Calculus Questions

Use a graph off to estimate lim f x or to show that the limit does not exist Evaluate f x near x a to support you X 8 conjecture f x 1 cos 5x 5 5 x 1 Each graph is displayed in a 1 3 by 0 5 window Use the graphing utility to estimate lim f x Select the correct choice below and if necessary fill in the answer box X 1 complete your choice a 1 A The limit appears to be approximately 2 5 Round to the nearest tenth as needed B The limit does not exist X b Evaluate f x for values of x near 1 to support your conjecture 0 9 0 99 0 999 Round to six decimal places as needed 1 001 1 01 1 1

unit consists of 100 chips Let R x denote the revenue in thousands of dollars generated from the production of x units of computer chips p Next question a Represent the following statement by equations involving R or R When 1700 chips are produced per day the revenue is 29 000 and the marginal revenue is 65 per b If the marginal cost of producing 1700 chips is 1 3 per chip what is the marginal profit at this production level a Choose correct equations below A R 1700 29 R 1700 065 B R 17 29 R 17 065 C R 17 29 000 R 17 65 b When 1700 chips are produced the marginal profit is Points 0 5 of 1 dollars per chip

dR Suppose a company finds that the revenue R generated by spending x dollars on advertising is given by R 1000 93x 03x for 0 x 1500 Find dx dR dx Simplify your answer Ix 1300

The third derivative of a function f x is the derivative of the second derivative f x and is denoted by f x Compute f x for the following function f x 5x5 4x4 7x f x

Let C x be the cost in dollars of manufacturing x bicycles per day in a certain factory If C 60 7000 and C 60 65 estimate the cost of manufacturing 61 bicycles per day The cost of manufacturing 61 bicycles per day would be approximately

If s Tx 9xG T find a a Tx 9xG T dx ds dx b ds dG and c ds dT

Compute the following dx d dx 2x 5 2 1x 5 2x 57 x05 2x 5 2 Simplify your answ

Find the first and second derivatives v 3t 5t 15

Compute the following 2x x 7x 3 dx 2x x 7x 3 1x 2 lx 2 Simplify your and

A solid has a cross section that is a square with side length x where x is the location on the x axis where the cross sectic is located Find the volume of the solid if it extends on the x axis along the interval 0 4 Provide your answer below

the first and second derivati y 4x 1 4 01

Find the following f x x 2 g x x 2 a f g x b f g x c fg x d f g x What is the domain of f g Enter your answer using interval notation

a Let A x denote the number in hundreds of computers sold when x thousand dollars is spent on advertising Represent the following statement by equations involving A or A When fourteen thousand dollars were spent on adve number of computers sold was 1800 and it was rising at the rate of 30 computers for each 1000 dollars spent on advertising b Estimate the number of computers that will be sold if 17 000 is spent on advertising a Which equations best represent the statement OA A 14 0 3 and A 14 18 B A 14 18 and A 14 0 3 OC A 14 0 3 and A 14 18 D A 14 18 and A 14 0 3 b The number of computers that will be sold if 17 000 is spent on advertising is approximately CILM

Find the following Simplify your answers completely f x 6x 7 g x 4 x a f g x b f g x c fg x d f g x What is the domain of f g Enter your answer using interval notation

Evaluate the function for f x x 3 and g x x 2 f g 0 f g 0

Consider the functions below f x x 1 g x x 1 Find each of the following if possible If it is not possible enter NONE a fog b g f c fog 0

Problem 4 Find the volume of the solid of revolution bounded by y 1 x 1 x 0 x 2 and the x axis and rotated around the y axis

Evaluate the function for f x x 3 and g x x 2 f g 9 f g 9

figure the shape of a small oam Assum the water level is at the top of the dam and that the shape of the dam is an isosceles triangle find the total force on the face of the dam Use 1000 kg m for the density of water and 9 8 m s for the acceleration due to gravity 4 A WY Y ELE Find the width function wty for each value of y on the face of the dam Place the origin at the vertex at the bottom of the dam with the y axis extending up in the vertical direction to the center of the top the dam and the x axis going in the horizontal direction Choose the correct answer OA Jay dy 27 OB w y y 4m 27 OD wy Y 27 m O c w y Set up the integral that should be used to find the total force on the face of the dam Use increasing limits of integration Select the correct choice and fill in the answer boxes to complete your choice Type exact answers

The figure to the right shows the shape and dimensions of a small dam Assuming the water level is at the top of the dam find the total force on the face of the dam Use 1000 kg m for the density of water and 9 8 m s for the acceleration due to gravity OA W y y 9 OB wy 18y c w y y 9 OA dy OB CA OD w y 81y Set up the integral that should be used to find the total force on the face of the dam Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers dx 12 m 6m 9m

OBJECT Use the disk method to find the volume of a solid of revolution around the y axis with a function defined as a function of y Question Define R as the region bounded by the graphs of g y 4y 2y 4 y 1 y 0 and the y axis Using the disk method what is the volume of the solid of revolution generated by rotating R about the y axis Enter your answer in terms of J Provide your answer below

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions If an answer does not exist enter D Round your answers to one decimal place Below enter your answers so that A is smaller than A a 22 c 17 C 22 A 38 3 48 38 3 38 3 b x x 0 0 B b

To find the distance across a river a surveyor chooses points A and B which are x 200 ft apart on one side of the river see figure She then chooses a reference point C on the opposite side of the river and finds that BAC 82 and ABC 52 Approximate the distance from A to C Round your answer to the nearest foot 219 A x ft 82 520 B

The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B which are 65 apart When the satellite is on one side of the two stations the angles of elevation at A and B are measured to be 87 0 and 84 2 respectively Round your answers to the nearest mile 87 0 84 2 B a How far is the satellite from station A 1324 mi b How high is the satellite above the ground

Bonus 1 10th deri f x x 10 10 x

Problem 2 Use the disk method to find the volume of the solid of revolution bounded by the graphs of f x x 2x 3 x 0 x 3 and the x axis and rotated about the x axis

lim 8 X 6 X 6 a Calculate f x for each value of x in the following table 5 99 X 5 999 5 9999 0 0 6 001 6 0001 36 x 0 x 6 Type integers or decimals rounded to four decimal places as needed X a Calculate f x for each value of x in the following table b Make a conjecture about the vi 36 x 6 5 9 0 6 1 6 01

For the function f x 9x x make a table of slopes of secant lines and make a conjecture about the slope of t tangent line at x 1 Complete the table Round the final answer to three decimal places as needed Round all intermediate values to four decimal place needed Interval 1 2 1 1 5 1 1 1 1 1 01 1 1 001 Slope of secant line 62 41 75 28 79 26 27 26 An accurate conjecture for the slope of the tangent line at x 1 is Round to the nearest integer as needed

The position of an object moving vertically along a line is given by the function s t 4 9t 32t 17 Find the velocity of the object over the following intervals a 0 4 b 0 3 c 0 2 d 0 h where h 0 is a real number a The average velocity on the interval 0 4 is 12 4 Simplify your answer b The average velocity on the interval 0 3 is 17 3 Simplify your answer Points 0 25 of 1 c The average velocity on the interval 0 2 is Simplify your answer

on of an object moving vertically along a line is given by the function s t 4 91 32t 17 F the object over the following intervals b 0 3 c 0 2 d 0 h where h 0 is a real number erage velocity on the interval 0 4 is your answer COLO

e function f x 9x x make a table of slopes of secant lines and make a conjecture about the slope ant line at x 1 plete the table nd the final answer to three decimal places as needed Round all intermediate values to four decimal pl ed Interval 1 2 Slope of secant line

graph off and use it to make a conjecture about the values of f a lim f x lim f x and lim f x or s X a x a X 3 not exist x 1 a 1 conjecture about the value of f 1 Select the correct choice below and if necessary fill in the answer bo e your choice The value of f 1 Type an integer or a decimal The value of f 1 does not exist conjecture about the value of lim f x Select the correct choice below and if necessary fill in the answ X 1 Olete your choice The limit exists lim X 1 X 1 The limit does not exist Type an integer or a decimal

The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B whic 65 mi apart When the satellite is on one side of the two stations the angles of elevation at A and B are measured to 87 0 and 84 2 respectively Round your answers to the nearest mile 87 0 84 2 B a How far is the satellite from station 47 1324 mi 0 How high is the satellite above the ground

Problem 6 Example Determine the area between the curves y x y 2 x and x x20 by integrating a with respect to y b with respect to x

volume of a solid of revolution around the y axis with a function defined as a function of y Question efine R as the region bounded by the graphs f g y 2y 2y 2 y 1 y 0 and ne y axis Using the disk method what is the olume of the solid of revolution generated by otating R about the y axis Enter your answer terms of

se the graph of h in the given figure to find the following values or state that they do t exist h 2 lim h x X 2 h 6 lim h x X 6 lim h x X 7 Type an integer or a decimal Find h 6 Select the correct choice and if necessary fill in the answer box to complete your choice A h 6 B h 6 is undefined Alim h x X 6 B d Find lim h x Select the correct choice and if necessary fill in the answer box to complete your choice X 6 Type an integer or a decimal Rich x OB The limit does not exist

graph of g to estimate lim g x or to show that the limit does not exist Evaluate g x near x a to suppe x a ture e11x11x 1 a 0 The limit appears to be approximately 60 5 Round to the nearest tenth as needed The limit does not exist graph is displayed in a 1 1 by 0 80 window the graphing utility to estimate lim g x Select the correct choice below and if necessary fill i x 0 lete your choice 0 1 Date g x for values of x near 0 to support your conjecture 0 01 0 001 0 0 001 0 01 x 0 1 in the answe

mph of f to complete parts a through 1 If a limit ist explain why 5 answer box if necessary A 1 2 4 Type an integer or a fract OB The value of f 2 is undefir b Find lim f x Select the corre X 2 and fill in the answer box if necess A lim f x 3 X 2 Type an integer or a fractic B The limit does not exist bec defined for all x 2 c Find lim f x Select the correct X 2 and fill in the answer box if necessa OA lim f x X 2 Type an integer or a fractions B The limit does not exist becam defined for all x 2

PROBLEM 5 Find values 4p lim 2n 9n 3 9n 1 11448 3p lim 21 n 2 1 1 21 3p n 2 2 2 2 x 4 x 2 Csin

Let f x 9 X 3 X 3 lim x x 3 X a Calculate f x for each value of x in the following table 2 999 2 9999 5 99 5 999 5 9999 3 01 3 001 3 0001 f x 6 01 6 001 6 0001 x 3 Type integers or decimals rounded to four decimal places as needed f x lim 9 a Calculate f x for each value of x in the following table b Make a conjecture about the valu X x 9 x 3 2 9 5 9 3 1 6 1 2 99 b Make a conjecture about the value of lim X 9 X 3 X 3 Type an integer or a decimal

etch a possible graph of a function that satisfies the conditions below f 0 3 lim f x 0 lim f x 3 X 0 X 0 hoose the correct graph below A B Q O C Q O D

Consider the behavior of values of sin near x 0 The graph of shown to the right Answer sin X parts a b and c a Create a table of values of sin sin X OB The value of sin 2 H O C The value of sin for the following values of x Complete the following table approaches 0 2 3x is constant IMA 2 5x Describe the pattern of values that can be observed in the table Choose the correct answer below A The value alternates between 1 and 1 2 7x Type an integer or a fraction SP N 9x talk 5 1 E

Determine the volume of a solid by integrating a cross section with a square or rectangle Question The solid S has a base described by the circle x y 9 Each cross section perpendicular to the x axis and the base is a square with one side in the base of the solid What is the volume of S Give the exact volume as your answer Do not include any units Provide your answer below

the graph of f in the figure to find the following values or state that they do not t If a limit does not exist explain why 3 im f x 3 im f x im f x 3 D A lim f x x 3 B B The limit does not exist because f is not defined for all x near 3 with x 3 Find the limit Select the correct choice and if necessary fill in the answer box within your choice Type an integer or a fraction OB The limit does not exist because lim f x lim f x X 3 X 3 z A lim f x 0 Type an integer or a fraction X 3 B The limit does not exist because f is not defined for all x near 3 with x 3 Find the limit Select the correct choice and if necessary fill in the answer box within your choice ME

jecture f x 1 cos 5x 5 5 x 1 x a a 1 e a graphing utility to graph f Select the correct graph below Q A n ch graph is displayed in a 1 3 by 0 5 window B Wh C EXIST O D Ins m

e graph of f to complete parts a through I If a limit ot exist explain why 7 6 5 4 3 2 4 3 Q CCT a Find f 2 Select the correct choice fill in the answer box if necessary A f 2 4 Type an integer or a fraction B The value of f 2 is undefined b Find lim f x Select the correct che X 2 and fill in the answer box if necessary OA lim f x X 2 Type an integer or a fraction OB The limit does not exist because defined for all x 2

near x 0 The graph of shown to the right Answer a b and c Create a table of values of sin sin X H X NIK 2 0 2 3 1 A for the following values of x Complete the following table IMAS 2 2 5t 2 7x 2 9x 2

ketch a graph of f and use it to make a conjecture about the values of f a lim f x lim f x and lim f x or x a x a x a they do not exist f x 1 X 1 A ketch a graph of f Choose the correct graph below 10 a 1 T Q Q S B 10 Q O C 10 A The value of f 1 2 Type an integer or a decimal OB The value of f 1 does not exist Q Q Q D 10 lake a conjecture about the value of f 1 Select the correct choice below and if necessary fill in the answer b Omplete your choice

Use a graph of g to estimate lim g x or to show that the limit does not exist Evaluate g x near x a to supp x a onjecture g x A e11x11x 1 x a 0 Use a graphing utility to graph g Select the correct graph below B C A The limit appears to be approximately Round to the nearest tenth as needed B The limit does not exist Q 17 D LULU Each graph is displayed in a 1 1 by 0 80 window Use the graphing utility to estimate lim g x Select the correct choice below and if necessary fill in the answ X 0 complete your choice

the graph of f in the figure to find the following values or state that they do exist If a limit does not exist explain why 3 im f x im f x m f x 3 3 ind the function value Select the correct choice and if necessary fill in the answer box within your A f 3 1 Type an integer or a fraction B The value of f 3 is undefined ind the limit Select the correct choice and if necessary fill in the answer box within your choice Alim f x 1 Type an integer or a fraction X 3 B The limit does not exist because f is not defined for all x near 3 with x 3 ind the limit Select the correct choice and if necessary fill in the answer box within your choice A lim f x Type an integer or a fraction X 3 B The limit does not exist because f is not defined for all x near 3 with x 3