Calculus Questions

Hal function is defined for positive integers as n n n 1 n 2 3 2 1 a Make a table of the factorial function for n 1 2 3 4 5 b Graph these data points and then connect them with a smooth curve c What is the least value of n for which n 106 a Complete the table below 1 n f n n 2 3 4 5

Determine the function f represented by the graph of the line y f x in the figure f x 10 10 5 5 0 2 10 5 10 5 6

Evaluate each expression using the graphs of y f x and y g x shown below a f g 7 b g f 3 c f g 4 f g f g 2 d g f 5 e f f 9 a f g 7 Type a whole number YA 10 9 8 7 6 5 4 2 12 3 4 5 y f x y g x 6 7 89 Xx

D If x 3y 3 y what is A 1 3 B dy dx at the point 2 8 5 C D 4 3

Question Watch Video Show Examples A quantity with an initial value of 980 decays exponentially at a rate of 0 3 every 6 decades What is the value of the quantity after 49 decades to the nearest hundredth

Use the table to evaluate the given compositions b g f 2 a h g 2 d g h f 1 g f h g 0 J f f h 0 e f f f 0 h g f h 2 a h g 2 b g f 2 c h h 3 d g h f 1 Simplify your answer e f f f 0 Simplify your answer f h h h 1 Simplify your answer g f h g 0 Simplify your answer h g f h 2 Simplify your answer 1 g g g 2 Simplify your answer j f f h 0 Simplify your answer Simplify your answer Simplify your answer Simplify your answer c h h 3 f h h h 1 I g g g 2 II X f x g x h x 3 2 1 1 0 1 1 3102 3 20 0 3 1 0 1 2 3 2 2 2

9 If f x In x then lim x 3 Im A B e f x f 3 x 3 C In 3 is D nonexistent

Use the graph of y 2 and transformations to sketch the exponential function Determine the domain and range Also determine the y intercept and find the equation of the horizontal asymptote f x 2x 6 Use the graphing tool to graph the function Click to enlarge graph 10 8 10 8 16 2 2 Ay

Use the graph of y 2 and transformations to sketch the exponential function Determine the domain and range Also determine the y intercept and find the equation of the horizontal asymptote f x 2 6 Use the graphing tool to graph the function Click to enlarge 0 10

For time 1 2 1 the position of a particle moving along the x axis is given by p t 7 2 At what time in the interval ISIS 16 is the instantaneous velocity of the particle equal to the average velocity of the particle over the interval Ist 16 A 1 B 121 25 C 2593 D 25

6 Let f be the function defined by f x What is the approximation for f 10 found by using the line tangent to the graph of fat the point 8 2 A B 25 12 C 13 D 7

7 The number of gallons of water in a storage tank at time t in minutes is modeled by w 1 25 1 for 0 STS 5 At what rate in gallons per minute is the amount of water in the tank changing at time t 3 minutes A 66 B 16 C 3 D 6

3 A particle moves along the y axis so that at time t 2 0 its position is given by y t t 41 41 3 Which of the following statements describes the motion of the particle at time 1 1 A The particle is moving down the y axis with decreasing velocity B The particle is moving down the y axis with increasing velocity C The particle is moving up the y axis with decreasing velocity D The particle is moving up the y axis with increasing velocity

4 At time 0 a storage tank is empty and begins filling with water For t 0 hours the depth of the water in the tank is increasing at a rate of W 1 feet per hour Which of the following is the best interpretation of the statement W 2 3 A Two hours after the tank begins filling with water the depth of the water is increasing at a rate greater than 3 feet per hour B Over the first two hours after the tank begins filling with water the depth of the water is always increasing at a rate greater than 3 feet per hour Two hours after the tank begins filling with water the rate at which the depth of the water is rising is increasing at a rate greater than 3 feet per hour per hour C D Over the first two hours after the tank begins filling with water the rate at which the depth of the water is rising is always increasing at a rate greater than 3 feet per hour per hour

the answer sheet You may write show work on 1 Which of the following is an equation of the line tangent to the graph of y cos x at x A y x 2 B y x C y x D y x

A 1 lim 0 sin x B C 0 D nonexistent

Evaluate each expression using the graphs of y f x and y g x hown below b g f 3 c f g 4 e f f 3 f g f g 2 a f g 8 d g f 7 a f g 8 Type a whole number YA 10 9 8 7 6 5 3 2 0 1 2 3 y f x y g x 4 5 6 7 8 9

Let a and b be positive numbers such that the function ax x 2 0 x 2 f x bx a Then write the value of f 8 3 x 2 is differentiable for all x 0

v t 1 For 0 St 11 seconds a particle moves along the x axis the velocity of the particle in meters per second can be modeled by the graph above which consists of 4 line segments a Find v 3 and a 3 b At what time s t does the particle change direction Give a reason for your answer c Find the average acceleration of the particle from t 2 seconds to t 7 seconds Include units of measure d Att 9 is the particle speeding up or slowing down Give a reason for your answer v t 2 4t 5 It is known that x 9 2 Is the particle moving towards or away from the origin reasoning at times a

Given that lim f x 4 and lim g x 5 find the limit lim lim 1 x sin x f x g x lim 1 lim f x 2 g x g x

5 0 4 Points DETAILS 50 feet The cup on the 9th hole of a golf course is located dead center in the middle of a circular green which is 40 feet in radius Your ball is located as in the picture below The ball follows a straight line path and exits the green at the right most edge Assume the ball travels 9 ft sec Introduce coordinates so that the cup is the origin of an xy coordinate system Provide numerical answers below with two decimal places of accuracy ball green PREVIOUS ANSWERS 40 feet 9 smallest x coordinate 40 20 Lav cup ball path rough a The x coordinate of the position where the ball enters the green will be 40 The ball will exit the green exactly 8 89 c Suppose that L is a line tangent to the boundary of the golf green and parallel to the path of the ball Let Q be the point where the line is tangent to the circle Notice that there are two possible positions for Q Find the possible x coordinates of Q MY NOTES X seconds after it is hit X

Identify the transformations of the graph of f x 1 x Then graph g x on the same coordinate plane as the graph of f x by applying the transformations on the asymptotic x 0 and y 0 to the reference points 1 1 and 1 1 Also state the domain and range of g x using inequalities g x 2 1

A ladder 25 feet long is leaning against the wall of a building Initially the foot of the ladder is 7 feet from the wall The foot of the ladder begins to slide at a rate of 2 ft sec causing the top of the ladder to slide down the wall The location of the foot of the ladder its x coordinate at time t seconds is given by x t 7 2t wall Y 1 L 25 ft ladder x t a Find the formula for the location of the top of the ladder the y coordinate as a function of time t The formula for y t 30r ground b The domain of t values for y t ranges from to c Calculate the average velocity of the top of the lander on each of these time intervals correct to three decimal places time interval ave velocity time interval ave velocity 0 2 2 4 8 9 6 8 d Find a time interval a 9 so that the average velocity of the top of the ladder on this time interval is 20 ft sec i e a e Using your work above and this picture of the graph of the function y t given below answer these true false questions Type in the word True or False 25 20 Put your cursor in the box click and a palette will come up to help you enter your symbolic answer

3 x Find the intervals of increasing decreasing for the function f x x e Find the minimum value s of the function f x e 2x The position of a particle as it moves horizontally is described by the given equation S s sint cost 0 t 2 If s is the displacement in metres and t is the time in seconds find the maximum and minimum displacements

An open topped storage box with a square base has a capacity of 5 m The material for the sides cost 1 60 m2 while the material for the bottom costs 2 00 m Find the dimensions that will minimize the cost of making the box A rectangular piece of paper with perimeter 100 cm is to be rolled to form a cylindrical tube Find the dimensions of the paper that will produce a tube with maximum volume

Draw the unit circle and plot the point P 7 2 Observe there are TWO lines tangent to the circle passing through the point P Answer the questions below with 3 decimal places of accuracy L1 12 a The line L is tangent to the unit circle at the point 0 141 0 993 b The tangent line L has equation y c The line L is tangent to the unit circle at the point d The tangent line L has equation y

The cup on the 9th hole of a golf course is located dead center in the middle of a circular green which is 40 feet in radius Your ball is located as in the picture below The ball follows a straight line path and exits the green at the right most edge Assume the ball travels 9 ft sec Introduce coordinates so that the cup is the origin of an xy coordinate system Provide numerical answers below with two decimal places of accuracy 50 feet ball green 40 feet 9 smallest x coordinate largest x coordinate cup ball path rough a The x coordinate of the position where the ball enters the green will be b The ball will exit the green exactly X seconds after it is hit c Suppose that L is a line tangent to the boundary of the golf green and parallel to the path of the ball Let Q be the point where the line is tangent to the circle Notice that there are two possible positions for Q Find the possible x coordinates of Q x X

Find the rational function s asymptotic x intercepts and holes Also sketch the graph of the rational function Then list the domain and range using interval notation Hint x 4 is a difference of two squares a b a b a b f x 3x 6x x 4 7

Determine which of the following graphs is the graph of the given function f x x 5x 6 A C 28 O N 7 10 W B D AN T

A tank holds 3 000 gallons of water which drains from the bottom of the tank in half an hour The values in the table show the volume V of water remaining in the tank in gallons after t minutes t min 5 v gal 2 058 1 320 765 336 81 0 a If P is the point 15 765 on the graph of V find the slopes of the secant lines PQ when Q is the point on the graph with t 5 10 20 25 and 30 Round your answers to one decimal place 5 2 058 10 1 320 20 336 25 81 30 0 Slope 10 15 20 25 30 b Estimate the slope of the tangent line at P by averaging the slopes of the two adjacent secant lines corresponding to the two points closest to P Round your answer to one decimal place

The graph of the quadratic function y 2x24x 1 is pictured below along with the point P 1 7 on the parabola and the tangent line through P A line that is tangent to a parabola does not Intersect the parabola at any other point We can use this fact to find the equation of the tangent line a If m is the slope of the tangent line then using the slope point formula the equation of the tangent line will be y m x b The values of x for which the point x y lies on both the line and the parabola satisfy the quadratic equation 2x bx c 0 where be and c b and c should depend on m c For most values of m the quadratic equation in part b has two solutions or no solutions The value of m for which the quadratic equation has exactly one solution is the slope of the tangent line This value is ma

Consider the circle of radius 10 centered at the origin Provide answers accurate to two decimal places a The equation of the tangent line to the cindle through the Joint 6 has equation yo X b Suppose that L is a tangent line to this cirde which is parallel to the line yar 7 and has a negative intercept Then the point of tangency of t with this cirde is

4 Points DETAILS The cup on the 9th hole of a golf course is located dead center in the middle of a circular green which is 40 feet in radius Your ball is located as in the picture below The ball follows a straight line path and exits the green at the right most edge Assume the ball travels 9 ft sec Introduce coordinates so that the cup is the origin of an xy coordinate system Provide numerical answers below with two decimal places of accuracy 50 feet green ball 40 feet 9 cup smallest x coordinate largest x coordinate ball pathi rough a The x coordinate of the position where the ball enters the green will be MY NOTES seconds after it is hit b The ball will exit the green exactly c Suppose that L is a line tangent to the boundary of the golf green and parallel to the path of the ball Let Q be the point where the line is tangent to the circle Notice that there are two possible positions for Q Find the possible x coordinates of Q

Calculate the composite functions fo g and go f and determine their domains 1 g x x 6 x 1 f g x Domain O x 0 Ox 0 g f x Ox20 O all real numbers Ox 0 Domain 4 O all real numbers Ox 0 Ox 0 Ox20 Ox 0

Find the equation of the tangent line to the function y In x 2 when x 3 Find the equation of the tangent line to the curve y cos x when x Your solution should use radian measure 3 x Find the intervals of increasing decreasing for the function f x x e

1 Find the derivative a y sin x b y sin x c y cos 4x 9 d y cos x 3 e y x sin x 5 f y 5e 5x g y e h y xe I 3x 1 i y ln x 3 j y 5 ln x 4 k y 5 1 y x 2 x 2x 1 m y In sin x n y x In x

Identify the transformations of the graph of f x that produce the graph of the given function g x Then graph g x on the same coordinate plane as the graph of f x by applying the transformations on the asymptotes x 0 and y 0 to the reference points 1 1 and 1 1 Also state the domain and range of g x using inequalities g x 2 1 2 6 5 4 3 4 5

3 Convert the following equation from Cartesian coordinates to Polar coordinates x 4 y 5 41 a r 10r sin 8 8rcos 0 b 7 Brsin 8 10r cos 0 41 8rsin 10rcos 80 10r sin 80 Br cos 0 0 c d e r 10rsin 8 8rcos 8 41 4 Let f x y x 2 Find f x y cos 3x 2y 2xsin 3x 2y x y x y cos 3x 2y 2x sin 3x 2y x x 3 x y cos 3x 2y 2x sin x 2y x y 3 a a b c d e 3 x y cos 3x 2y 2x sin 3x 2y x y x y cos 3x 2y sin x 2 x x2 5 Given f x y 3x y3 2xy Find the critical points and classify them a 0 0 is a saddle point is a local maximum point b 0 0 is a local maximum point is a local minimum point c 0 0 is a saddle point is a local minimum point d 0 0 is a local maximum point is a saddle point e 0 0 is a local minimum point is a saddle point 6 Which of the following integral do we get if we interchange the order of integration Sony dydx f x y dxdy b fff x y dxdy c fff x y dxdy d ff x y dxdy e fff x y dxdy S SVEY

Graph the equation using the slope and the y intercept 10 y 3x 2 Use the graphing tool to graph the line Use the slope and y intercept when drawing the line Click to enlarge graph 20 16 12 8 20 16 12 2 ob 8 4 4 8 12 8 12 16 X 20 Q

Use the graph of f x log 3x to graph the function f x log 3 x 2 4 Use the graphing tool to graph the function Click to enlarge graph

Use the graph of y e and transformations to sketch the exponential function f x e 7 Determine the domain and range Also determine the y intercept and find the equation of the horizontal asymptote 4 Click to enlarge graph I What is the domain of f x e 7 0 Type your answer in interval notation What is the range of f x e 7 Type your answer in interval notation What is the y intercept of f x e 7 Type an integer or a simplified fraction What is the horizontal asymptote of f x e 7 CIL 10 8 6 4 2 10 8 6 2 2 6 Ay 8 40 6 X 10 G E D

21 Left Handed People In a sample of 1000 people 120 are left handed Two unrelated people are selected at random without replacement a Find the probability that both people are left handed b Find the probability that neither person is left handed c Find the probability that at least one of the two people is left handed 22 Light Bulbs Twelve light bulbs are tested to see if they last as long as the manufacturer claims they do Three light bulbs fail the test Two light bulbs are selected at random without replacement a Find the probability that both light bulbs failed the test b Find the probability that both light bulbs passed the test c Find the probability that at least one light bulb failed the test

0 The function f x logx is transformed into the equation f x log x Identify the parameter and the effect the parameter has on the parent function

4 K 2 D C 7 4 A B X Analitik d zlemde verilen ABCD karesinde C 7 4 noktas veriliyor K ve D noktas d do rusu zerinde oldu una g re K noktas n n apsisi a a dakilerden hangisidir A 7 B 6 C 5 D 4 E 3

Does each equation represent exponential decay or exponential growth Drag and drop the choices into the boxes to correctly complete the table Note If an equation is neither exponential growth nor exponential decay do not drag it to the table Exponential Decay Exponential Growth

Find the derivative f ex for each function given be a f x 1 e 1 ex x b f x Inx fex 5x3 Cosx sinx 4

3 Compute the following limit for the functio f x x 2x Lim f 3 h f 3 4 0 h

Evaluate the following limit a lim 2x 3 X 1 X L b lim c lim 818 x 7x 6 X 6 V x 3x 2x 1

Let f x be the following piece wise f x x 1 if 0 x 1 1 x if x 1 Show lim f x does not exist X 1 the function function without grap

The point 1 1 can be represented in polar coordinate as