m axis and with opposite vertex lying on the curve y 8x 7 with z gt O 1 a 1 b 2 C 3 d 4 e 6 102 7 8 9 10 11 Find the area of the largest rectangle with one corner at the origin the opposite corner in the rst quadrant on the graph of the line fz 6 7 32 and sides parallel to the axes a 1 3 e 5 What is the maximum area of the rectangle with sides parallel to the coordinate axes one corner at the b 2 d 4 origin and the opposite corner in the rst quadrant on the ellipse given by the equation 2x2 y2 r2 2 72 72 d 72 72 a b 7 i 7 i T C2 UN e4 A rectangular eld as shown below is constructed using 2400 feet of fencing There are six parallel fences in the vertical direction What is the maximum possible area in square feet of the rectangular eld a 100 000 b 110 000 120 000 d 130 000 e None of the above Find the point mo yo in the rst quadrant that lies on the hyperbola y2 7 2 5 and is closest to the 34 point A40 Then 0343 is a 17 V3 b 273 0 A c 25x1125 d 37 v14 393 47 v21 Suppose you want to nd the shortest distance between the point 10 on the m axis and a point on the ellipse 2 4342 16 Which problem do you need to solve 2 167 2 712ltHTgt where74 z 4 2 7 2 m2lt 1671 where74 z 4 2 c Minimize D an 7 M16 7 4x2 71 where 72 g x g 2 Minimize D a b Minimize D 2 d Minimize D x 71 7 M16 7 4x2 where 72 g x g 2 13 None of the above 103 12 13 14 15 16 17 18 19 32 Suppose y 7 What is the minimum sum of z and y if z and y are both positive 1 6 Suppose that the sum of z and y is 127 z and y both positive What is the value of x that gives the b g c 3 d 2 e 4 largest possible value of y a 6 W W 8 d v2 e 4 Suppose the product of z and y is 64 and both z and y are positive What is the minimum possible sum of z and y a 9 b 12 c 15 d 16 e 20 Find the area of the rectangle of maximum area with one vertex corner at 07 0 and opposite corner on the ellipse 2 4342 4 1 a 34 b m4 c m4 Let T be the triangle enclosed by the z axis7 the y axis7 and the line y 4 7 2m Find the area of the e m4 largest rectangle with sides parallel to the coordinate axes that can be inscribed in T 2 square units 1 6 square units b 8 square units c 4 square units e 3 square units Let 017 b be the point on the line y 4 7 2x that is closest to the origin 07 0 What is the distance from 071 to 00 Hint Draw a picture a NEs b B s c AME5 d 5x55 e MS5 Related rate problems At 1200 noon a ship sailing due East at 20 miles per hour passes directly North of a lighthouse located on the coast exactly one mile South of the ship How fast is the distance between the ship and the lighthouse increasing at 100 pm 100 200 300 400 500 7 b 7 7 d 7 7 a 101 201 C 301 401 e 501 Water is evaporating at a rate of 5 cubic feet per day from a cylindrical tank The circular base of the tank parallel to the ground has a radius of 4 feet How fast is the depth of the water decreasing when the tank is half full measured in feet per day a c 1 1 1 1 m E d 87r e E 104 20 21 22 23 24 25 26 A triangle has a base of length 5 on the z axis The altitude of the triangle is increasing at a rate of 3 units per second How fast is the area of the triangle increasing when the area of the triangle equals 14 square units a 475 35 25 Q 2 L5 b C d e 2 A train travels along a straight track at a constant speed of 50 miles per hour A straight road intersects the track at right angles and a truck is parked on the road one mile from the track How fast is the train traveling away from the truck when the train is 3 miles past the intersection 31 10110 15110 a 2010 d 5110 13 205 Water is evaporating from a pool at a constant rate The area of the pool is 5000 square feet Assume the sides of the pool drop straight down perpendicular from the edge The water in the pool drops 5 feet in one day How fast is the water evaporating in cubic feet per day 2500 A point moves along the line y 4 3x so that the y coordinate of the point increases at a constant rate a 2000 c 3000 d 3500 e 4000 of 2 units per second How fast is the x coordinate of the point increasing a 23 b 1 c 32 d 2 e 3 Two trains leave a station at the same time One travels north on a track at 30 miles per hour The second travels east on a track at 40 miles per hour How fast are the trains travelling away from each other in miles per hour when the northbound train is 60 miles from the station 50 miles per hour 13 50x5 miles per hour a 60 miles per hour d b 40 miles per hour 130 miles per hour A sandbox with square base is being lled with sand at the rate of 9 cubic feet per minute The sandbox is 9 feet long and 9 ve wide How fast is the level of sand in the sandbox rising a 35 feet per minute d b 29 feet per minute c 25 feet per minute 19 feet per minute 13 15 feet per minute The length of the horizontal side of a rectangle is increasing at a rate of 3 inches per minute Suppose the instantaneous rate of change of the area of the rectangle equals zero at the instant that the length of the horizontal side is 2 inches and the length of the vertical side is 5 inches How fast is the length of the vertical side decreasing at this instant a 132 b 152 a 172 d 192 13 212 105 27 28 29 30 31 32 33 34 At 1200 noon a boat is 15 miles due north of a lighthouse The boat is moving east at 20 miles per hour How fast is the distance from the boat to the lighthouse increasing one hour later b 13 miles per hour c 128 miles per hour 1 25 miles per hour 16 miles per hour a 20 miles per hour The relationship between degrees Celsius7 C7 and degrees Fahrenheit7 F is 9 F 32 30 Suppose you heat water at a constant rate of 9 degrees F per minute How fast are you heating the water measured in degrees C per minute a b 1 5 c g d 59 3 95 A rectangular pool of dimensions 10 ft x 20 ft x 6 ft length x width x height is lled with water at a rate of 15 ft3min How fast is the level of the water rising when the pool is half full 00075 ftmin b 00085 ftmin c 0006 ftmin d 0005 ftmin 13 0009 ftmin Two trains leave a station at the same time One travels north on a track at 50 mph The second travels east on a track at 120 miles per hour How fast are they traveling away from one another in miles per hour when the northbound train is 100 miles from the station a 130 c 01 Two trains leave a station at the same time One train travels east at a speed of 15 miles per hour The 125 1324 135 13 1386 other train travels north at a speed of 20 miles per hour How fast in miles per hour are the trains traveling away from each other when the eastbound train is 30 miles from the station 25 lt8 A stone is dropped into a pond and causes a circular ripple If the radius of the circle increases at a rate b 35 c 40 d 50 100 of 025 ftsec7 how fast does the area increase in ftZsec when the radius equals 04 ft 027139 An expanding rectangle has its length always equal to three times its width The area is increasing at b 045 c 085 d 0165 e 0645 a rate of 42 square feet per minute At what rate in feet per minute is the width increasing when the width is 4 feet 175 a 150 An expanding rectangle has its length always equal to twice its width The area is increasing at a rate of c 200 d 225 e 275 40 square feet per minute At what rate is the width increasing when the width is 2 feet a 10 b s c 6 d 5 e 4 106 Computing some derivatives Chapter 4 Practicereview problems The collection of problems listed below contains questions taken from previous MA123 exams 1l 2l 3 4 5 Computing so me derivatives If v 20 32 then M a 2xh b 2z3h c 2z3h d 2x3 e 2z8h If mg 32 nd fmhfx h a 2x2h12 b 2zh72 2zh12 c 2z2h2 e 2zh712 lf Ft Hi1 then the slope of the tangent line to the graph of Ft at t 2 is a 13 b 12 C 0 d 13 e 12 Suppose that fz m i 3 Find 2 72 2 a z 32 b hz 32 C x h 3z 3 72 2 zh3z3 e 32 Evaluate the limit haO h where a 16 b 15 c 14 d 13 3 12 lf Fs m nd F 1 1 1 3 3 5 b m C E d m 3 5 7 8 9 10 11 The equation of the tangent line to the graph of w xt 1 at t 3 is a w213t73 b w214t73 d w3 16 2578 c w 3 14t 7 3 e w 3 13t 7 8 Approximating some derivatives Suppose x 2m Use the de nition of the derivative and a calculator to nd the approximate value of the derivative of f at z 4 Select the answer that best approximates the derivative 31 43 b 53 c 63 d 93 e 113 Suppose x logz where logm denotes the base 10 logarithm Use the de nition of the derivative and a calculator to nd the approximate value of the derivative of f at z 2 Select the answer that best approximates the derivative 31 102 b 145 c 180 d 217 e 378 Let 105 2 Use a calculator and the de nition of the derivative as a limit to estimate the value of f 1 1386 Let x lnm 2 1 Use the limit de nition of the derivative and a calculator to estimate f 4 b 2296 c 4768 d 5545 13 8047 Your answer should be correct to four decimal places 01667 b 02500 c 01429 d 02000 13 10000 Algebra Review Chapter 1 Practicereview problems The collection of problems listed below contains questions taken from previous MA123 exams 1 A rectangular solid has edges of lengths 4 ft7 5 ft7 and 8 ft Suppose we double the length of M of the sides What is the volume of the new rectangular solid a 80 ft3 b 160 ft3 c 320 ft3 d 640 ft3 e 1280 ft3 2 If we simplify the expression a5b8 7 a4b7 aab22 to the form anQ 7 LRbS7 then R a 0 b 1 C 2 d 3 e 4 Domain and inequalities 3 Suppose 1m2 7 5 What is the largest value of A such that is de ned on the interval 710 A 7 a 45 b 71 c 0 d 1 e V5 4 What is the largest value of A such that the function ft is de ned on the interval 07 A where 1 m 5 8 a 1 b 2 C 3 d 4 e 5 5 The inequality 2 z 7 2 gt 0 is equivalent to a mlt72orzgt1 b 72ltmandzlt1 c m72orz1 d zlt7 orzgt1 e m7 2andz1 6 The inequality 2 2x 7 15 g 0 can be rewritten in the form a m 730rz25 b 75 m 3 c m2152 d m 750rz23 e 73 m 5 7 The values of z satisfying the inequality 2 5x 7 24 lt 0 are a z lt 78 and z gt 3 b x lt 8 and z gt 73 c Cannot be determined d 73ltzlt8 e 78ltzlt3 8 Suppose xmz 7 2x 7 3 What is the largest value of A such that is de ned on the interval 757 A a 4 b 3 C 2 d 1 e 0 9 Find the domain of the function 1 F s 82 7 1 a All 3 such that either 700 lt s lt 71 or 1 lt s lt 00 b All 3 such that 71 lt 3 lt1 V c All 3 such that 700 lt s lt 00 d All ssuch that either 7oolt slt1or1lt slt 00 V 13 All 3 such that 0 lt s lt 1 10 The inequality z 7 1 gt 2 is equivalent to a mlt2ormgt1 b 2ltmandzlt1 c mgt30rzlt71 d mgt30rmlt1 e zgt2andzgt1 Composition of functions 11 If hz xmz 1 and gz 2x 7 1 then hgz a 4m b m c 2W4 d V4m2 7 4m 1 8 2m2 71 12 If my 352 and 93 71 then hg3 a 110 b 15 c 12 1 13 13 unde ned 13 If Ps 52 1 and Rt t7 27 then PRm a x274z5 b x24z3 C 271 d 902 5 393 952 1 2 14 If ut t 7 then m if 11m a 7 b 1 c 377 d 0 e m um um um pa um pm pm pm If at t 7 4 nd a function bt such that abt t a btt b bt4 c btt74 d btt4 e bt47t lf Rt t 2 and RQt t then 0 C205 275 b C205 75 C C205 t i 2 d CW 75 2 393 C205 2 75 If i 1 h i if ut 7 m t en 7 m 1 11m 7 a 190 1 b 190 1 C 190 1 d 190 1 393 90 Lines and parabolas An equation of a line through the points 3 5 and 8 7 in the s t plane is a b m 6W7 s65t75 t65s75 e c 2t65s75 s 5 6t 7 5 If the equation of the line through the points 30 and 2 1 is written as y A Bx 7 2 then a A1andB71 b A3andB71 c A71andB3 d A71andB1 e A5andB71 Find A and B such that the equation of the line through 13 and 2 7 can be written as y A Bx 7 2 a A4andB7 b A7andB4 d A7andB2 c A3andB1 13 This is not possible The line de ned by the equation y 2 Am 7 1 passes through the point 5 3 The slope of the line is a 0 b 14 C 12 d 2 e 4 If the line given by s A Bt 7 1 is perpendicular to the line 3 t and contains the point 16 in the t s plane then ALB4 m AlB1 wAlB4 ALB6 AampBA 23 24 25 26 The following table gives the Median Weekly Earnings in dollars for wage and salary workers in the US from 2000 to 2006 Data from the Bureau of Labor Statistics Use the given values from the table for the two years 2000 and 2005 to express earning amount as a linear function of time in years Let A denote amount earned in dollarsand let t denote time in years l Yr l 2000 l 2001 l 2002 l 2003 l 2004 l 2005 l 2006 l l Amount l 576 596 l 608 l 620 l 640 l 651 l 671 a A 576 12t 7 2000 d A 576 15t 7 2000 b A 576 13t 7 2000 c A 576 14t 7 2000 e A 576 16t 7 2000 Find an equation for the line that is perpendicular to the line y 2x and contains the point 71 5 c y5lm1 a y52z1 2 b y52z71 d y572ltz1gt e 5757671 Suppose the parabola given by the equation y A Bm 1 Cm 1m 7 2 contains the points 713 26 and 37 What is the value of B a 72 b 71 c 0 d 1 e 2 Systems of quadratic equations The area of a right triangle is 8 The sum of the lengths of the two sides adjacent to the right angle of the triangle is 10 What is the length of the hypotenuse of the triangle a 10 b m c 8 d m e 6 Exponential and logarithmic functions Supplement Practicereview problems The collection of problems listed below contains questions taken from previous MA123 exams 1 2 Bl 4 5 Suppose that fz lngz Assume that 95 3 and g 5 4 Find f 5 a 53 b 35 c 43 1 34 13 Does not exist Suppose that fz 59m Assume that 95 3 and g 5 5 Find f 5 a 554 553 c 355 d 453 e 354 Find 10471 Where 1035 a 5 2 b 7572 25 d 725 e 7571 Find the equation of the tangent line to the graph of fz zzem at z 1 y3em72e e y25xie a y3em25 b y25z3e d y 25m73e Suppose that Qt Q05quot Assume that 07 5 lies on the graph of Qt Assume also that the slope of the tangent line to the graph of Qt at t 0 is 10 Find r a 1 2 c e d 5 e 10 The number of bacteria in a sample 25 hours from now is given by Qt Q05 lf Q0 107 000 and Q 0 207 0007 how many bacteria are there in 4 hours a 1000056 1000058 c 10000510 d 10000512 13 10000516 How many years will it take an investment to triple in value if the interest rate is 4 compounded continuously ln3 ln04 ln3 w b 75 C T d 393 T 3 m

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