Application of derivatives Questions and Answers

Curriculum 2 0 Precalculus Unit 3 Topic 1 SLT Evaluating Trigonometric Functions B x y TC G 6 7 Determine the exact values of cos A 1 0 5 Use the diagram above to approximate the values of cos TC 6 Use a calculator to find the values of cos and sin Name Aman Date cosand sin

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Question Consider the region bounded below by f x 0 8e0 9x above by g x 12 on the left by x 5 and on the right by x 9 x 0 If this region is rotated about the x axis what is the volume of the resulting solid Select the correct answer below Oxf 0 8e09 12 O 0 OAL JT 12 x 9 O JT 2 5 0 8e0 x dx 144 0 64e 8x dx X 9 5 0 64e 8x dx 144 dx

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1 Let f x be the mounthly charge in dollars for using x cubic feet of water Explain the meaing of The statement fC1000 55

6 Consider f x x 2 41 a write an decreasins C of X b find all of the mtercepts of f Curite answers as orderd pairs con cave down on which fis where d write two Statements to describe The long ron end behavior of f

Find y if y 3 X

2 f x tan 3x for 0 x 2

For questions 5 6 use the graph of f x shown below 5 List and explain each type of discontinuity and where it is located 3 5 1 b 8 0 1 2 1 6 Determine whether f x is continuous on the given interval If not explain why a 8 x 4 6 7 C 8 0 8 9

Select True or False for each statement You must get all of the answers correct to receive c 2 2 V V V V In 23 In 25 In 2 In a 5c 5c ln a In x a ln x In 64 In 6 4

Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false You must get all of the answers correct to receive credit 1 In ab ln a ln b 2 lnx lny 3 ln x y ln x ln y 4 In a b ln a

f x 2 ln sin

A hemispherical bowl of radius 7 inches is filled to a depth of h inches where Oshs7 Find the volume of water in the bowl as a function of h Check the special cases h 0 and h 7 The volume of the water in terms of his V 7 686 3 Type an exact answer using as needed 7h in h 3 in Type an exact answer using as needed What is the volume of water when h 0 V 0 0 in Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression What is the volume of water when h 7 C

Find y if y 9 4 x

Find y if y 2x5 3x4 9 2

Due Wednesday by 11 59pm Submitting an external tool Points 100 CURRENT OBJECTIVE Use the washer method to find the volume of a shifted solid of revolution Question Define R as the region bounded above by the graph of f x 3x 1 below by the x axis on the left by x 1 and on the right by x 4 What is the volum the coli

Use the washer method to find the volume of a shifted solid of revolution Question Define R as the region bounded above by the graph of f x 3x 1 below by the x axis on the left by x 1 and on the right by x 4 What is the volume of the solid obtained by rotating R about the line y 4 Enter an exact value in terms of

a Write and simplify the integral that gives the arc length of the following curve on the give b If necessary use technology to evaluate or approximate the integral y cos 3x for 0 SXS a Which of the following is a simplified form of the integral OA 1 3 sin 3x dx OB 1 9 sin 3x dx S 1 3c0 OD 1 9 cos 3x dx O C 3 cos 3x dx b The arc length of the curve is approximately Bound to three decimal

Evaluate lim 2x sin OTH 5x

Suppose f x has the following values x f x 912 lim f x appears to be 06 69 8 Undefined 6 4 9 7 7 lim f x appears to be 2 5 8 9 Undefined 06 4 99 7 97 lim f x appears to be 2 5 8 09 Undefined 4 999 7 997 5 6 5 001 5 01 9 03 9 003 5 1 9 3

Evaluate the following limits If needed enter oo for co and oo for oo and DNE if the limit exist 4x lim 2 3 x 3 4x lim 2 3 x 3 4x lim z 3 x 3

Rationalize the denominator 3 10 7 A Written Examp

Simplify 6 6 20 6 3 62 56 24 b 5 Written Example

and direction What is the velocity vector 6 j 4t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t 0 as a product v t 3 i 16t j 4 k What is the acceleration vector a t 0 i 16 j 0 k Write the velocity vector at t 0 as a product of the speed and direction v 0 5 3 0 4 k Type exact answers using radicals as needed

Question Consider the region bounded by the graphs of f x 0 8e0 6x g x 10 x 4 3 and Select the correct answer below x 2 If this region is rotated about the x axis what is the volume of the resulting solid 2 0T13 O 3 2 07 Off 0 8e0 6x 3 0 64e 2x 2 0 O 3 2 10 x 4 O 2 100 T L 0 8e0 6x dx 10 x 4 X 2 100 x 4 0 64e 2x dx dx dx

Use the graph of y f x to graph the function y f 4 x 1 Label the points on the sketch For each point on the graph of f x compute the location of the corresponding point on the graph of y f 4 x 1 Point on f x 3 4 Point on f 4 x 1 Type an ordered pair 10 10 16 f x

blem 1 20 Points Use the graph of each function to identify its domain and its range N f x x 2 f x 4x 3 5 0 12 10 7 2

Problems 19 26 refer to right triangle ABC with C 90 Write all answers that are angles to the nearest tenth of a degree 19 If A 40 and c 16 cm find a 20 If A 20 and c 42 cm find a 21 If B 20 and b 12 ft find a 22 If B 20 and a 35 ft find b

wire 18 meters long is to be cut into two pieces One ece will be shaped as a square and the other piece ill be shaped as a circle see the figure Answer arts a c 18 m 4x 18 4x O a Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the square A x Simplify your answer b What is the domain of A domain Type interval notation c Graph A A x For what value of x is A smallest meters Use the MINIMUM function Round to two decimal places

Let P be a point on the graph of y x The tangent line at P will intersect the graph at the point Q a If P has x coordinate a then what is the equation of the tangent line at P y x b The line in a will intersect the curve y x in two places one of these is P and the other point is Hint This part involves factoring a cubic in x In general this may be difficult However in this case since we know point P is an intersection point x a must be a solution so x a must be a factor You should now be able to f c The ratio of the slope of the curve at Q to the slope at P is

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

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

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

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

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

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

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

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

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

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

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

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

oes not exist explain why 7 6 5 3 2 1 3 4 10 to S in the answer box if necessary A f 2 Type an integer or a fra B The value of f 2 is unde

Use the disk method to find the volume of a solid of revolution around the x axis with other functions Question A solid of revolution is generated by rotating the region between the x axis and the graphs of f x 2e2x x 0 and x ln 9 about the x axis Using the disk method what is the volume of the solid Enter your answer in terms of Sorry that s incorrect Try again 407

Use the graph of h in the given figure to find the following values or state that they do not exist a h 2 b lim h x X 2 c h 6 d lim h x x 6 e lim h x X 7 h 2 5 Type an integer or a decimal B h 2 is undefined a Find h 2 Select the correct choice and if necessary fill in the answer box to complete your choice 10 8 OA lim h x X 2 Type an integer or a decimal OB The limit does not exist ay h x b Find lim h x Select the correct choice and if necessary fill in the answer box to complete your choice X 2 X 10 E L

CURRENT OBJECTIVE Use the disk method to find the volume of a solid of revolution around the x axis with other functions Question A solid of revolution is generated by rotating the region between the x axis and the graphs of f x 3 2sin x x 37 and x about the x axis Using the disk method what is the volume of the solid Enter your answer in terms of Provide your answer below