Application of derivatives Questions and Answers

26 If y x 8 widw then 0 y x 6x6 8 x 4 24 Question Help Video X

If q r Syr d r cost9 t dt then

If y x 1 V8 widw then

4 I If h x h x 5 0 84 dw COS dw then

If f t n s 49 7 ds then f t 1 8

Find the derivative of y with respect to x t or 0 as appropriate In x x6 27 y A 1 6ln x x12 B 6ln x 1 x7 1 6In x x7 D 1 6lnx x12

Next question 17 If f x t Get a similar question You can retry this to dt then f x 17 Hint Remember that f a t d 1 to to dt 17 Find the derivative of g a sin 1 dt g x sin x

Answer the following questions about F x x 43x 486 B Graph F x and use geometric formulas to calculate the area between the graph of F x and the x axis from x 10 and x 15 Choose the correct graph below 20 10 15 OB 60 40 20 10 20 OC 20 10 10 20 Q OD 1000 500 50 100

ind the derivative of the function 25 r sec 0 tan 0 2 A 2 sec 0 tan 0 3 tan2 0 sec 0 tan 0 C 2 sec 0 sec 0 tan 0 2 B 2 sec 0 tan 0 sec 0 D 2 sec 0 tan 0 3

Use geometric formulas to find the unsigned area between the graph of y f x and the x axis over the indicated interval f x x 8 2 9 The area between f x and the x axis is Type an exact answer using a as needed

Find the critical point s of the function f x y y6 6xy 12x and determine whether each is a local maximum local minimum a saddle point or none of these Enter the critical point s as point s a b Number of critical point s Choose one

Current Attempt For f x y A x Bx y Cy what values of A B and C give f x y a local maximum value of 17 at the point 6 7 A B C i

Find an equation for the tangent to the curve at the given point 20 f x 8 x x 9 64 9 A y x 9 B y 9 C y x 41 1 D y x

etermine where the given function is concave up and where it is concave dow 19 f x 4x x3 A Concave up on 0 concave down on 0 B Concave up on 0 and 1 concave down on 0 1 C Concave up on 0 concave down on 0 D Concave down for all t

2 Get a formula for the derivative of the following functions f x 2x 5x 3x 1 k x x 3x 2 x 1 g x m x x 1 tan x x 1 h x n x 1 x 5 3x cos x

p x x 4 sin x s x sin x cos x sin x v x X x 2 tan 3x 1 q x t x w x X cos x sin x r x 2 2 x 2 x 4x 5x x 3x 2 u x y x 2x 5x 1

A chemist has a 100 gram sample of a radioactive material He records the amount of radioactive material every week for 6 weeks and obtains the following data Complete parts a f Week x 0 1 2 3 4 5 6 Weight y in grams 100 0 82 7 66 5 56 9 45 3 37 2 30 5 a Treating week x as the independent variable use a graphing utility to fit an expone function to the data y ab Round to four decimal places as needed

A person wants to purchase both parcels of land shown in the figure to the right and make a check on their combined area There is no equation for the river frontage so the person uses the average of the left sum of rectangles covering the area The 1 000 foot baseline is divided into 10 equal parts At the end of each subinterval a measurement is made from the baseline to the river and the results are tabulated Let x be the distance from the left end of the baseline and let h x be the distance from the baseline to the river at x Use L 0 to estimate the combined area of both parcels and calculate an error bound for this estimate How many subdivisions of the baseline would be required so that the error incurred in using L would not exceed 2 500 square feet X 0 h x 0 700 200 100 300 400 500 600 183 235 240 324 390 255 284 The combined area of both parcels is 1 ft 800 445 900 480 River Parcel 1 1 000 ft 1 000 500 Parcel 2 h x 500

3 2 6 5 32 2 2 3 5 6 At the point shown on the function above which of the following is true Of 0 f 0 of 0 f 0 Of 0 f 0 f 0 f 0

Find the center vertices and foci of the hyperbola y 5 x 3 1 25 center x y vertices x y foci x y x y x y 1 8 smaller y value larger y value 2 smaller y value Find the asymptotes of the hyperbola Enter your answers as a comma separated list of equations larger y value Sketch its graph using the asymptotes as an aid Use a graphing utility to verify your graph y y 10

The filament of an automobile headlight is at the focus of a parabolic reflector which sends light out in a straight beam see figure 8 in 1 4 in a The filament of the headlight is 1 4 inches from the vertex Find an equation for the cross section of the reflector Assume that the vertex i inches b The reflector is 8 inches wide Find the depth of the reflector Round your answer to two decimal places in

Calculate the indicated Riemann sum S5 for the function f x 26 2x Partition 2 8 into five subintervals of equal length and for each subinterval XK 1 xk let CK S5 let Ck xk 1 Xk 2

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin 10 4 0 5 y 10 5 5 0 3 2 4 0 0 3 2 5 10 X

a Given that u 2 5 and 1 3 find the new vector w 2 v 5 9b What is the component form of

8 Consider the vector whose initial point is P 3 2 and terminal point is Q 5 1 Find the position vector and graph both vectors in the space provided

Use the Binomial Theorem to expand and simplify the expression x 1 5

Calculate the binomial coefficient 100 2

Given u x t t e compute

Find all the first and second order partial derivatives of f x y 8 sin 2x y 10 cos x y A f fx B af fy 8cos 2x y 10sin x y c f fxx E D fay dady F r 1 32sin 2x y 10cos x y 8sin 2x y 10cos x y 16sin 2x y 10cos x y 16sin 2x y 10cos x y B

Suppose f x y is a differentiable function Match the names of its derivatives in Newtonian and Leibnitz notation 1 2 A 3 81 5 82 Dy Of 4 8x dydn A f B fam c fry D fy E fym F fyy

Consider the following function f x 1622 3 Find f 64 and f 64 f 64 64 Find all values c in 64 64 such that f c 0 Enter your answers as a comma separated list If a answer does not exist enter DNE C Based off of this information what conclusions can be made about Rolle s Theorem O This contradicts Rolle s Theorem since f 64 f 64 there should exist a number c in 64 64 such that f c 0 This does not contradict Rolle s Theorem since f 0 does not exist and so f is not differential on 64 64 O This does not contradict Rolle s Theorem since f 0 0 and 0 is in the interval 64 64 O Nothing can be concluded This contradicts Rolle s Theorem since f is differentiable f 64 f 64 and f c 0 e but c is not in 64 64

Score 3 9 3 9 answered Question 3 C 3 Save progress Done 0 1 pt 6 Deta Verify that the function satisfies the three hypotheses of Rolle s Theorem on the given interval Then find all numbers c that satisfy the conclusion of Rolle s Theorem Enter your answers as a comma separated list f x cos 52 20 7 20

Score 2 9 279 answered Question 8 f 5 Suppose that 4 f x 5 for all values of x What are the minimum and maximum possible values of 5 f 3 0 1 pt 6 f 3

Question 7 If 1 5 and f x 1 for 1 x 4 how small can f 4 possibly be

Calculate the indicated Riemann sum S3 for the function f x x 4x 5 Partition 0 3 into three subintervals of equal length and let c 0 7 c 1 3 and c3 2 8 S3 Simplify your answer

Compute L3 and R3 for Graphs A and B The value of L3 for Graph A is Simplify your answer AY 16 14 12 10 86 4 2 04 0 Ay 16 14 12 10 8 6 4 2 y g 4 6 Graph A 8 x 10 Q E Q

Use the Binomial Theorem to expand and simplify the expression a 7 4

Calculate the binomial coefficient 2

Write an expression for the apparent nth term of the sequence Assume that n begins with 1 10 12 14 16 18 an 11

Write an expression for the apparent nth term an of the sequence Assume that n begins with 1 4 5 6 7 8 1 3 5 7 9 an 11

The owner of a play equipment store wants to construct a fence to enclose a rectangular outdoor storage area adjacent to the store using part of the side of the store which is 250 feet long for part of one of the sides See the figure below There are 420 feet of fencing available to complete the job Find the length of the sides parallel to the store and perpendicular that will maximize the total area of the outdoor enclosure Length of parallel side s Length of perpendicular sides Not parallel side fence X perpendicular side STORE rged view

A car rental agency rents 200 cars per day at a rate of 30 dollars per day For each 1 dollar increase in the daily rate 5 fewer cars are rented At what rate should the cars be rented to produce the maximum income and what is the maximum income Rate Maximum Income

Use the Fundamental Theorem to evaluate the definite integral exactly 8 x x dx

4 Find the point at which the given function h x x2 equals its average value on the interval 2 4 Hint use Mean Value Theorem for integral

Find Jes e3x cos 2x dx 2 2 points

1 2 1 point You hold two sheets paper vertically facing each other You blow air between the sheets and observe that they are moving closer toward each other Which statement is correct As you blow the air it moves and it causes the pressure between the sheets the increase above the ambient atmospheric pressure As you blow the air between the sheets its movement causes the pressure between the sheets to drop As you blow the air between the sheets its movement creates vacuum between the sheets As you blow the air between the sheets it displaces all the air between the sheets and they collapse 1 point In a section of horizontal pipe with a diameter of 50 cm the velocity of the water flowing is 4 0 m s and the pressure is 4 3 kPa What volume of water flows per unit time in the pipe Give your answer in m 3 s but do not include the units in the answer box Type your answer

Consider the concentration C of a drug in the blood as a function of a the amount of the drug given and t the time since the injection For 0 x 4 and t 0 we have C f x t xte t The units of Care milligrams per litre the units of a are milligrams and the units of t are hours Sketch the following two single variable functions on a separate page Pay attention to the domain given at the start of the problem a f 2 t b f x 1 5 Using your graph in a where is f 2 t a maximum t a minimum t Using your graph in b where is f x 1 5 a maximum x a minimum x Give a practical interpretation of the function f 2 t Choose the best answer A the time after injecting 2 mg as a function of concentration B the time after injection until the concentration in the blood is 2 mg L as a function of the amount injected C the concentration of the drug in the blood 2 hours after injection as a function of the amount injected D the initial injection necessary for concentration to be 2 mg L as a function of time E the concentration of the drug in the blood resulting from a 2 mg injection as a function of time Give a practical interpretation of the function f x 1 5 Choose the best answer A the time after injection until the concentration in the blood is 1 5 mg L as a function of the amount injected OB the initial injection necessary tration to be 1 5 mg L as a function of time C the concentration of the drug in the blood 1 5 hours after injection as a function of the amount injected D the time after injecting 1 5 mg as a function of concentration E the concentration of the drug in the blood resulting from a 1 5 mg injection as a function of time

The marginal price for a weekly demand of x bottles of shampoo in a drugstore is given by the function shown below Find the price demand equation if the weekly demand is 150 when the price of a bottle of shampoo is 8 What is the weekly demand when the price is 6 50 p x 9 000 3x 50 Find the price demand equation Type an equation www

A yeast culture is growing at the rate of W t 0 40 1 grams per hour If the starting culture weighs 4 grams what wi be the weight of the culture W t after t hours After 9 hours The weight of the culture after t hours will be W t

Let s solve the logarithmic equation log 3 log x 8 log x a First we combine the logarithms on the LHS to get the equivalent equation b Next we use the fact that log is one to one to get the equivalent equation c Now we find x log x X