Application of derivatives Questions and Answers

Use Theorem 9 11 to determine the convergence or divergence of the p series 1 16 2 p 1 converges diverges 1 81 3 1 256 4 1 625 5

Find the sum of the convergent series Round your answer to four decimal places n 1 sin 2

Consider the following repeating decimal 0 8 a Write the repeating decimal as a geometric series 0 8 1 0 1 1 0 1 00 n 0 X b Write the sum of the series as the ratio of two integers 8 9

Compute uxv if u 7 and v 6 and the angle between u and vis uxv Simplif 73 3

Graph the function f x x 3 x 2 x 1

Find the horizontal asymptote Give your answer in the form y b or enter if there is none 1 5 X f x

Find the equation of the slant asymptote for the function f x Provide your answer below x 1 X 1

sketch the graph a function of that has all of the properties listed your graph need not be given by a algebrak Continuous for all real number 7 Decreasing on 00 2 an 2 5 Thereasing on 2 2 and 5 J Concare upward on 0 4 and 3 7 00 J Concave dounward on 0 4 3 7 11y intercept at 0 2 graph please

Find the abso minimum off x x 2x 3x 4 5 option A 2 61 B 2 62 107 C 2 12 J C2 61

9 find the locted in of the absolute maximum extrema for the function 00 14 onl 3 kul At 2 2 Po X option A 1 4 2 4 B 0 5 19 09 09 42 D Nore 1 10 0 010400 164 TOT

Suppose that the price PCin dollars and they weekly sales X in thousand of units of a certain commodity clatory the 8p x 1025 Determine Que rate at which sales are changing at a time when x 5 P 5 7 demand equation of the rate of 20 per week changing is and the price is Fallaing The rate at which sales ate thousand units per week

11 Find dy by impicit differentiate ty option A EX S B X C 9 0 87XXX X

Here is ph of the function h K 5 3 44 W 2 M 4 2 W H 2 Use the graph to find the following If there is more than one answer separate them with commas a All local maximum values of h al maximum 0 0

Here is 5 on or the function n 3 2 S fo 3 2 1 24 W A 2 3 5 M Use the graph to find the following If there is more than one answer separate them with commas a All values at which h has a local minimum 0 0

If d z d z J6 cos s ds then Question Help Video Message instru

The temperatue of a town t months after January can be estimated by the function 7 f t 26 cos t 49 Find the average temperature from month 3 to month 5 MUCI

A The graph of the derivative of f is shown in the figure above Which of the following could be the graph of f D Act B f x X E A VA C

8 ft 10 ft Q Find the area of the trapezoid pictured above Select an answer Question Help Video Message instructor Calculator

Find f f t f t sec t sec t tan t t f 2 7

Let r t t sin t and y t 1 cos t At t 1 z 1 y 1 da dt dy da 1 1 t 1 tangent slope speed 1 Question Help Video Message instructor 27 1 55 13 0 5 0 0 2 CYCLOID 3 4 5 6

If the rate of change of N is inversely proportional to sec x N where x If N 0 3 and N 5 Find k the proportionality constant A

graphs of four derivatives are given below Match the graph of each function in a d with the graph of its derivative in I IV V O II a th A 4 III e TV d M O

Applied Max Min 20 points a Find the dimensions of the rectangle with the greatest area that can be built so the base of the rectangle is on the x axis between 0 and 2 0 x 2 and one comer of the rectangle is on the curve y 2 What is the area of this rectangle Hint The base of the rectangle is 2 x Write the area function A X Find the derivative A x Set A x 0 and solve for x Using your answer above state the optimal area b Generalize the problem in part a for the curve y CV with C 0 and 0 x 2 Write the area function A X Find the derivative A X y 2 x Set ATX 0 and solve for x x 2

The slope of the line through the distinct points x1 Y1 and x2 Y2 can be interpreted as the rate of change in GETTE The slope of the line through the distinct points x1 Y1 and x2 Y2 can be interpreted as the rate of change in with respect to with respect to

Calculate the gradient of h x y z x Vh x y z Write your vector using brackets and separate your entries with a comma e g 3x 5y z 9

se implicit differentiation to find an equation of the tangent line to the curve at the given point x y 3x 4x x 0 0 5 y 0 25 0 25 0 5 cardioid X

Determine the interval over which h is continuous and the interval over which h is differentiable h is continuous on 00 00 08 h is differentiable on 00 00 0 Part 2 of 4 Use the above information to determine if the Mean Value Theorem may be applied to h over 3 8 Select all that apply Yes the Mean Value Theorem may be applied to h on 3 8 No the Mean Value Theorem may not be applied to h on 3 8 since h is not continuous on 3 8 No the Mean Value Theorem may not be applied to h on 3 8 since h is not differentiable on 3 8 Determine the average rate of change of h over 3 8 Part 3 of 4

Let f be a function continuous on 10 o with a first derivative defined by 1 x 10 Consider the interval 11 18 where f 11 20 f 18 16 Determine the value of a in the interval 11 18 guaranteed by the Mean Value Theorem x

The graphs of y log x and y tanx are shown below The point a is marked the graph of y log r and the point k a is marked on the graph of y tan x y JH X 31 The value of is Select one O a 210 Ob 240 O c 120 Od 150

Determine h x for the following function Answer h x 4x x 9x 20

etermine h x for the following function Answer h x 2x x 10x 25

Consider the following relation Enable Zoom Pan To X

Each of 7 students reported the number of movies they saw in the past year Here is what they reported 4 20 8 17 7 19 15 Find the mean number of movies that the students saw If necessary round your answer to the nearest tenth movies X 5

A track and field playing area is in the shape of a rectangle with semicircles at each end See the figure The inside perimeter of the track is to be 2000 meters What should the dimensions of the rectangle be so that the area of the rectangle is maximized The dimensions of the rectangle are and Use a comma to separate answers as needed Use ascending order Type an integer or decimal rou hundredth as needed

Write the composite function in the form f g x Identify the inner function u g x and the outer function y f u Use non identity functions for f u and g x y 6 x5 f u g x Find the derivative dy dx dy F dx

Find the dimensions that produce the maximum floor area for a one story house that is rectangular in shape and has a perimeter of 143 ft OA 35 75 ft x 143 ft OB 11 92 ftx 35 75 ft OC 35 75 ft x 35 75 ft OD 71 5 ft x 71 5 ft

dy ssume x and y are functions of t Evaluate for the following dt dy dx y 2x 93 3 x 5 dt 3 x 5 y 7

he velocity of a particle in ft s is given by v t 5t 9 where t is the time in seconds for which it has traveled Find the time at which the velocity is at a minimum A 4 5 s OB 5 s C 2 5 s D 9 s

If f x 2x x3 find f x F x f x and f x F x f x f 4 x Graph f f f and f on a common screen y 10

Find y by implicit differentiation x 8y 8

Find the volume V of the described solid S a frustum of a pyramid the portion of a pyramid that remains after the top part has been cut off by a plane parallel to the base with square base of side b square top of side a and height h V b ha ab b 3 What happens if a b Enter your answer in terms of b and h We get a square pyramid x with volume V h 2 What happens if a 0 Enter your answer in terms of b and h We get a rectangular solid X with volume V 0 X x

Simplify the ratio of factorials 3n 1 3n 1 1 3n 3n 1

that the price p in dollars and the weekly sales x in thousands of units of a certain commodity satisfy the demand equation 2p x 650 Determine the rate at which sales are chang at a time when x 20 p 5 and the price is falling at the rate of 80 per week Suppose The rate at which sales are changing is thousand units per week

he slope of the tangent line to a curve is given by f x 8x 9x 6 If the point 0 8 is on the curve find an equation of the curve

Sketch the graph of a function f that has all of the properties listed Your graph need not be given by an algebraic formula a Continuous for all real numbers b Decreasing on o 1 and 1 4 c Increasing on 1 1 and 4 00 d Concave upward on oo 0 1 and 2 8 00 e Concave downward on 0 1 2 8 1 y intercept at 0 3 Choose the correct graph below OA A OB SCITE O C Q 4 143 Q OD

dy Find by implicit differentiation dx x 3x y y 8 O A OB O C COD x 3xy x y 2 x 3xy 2 x y x 2xy x y 2 2 x 2xy 2 x y

Find dy 53 dx by implicit differentiation 7xe 16

dy Find for the equation 7x 3y4 4 using implicit differentiation dy 11

Find the limit of the sequence 37 n 6 an

7 Find each of the following a b tan 1 cos c d sin e sin Cos 2 2 2 2