Application of derivatives Questions and Answers

onsider an object moving along a line with the following velocity and initial position v t 6 3t on 0 4 s 0 0 etermine the position function for t20 using both the antiderivative method and the Fundamental Theorem of Calculus Check for agreement between the two methods to determine the position function for t20 using the antiderivative method first determine how the velocity function and the position function are related Choose the correct answer below YA The position function is the antiderivative of the velocity function OB The position function is the absolute value of the antiderivative of the velocity function OC The position function is the derivative of the velocity function D The velocity function is the antiderivative of the absolute value of the position function Which equation below will correctly give the position function according to the Fundamental Theorem of Calculus t A s t s 0 v x dx 0 b OC s t s 0 v t b OB s t v t dt Sv t d a Sv x dx 0 OD s 0 s t 0

Consider an object moving along a line with the following velocity and initial position v t 6 3t on 0 4 s 0 0 Determine the position function for t20 using both the antiderivative method and the Fundamental Theorem of Calculus Check for agreement between the two methods CHI To determine the position function for t20 using the antiderivative method first determine how the velocity function and the position function are related Choose the correct answer below OA The position function is the antiderivative of the velocity function OB The position function is the absolute value of the antiderivative of the velocity function OC The position function is the derivative of the velocity function D The velocity function is the antiderivative of the absolute value of the position function

Simplify the difference quotient f x x 15x f x f a x a f x f a x a Simplify your answer for the following function

If cos 0 3 5 and 0 0 then tan 0 O 3 4 O 4 5 O 4 3 O 5 4

For what value of t does 2 t 2 3 5 3 t 1 O There is no value of t that works O 1 5 05 O 1 5 0 5 O 3 5

An equation of the line passing through the points 1 3 and 5 2 is given by Oy 19 x 10 6 O none of these Oy x Oy fx 3 28 6 O y 5 fx 1 13 il 6

At noon t 0 Alicia starts walking along a long straight road at 5 mi hr Her velocity decreases according to the function v t 5 t 1 for t 0 At noon Boris also starts walking along the same road with a 1 mi head start on Alicia his velocity is given by u t 3 t 1 for t20 a Find the position functions for Alicia and Boris where s 0 corresponds to Alicia s starting point b When if ever does Alicia overtake Boris

The figure shows the velocity function for motion along a line Assume the motion begins with an initial position of s 0 0 Determine the following a The displacement between t 0 and t 10 b The distance traveled between t 0 and t 10 c The position at t 10 4 2 0 2 t G

Consider the graph shown in the figure which gives the velocity of an object moving along a line Assume time is measured in hours and distance is measured in miles The areas of three regions bounded by the velocity curve and the t axis are also given Complete parts a through e a On what intervals is the object moving in the positive direction Select the correct choice and if necessary fill in the answer box to complete your choice OA The object is moving in the positive direction on the interval s Type your answer in interval notation Use a comma to separate answers as needed The object is never moving in the positive direction OB M 6 A 15 11 15

Use the given graph off to state the value of each quantity if it exists If an answer does not exist enter DNE a lim f x X 3 b 2 lim f x x 3 c lim f x d lim f x X 7 0 0 e f 7 y 6 N 2 4 6 8

2 log b log b xy xy N Z Type an integer or a decimal www

First rewrite the expression log c using base 10 log c

are associated with the points P and Q on the graph of the function Answer parts a and b 10 The volume V of a gas in cubic centimeters is given by V P atmospheres and 0 5 sps4 a Sketch a graph of the function and the secant line through P and Q b Find the slope of the secant line in part a and interpret your answer in terms of an average rate of change over the interval Include units in your answer a Choose the correct graph below A AV 20 16 12 8 00 4 0 0 1 2 3 4 5 6 p B AV 20 16 12 8 4 P 0 0 1 2 3 4 5 6 b The slope of the secant line is rate of Type integers or decimals OC 20 16 6284 AV 12 where p is the pressure in This means the over the interval 0 5 p 4 p Q 0 1 2 3 4 5 6 G D AV 20 16 12 8 4 OFFI 0 1 2 3 4 5 6 changes at an average

A function and an interval of its independent variable are given The endpoints of the interval are associated with the points P and Q on the graph of the function Answer parts a and b 10 The volume V of a gas in cubic centimeters is given by V where p is the pressure in atmospheres and P 0 5 sps4 a Sketch a graph of the function and the secant line through P and Q b Find the slope of the secant line in part a and interpret your answer in terms of an average rate of change over the interval Include units in your answer a Choose the correct graph below O A 20 16 12 628 4 0 0 1 2 3 4 5 6 O B Q 20 Q16 12 8 4 ABE 0 0 1 2 3 4 5 6 Q O C 20 Q16 12 8 4 C 0 0 1 2 3 4 5 6 O D Q AV 20 Q16 12 8 N 4 0 0 1 2 3 4 5 6 P Q

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

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 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

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

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

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

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

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

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

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

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

Rewrite as an augmented matrix and solve by row reduction x 5y 7 2x 7y 20

Consider the function f x 2x 3x 72x 2 on the interval 6 8 Find the average or mean slope of the function on this interval By the Mean Value Theorem we know there exists a c in the open interval 6 8 such that f c is equal to this mean slope For this problem there are two values of c that work The smaller one is and the larger one is

Given this matrix 10 113 01 5 a Rewrite as a system of equations b How many solutions does it have

Two books and three DVDs costs 54 Three books and one DVD costs 46 Find the cost of a single book and a single DVD SET UP AS AN AUGMENTED MATRIX BUT DO NOT SOLVE

Determine the dimension of the matrix 3 2 7 5 80

2 Solve each scenario using a different method graphing substitution elimination

1 Write 7 3 without exponents Use the equation editor by clicking insert math above or take a photo of what you wrote and upload it

1 a Find the vertical Asymptote b Find the Horizontal Asymptote end behavior 5x 6 4 2x 1 x 5 2 3 x 4x 10x 2 x 5x x 4 7 2 x

Use division to justify the expression that describes the end behavior for f x a Approaches y 0 b Approaches y 3x 10 C Approaches y 3x 2 d Approaches y 3 3x 4x 8x x 2x

In a biology lab 500 bacteria reproduce by splitting Every hour on the hour each bacterium splits into two bacteria 30000 20000 10000 What is a reasonable domain for the given function Explain What is a reasonable range for the given function Explain HHH Submit

3 Match the graph of each function in a c with the graph of its derivative in I III 3 b 0 X c 0 111 0 0 II

Reduced the BVP y xy x with y 0 y 1 0 into a variational problem