Application of derivatives Questions and Answers

beginning of a race Velocities of the two cars during the race are given in the tables below where velocities are in miles per hour and times are in seconds Sappho s car timer velocity vs 0 1 2 4 5 6 7 8 9 10 0 20 35 48 85 93 99 106 Homer s Car time 0 1 2 4 6 7 8 10 velocity VII 0 18 43 68 79 86 93 95 96 A Use ten trapezoids to approximate the area between the curves y vs t and y vu 1 on the interval from 0 to 1 10 What definite integral are you approximating B If you haven t done so convert your answer from part A into units that make good common sense for the physical situation described Then explain what practical meaning this answer has for the physical situation

1 Explain in your own words what is meant by the equation lim f x 5 x 2 Is it possible for this statement to be true and yet f 2 3 Explain

If a rock is thrown upward on the planet Mars with a velocity of 10 m s its height in meters t seconds later is given by y 10t 1 86t 1 a Find the average velocity over the given time intervals ii 1 1 5 iii 1 1 1 i 1 2 iv 1 1 01 v 1 1 001 b Estimate the instantaneous velocity when t 1

5 The deck of a bridge is suspended 275 feet above a river If a pebble falls off the side of the bridge the height in feet of the pebble above the water surface after t seconds is given by y 275 16t a Find the average velocity of the pebble for the time period beginning when t 4 and lasting i 0 1 seconds ii 0 05 seconds iii 0 01 seconds b Estimate the instantaneous velocity of the pebble after 4 seconds

3 Use the power rule to find the derivative of f x x fi

Use limits to compute the derivative 3 f 0 where f x x 6x 2 0 Simplify your answer

Find the equation of the tangent line to the graph y f x at the given value of x f x 10x 8 x 6 What is the equation of the tangent line Type an equation CO

Use the power rule to find the derivative of f x 2 x f x

Find the equation and sketch the graph of the following line With slope 2 and y intercept 0 5 www

Find an equation of the line below Slope is 4 8 4 on line

What does the transformation fly 26 da

What does the transformation f x f x do to the graph of f x reflects it across the y axis shrinks it vertically stretches it vertically reflects it across the x axis

1 Consider the region bounded by the curves y x x 12 x 5 and x 5 and the x axis A Set up a sum of integrals not containing an absolute value symbol that can be used to find the area of this region I B Find the area of the region by using your answer from part A Don t approximate with

4 Consider the curves y x and y mx where m is some positive constant No matter what positive constant m is the two curves enclose a region in the first quadrant Without using a calculator find the positive constant m such that the area of the region bounded by the curves y x and y mx is equal to 8

3 Consider the function f x x sin x on the interval 0 which is graphed below 0 8 0 6 0 4 0 2 0 4 0 6 0 8 I y xsin x A Find a constant function g x constant on the interval 0 such that the area under the graph of g is the same as the area under the graph of fon the interval 0 Hint Remember that a constant function g x c is just a horizontal line

sing of a certai is given b where P is in dollars and M x is marginal cost in dollars per unit A Suppose the marginal cost at a production level of 500 units is 10 per unit and the cost of producing 500 units is 12 000 that is M 500 10 and P 500 12000 Use a tangent line approximation to estimate the cost of producing only 497 units B Suppose the production schedule is such that the company produces five units each day That is the number of units produced is x 51 where is in days and 0 corresponds the beginning of production Write an equation for the cost of production P as a function of time t

Find the area of the region IN THE FIRST QUADRANT upper right quadrant bounded by the curves y sin x cos x y 2x cos x and y 4 4x Use the area to the left of curve y 4 4x You get O A 0 16165 OB 1 7281 OC 0 36974 O D 1 8467 OE 0 37859

Find the area between the curve y 2 and y x 3 on the interval 4 x 4 You get O A 184 OB 16 O c 212 OD 24 40 OE col

1 Solve each of the following trig equations 0 ses 360 Draw diagrams 3 b tan 0 0 a sin 3 marks eas 2 c cos 8

Assume that a couple invests 750 upon the birth of their daughter Assume that the investment earns 6 4 compounded annually What will the investment be worth on the daughter s 10th birthday The investment will be worth approximately on the daughter s 10th birthday Do not round until the final answer Then round to the nearest cent as needed

Factor the following polynomial 4x 40x 100 4x 40x 100

Let g x h g x X 2 x and h x x 4x 8 Calculate the function h g x

Suppose a 26 2 is expanded in descending powers of a Then the 5th term in the expansion is i 22 a 26 12 4 ii 12 a 26 12 1 a 26 5 iv 12 1 a 26 5 V None of the options

Try it Evaluate the step function for the given input vale 4 3 x 1 1 1 x 2 2 x 4 X 24 g x a Step 3 5 g 2 19 g 2 g 5

1 point Find the Laplace transform F s L f t of the function f t est 10 h t 2 defined on the interval t 0 Here h t is the unit step function Heaviside F s L est 10h t 2 help formulas

The volume of an open topped box with a square base is 245 cubic centimeters Find the height in centimeters of the box that uses the least amount of material OA 6 centimeters OB 3 centimeters OC 7 883 centimeters OD 2 centimeters OE 3 942 centimeters

X sin 2x 2 dx OA cos 2x 3 2 C OB 3 cos 2x 3 2 C OC cos 2x 3 2 C OD cos 2x 3 2 C E None of these This one can t be done using the metho substitution

x 4x 7 Ad O A 2 42 7 C O B 2 42 7 C O C C O E 2 5 x 4x 7 5 OD None of these This one can t be done using the method substitution 4 5 x 4x 7 5 C

1 point Consider the initial value problem y 9y 0 y 0 3 y 0 0 a Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation Denote the Laplace transform of y t by Y s Do not move any terms from one side of the equation to the other until you get to part b below b Solve your equation for Y s Y s L y t c Take the inverse Laplace transform of both sides of the previous equation to solve for y t y t help formulas

1 point Consider the rational function A Select ALL terms below that occur in the general form of the complete partial fraction decomposition of F s The capital letters A B C L denote constants B C D E F G H Hs I s 11 A s 6 Bs C s 6 s 11 2 Es F 5 6 G s 11 Ks L F s s 11 D s 6 5 2 s 6 2 s 11

9 1 point Find the inverse Laplace transform f t L F s of the function F s 9 2 f t 1 2 5 1 5 5 2 S 5 help formulas

1 point Find the inverse Laplace transform f t L F s of the function F s 9s f 0 L 9 7 64 9s 7 s 64 help formulas

A tan x C B None of these This one can t be done using the method of substitution OC sec x C D 2 sec x C

Jx 2 x 2 dx 3 x 4x x 2x O B x 4x C O C c x 2x C O A 1312 O D 1 3 C x 4x C x 2x C O E None of the above

A particle s position in meters is given by the equation s t 5x sin 2x on the interval 1 4 minutes What is its instantaneous velocity at t 0 OA TT OB 3 O C 1 OD O E id3363061

A paperweight is thrown straight up with an initial upward velocity of 60 meters per second and an initial height of 6 meters The acceleration due to gravity of the object is given by a t 9 8 meters per second per second where tis in seconds The equation that describes the paperweight s velocity as a function of time tis A v t 4 9t 60 B v t 9 8t C c v t 4 9t 60 D v t 9 8t 60 E t 9 8t 60

The absolute maximum value of f x x 12x on the closed interval 1 4 occurs at x OA 2 OB 0 OC 2 OD 4 E Cannot be determined

Use your calculator to estimate 4 sin 22 dx 4 OA 3 628 OB 40 078 O C 4 734 OD 7 875 OE 4 320

The population of ground squirrels in a public park is measured to be 100 The growth of the population over time can be modeled by P t 600 1 5e 0 5t with P representing the number of squirrels and t given in months after the initial population measurement According to this model what is the maximum number of squirrels the park can support

Use your calculator to estimate OA 3 315 B 4 021 O C 0 417 OD 0 221 OE 0 767 1 dx

O A 1 000 OB 0 693 O C 2 000 OD 0 632 O E 2 914 LO J ax

Given that f x is continuous f x dx 7 f f x dx 3 and 4 S f x dx 2 Then f f x dx A 2 B 6 C 3 D 4 E O

Given that fx dx 63 0 and 3 xdx A 121 5 OB 49 5 OC 256 5 OD 274 5 13 5 then f 3x 5x dx E Can t tell from the information given

1 Use the graph to estimate the intervals on which the 2 x function is increasing f x x 3 2 For each function graphed estimate the intervals on which the function is increasing and decreasing 4 4 4 B 6 my mi 5 EN P 220 SEN

graph y f x 10T 81 41 21 2 3 III 4 Consider the following statements 1 f x dx f x da I below 5 II 1 f z dz f z dz dx dx 5 OA I only B II only C III only D II and III only c5 3 f ar da f x d dx Which of the above statements is true 6

Find limx 6 x 12x 36 x 6 O A Does not exist OB 36 O C 0 OD CO O E co 8

Find lim 6 x 12x 36 x 6 O A Doe s not exist OB 36 O C O O D D 00 OE 8

If the graph of f x looks like M Which of the following represents the graph of f lxl A None of these B C m VV

A metai bar is heated and then allowed to cool Its temperature T in C is found to be T 5 25e 0 30t where t in min is the time of cooling Find the time rate of change temperature after 10 0 min After 10 0 minutes the time rate of change is degrees per minute Round to three decimal places as needed

The vertical displacement y in cm of the end of an industrial robot arm for each cycle is y 3 2 Its vertical velocity for t 18s is cm s Round to the nearest tenth as needed tan 0 5t where t is the time in s Find its vertical velocity for t