Application of derivatives Questions and Answers

approximate the amount of work needed to pump liquid out of a tank 3 A tank is in the shape drawn below its boundary is obtained by rotating the parabola y 2x for 0 x 1 around the y axis Units are in meters Suppose the tank is full of a liquid whose density varies according tc the distance above the bottom of the tank 8 y 8 5 y kg m a Set up a Riemann sum that approximates the Work to pump all the liquid out a spout 1 meter above the top of the tank RS b Set up the integral to compute the total work Do NOT Evaluate W 34 y 2r LA 1

9 Joey enlarged a 3 inch by 5 inch photograph on a copy machine He enlarged it four times he table below shows the area of the photograph after each enlargement What is the average rate of ange of the area from the original photograph to the fourth enlargement to the nearest tenth 0 1 2 3 4 Enlargement Area square inches 15 18 8 23 4 29 3 36 6 c 5 4 a 4 3 Process Explanation b 4 5 atsulava 3 d 6 0

13 The functions f x g x and h x are shown below Which inequality the ordering of the functions according to the average rate of change on the interval 3 2 10 b f x 20 16 6 12 16 6 A g x h x f x B f x h x g x C g x f x h x D f x g x h x X 3 2 2 0 1 2 g x 38 28 20 14 10 8 h x x 2x 22 ww

The price of gasoline purchased varies directly as the number of gallons of gas purchased Suppose 24 gallons are purchased for 94 20 Step 2 of 2 Using the equation from the previous step find the price of purchasing 34 gallons Round your answer to two decimal places if necessary Answer 2 Points Keypad Keyboard Shortcut

Given that 44i is a zero factor the following polynomial function completely Use the Conjugate Roots Theorem if applicable f x x4 2x 8x 128x 256 Answer 2 Points 11

Find all solutions to the following polynomial equation x 23x 169x 145x 0 Separate your answers with a comma Write the exact answer Do not round Answer 2 Points

Consider the following function x if x 3 ifx 3 Step 1 of 3 Identify the general shape and direction of the graph of this function on the interval o 3 Answer 2 Points Ay Ay X t x Ay x X 588 60660 Ay x X Keyb

Answer 2 Points 3 Step 3 of 4 Graph the given function by indicating how the more basic function has been shifted reflected stretched or compressed Vertical Stretch Compress Stretch Select the type of transformations to create the correct graph Any further inputs required to complete the transformations will appear when the appropriate selection is made Horizontal Shift O Left Right x Axis Reflection O Yes Vertical Shift Up y Axis Reflection Yes O Compress None Units No g x No O None Down 1 None Units 1 10 5 S Keyp Keyboard Shorm Enable Zoom Pan 5 10 A X

Suppose f is a function to which we can apply the derivative Find the derivative of Type your solution step by step and mention the name of the rule that you use in each step h t In t f t

The half life of gold 194 is approximately 1 6 days Step 1 of 3 Determine a so that A t Apa describes the amount of gold 194 left after t days where Ao is the amount at time r 0 Round to six decimal places AnswerHow to enter your answer opens in new window 2 Points Keyp Keyboard Short

Simplify completely assuming both x 0 and y 0 32x6y3

Simplify the following assuming x 0 y 0 and z 0 V 375x0y z 60x6y4z

Simplify completely assuming both x 0 and y 0 36x y Prou

Simplify assuming that x 0 and y 0 225x342 9x y12

Simplify completely assuming both x 0 and y 0 81x y8

Simplify the following assuming x 0 y 0 and z 0 288x y z 2 18x y z

A set of 125 golf scores are normally distributed with a mean of 76 and a standard deviation of 3

Let f x 7 2 x 8 For what value of x does function f have the given value f x 2 3

28 An improper integral ex dx converges if E r 0 F G r 1 r 0

Find the infinite sum if it exists for this series 3 0 6 0 12

For the year ending December 31 2017 sales for Company Y were 65 71 billion Beginning January 1 2018 Company Y plans to invest 6 5 of their sales amount each year and they expect their sales to increase by 4 each year over the next three years Company Y invests into an account earning an APR of 1 9 compounded continuously Assume a continuous income stream How much money will be in the investment account on December 31 2020 Round your answer to three decimal places billion dollars How much money did Company Y invest in the account between January 1 2018 and December 31 2020 Round your answer to three decimal places billion dollars How much interest did Company Y earn on this investment between January 1 2018 and December 31 2020 Round your answer to three decimal places If intermediate values are used be sure to use the unrounded values to determine the answer billion dollars

path star at a A planet orbits a star along an ell The planet s path can be modeled by the ellipse shown below centered at the origin with a horizontal major axis If the minimum distance from the planet to the star is 70 million km and the maximum distance is 280 million km write the equation of the ellipse in millions of km

the line below KEREKE THA BEDDE MEKLEREENEEM XTREMENTMINERIN DESCRE

Without using a calculator find the value of sin 24 cos 2 24 sin 85 cos 50

X 1 2 To plot points click on a point on the graph and drag it to the desired location Graph the equation y 4 by plotting points Provide your answer below 9 8 7 6 5 4 3 2 A 8 7 6 5 4 3 2 10 1 2 3 4 5 6 678 8 ok 1 2 3 4 5 6 7 8 9 FEEDBACK MORE IN

For many species of fish the weight W is a function of the length L that can be expressed by KL 3 k constant W W L Suppose that for a particular species k 0 01 that for this species the length in centimeters is a function of the number of years t the fish has been alive an that this function is given by L L t 45 t 16 10 0 t 16 Find Wo L t to express W as a function of the age t of the fish WOL t

The solution of the differential equation y 4y 32 cos 2x 36 sin 2x subject to the initial conditions y 0 0 and y 0 y x

F x f 3 tan t dr find all critical numbers of F in the interval 3 3 Type your answer in the space provided and show your work that is show how you found your answers

f x x a 0 n 3 0 x 0 2 Exercise a Approximate f by a Taylor polynomial with degree n at the number a Step 1 The Taylor polynomial with degree n 3 is The function f x 2x has derivatives Step 2 T3 x f a f a x a f a x a f a x a 21 31 Step 3 f x H f x H Therefore T x 4x With a 0 f 0 1 16x 4 64x 48r Submit Skip you cannot come back 4r 16x 4 1 f 0 0 Step 1 If Ir 4 x M then we know that M R x 64 48 Tx9 x and Je x 0 F 0 4 Exercise b Use Taylor s Inequality to estimate the accuracy of the approximation fx T x when x lies in the given interval and f 0 0

Step 3 of 3 Identify the correct graph of this piecewise defined function Answer 4 p x 9 10 4 Enable Zoom Pan 4 HA if x 2 4 Enable Zoom Pan HA

Consider the following function p x Answer 2x 4 9 10 if x 2 if x 2 Step 1 of 3 Identify the general shape and direction of the graph of this function on the interval 2 X x Ay Ay X

Consider the following function if x 2 2 Step 1 of 3 Identify the general shape and direction of the graph of this function on the interval o 2 Answer g x Ax x

Find the area of the region bounded by the given curve r 9e 2 on the interval T 0 2 9

8 Find the global maximum and minimum of the function f x y x y 2x 4 on the triangle D x y x20 0 y 3 y2x

Find the area of the region bounded by r 8 cos 60 0 0 2

Rewrite the polar equation r 4 sin 0 as a Cartesian equation

Find the length of the spiraling polar curve r 2e 6 From 0 to 27 The length is

The atmospheric pressure p on a balloon or an aircraft decreases with increasing height This pressure measured in millimeters of mercury is related to the height h in kilometers above sea level by the formula p 760 e 0 145h a Find the height of an aircraft if the atmospheric pressure is 243 millimeters of mercury b Find the height of a mountain if the atmospheric pressure is 634 millimeters of mercury a The height of the aircraft is 7 86 kilometers Round to two decimal places as needed kilometers b The height of the mountain is Round to two decimal places as needed

5 12 Sketch the region enclosed by the given curves Decide whether to integrate with respect to r or y Draw a typical approx mating rectangle and label its height and width Then find the are of the region 5 y x 1 y 9 x x 1 x 2

Built is the British Airways London Eye which was completed in 2000 The diameter is 135 m and passengers get on at the bottom 4 m above the ground The wheel rotates once every 3 minutes a Draw a graph which represents the height of a passenger in metres as a function of time in minutes b Determine the equation that expresses your height h as a function of elapsed time b c How high is a passenger 5 minutes after the wheel starts rotating 2 marks 2 marks c d How many seconds after the wheel starts rotating is a passenger 85 m above the ground for the first time Answer to the nearest tenth 2 ma

A spinner with 10 equally sized slices has 3 blue slices and 7 yellow slices The dial is spun and stops on a slice at random What is the probability that the dial stops on a red slice Write your answer as a fraction in simplest form 0 8 08 X P

iven f x below find f x f x sin x

1 4 Find the area of the shaded regie 1 VA 3 0 1 1 y x y 1 x x y 1 y y 1 X x 4 X X

Functions Course Packet of Let f x 1 and g x 21x 3 Determine the rule for the composite functions fo g and go f 4x 7 fog x gof x 4 21x 3 3 3 7 21 4x 7 X

Let f x 4 x Simplify each of the following expressions f 3 4 3 f 3 h 4 3 h f 3 h f 3 4 3 h 4 3 f 3 h f 3 h 4 h 3 3 h X Make sure to cancel out the factor of h in the denominat

Let f x 5 Simplify each of the following expressions X f 8 f 8 h f 8 h f 8 f 8 h f 8 h Make sure to combine terms by getting a common denominator Make sure to cancel out the factor of h in the denominator

Determine the domain of each of the following functions Enter your answer using interval notation a Domain of b Domain of c Domain of d Domain of 2x 1 x 25 2x 1 x 25 2x 1 x 25 2x 1 x 25

Write each collection of numbers using interval notation a All real numbers greater than or equal to 0 i e x 0 b All real numbers less than 0 i e x 0 c All real numbers greater than 0 and less than or equal to 4 i e 0 x 4 d All real numbers less than 0 or greater than or equal to 4 i e x 0 or x 4 e All real numbers less than or equal to 4 excluding 0 i e x 0 or 0 x 4

2 State whether the following expressions will be positive or negative 151 185 X 157 142 280 260 842 384 a 147 183 c 180 168 635 845 g 292 605 b d f h

Compute 2x y dA where D is the region bounded between y and 3 MAN 3 28 O 1 28