Application of derivatives Questions and Answers

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

In this question we will approximate 82 with a rational number a To use the linear approximation there is an obvious choice of function f x and point ao What are they f x 20 82 f x f x f xo x xo 999 b Use a linear approximation of f x at x xo to find a rational number approximating 82 Give your answer correct to at least 4 decimal places c Newton s Method finds roots zeroes of functions In order to use Newton s Method to approximate 82 we need a function g x such that 9 82 0 and g x has rational coefficients There is an obvious choice for g x For example to approximate 50 the obvious choice is g x x 50 to approximate 10 the obvious choice is g x x7 10 What is the obvious choice of g x to approximate 82 20 d Which integer zo makes g zo as close as possible to 0 on two iterations of Newton s Method using afa and n found above to approximate 82 Give your answers correct to at least 4 decimal plac

The question asks us to sketch the function y cos 3t First note that the function is a cosine function and will have that basic shape From the text the general form of a non shifted cosine function is y a cos bt The text also tells us that a is the amplitude that is the maximum height above zero that the function attains Note that the amplitude must be a positive number Looking at our function the amplitude is

Suppose f x y is a differentiable function Match the names of its derivatives in Newtonian and Leibnitz notation 1 2 3 4 5 of By of Da dyda 6 Drby A fm B fan c fry D fy E fym F fu

Problem 8 I t 24 77e kt where k 48 Find the average temperature of the coffee during the first 37 minutes after it is poured Enter your answer symbolically as in these examples Just Save Submit Problem 8 for Grading

Find the partial derivatives of the function fx x y fy x y fay x y f ra f x y xye B

Given u x t t e compute Uxx H

Find the first partial derivatives of 2x 3y 2x 3y f x y af x af 4 1 4 1 dy at the point x y 4 1

Compute the partial derivative f x y cos x 6y fy 0

Suppose f x y xy 2 Compute the following values f 0 2 2 0 f 0 0 1 2

5 Solve the triangle rounding to the nearest tenth assuming a is opposite side a 3 is opposite side b and y is opposite side c a 4 b 6 c 8

Use the parent function f x logx and transformation techniques to graph h x log x 7 3A dicate which transformation s is are applied Determine the domain range and vertical asympto lect all transformations that apply Vertically compress the graph off by a factor of 3 Shift the graph of f 7 units to the left C Shift the graph of f 7 units to the right Reflect the graph of f about the y axis E Horizontally stretch the graph of f by a factor of 7 F Horizontally compress the graph of f by a factor of 7 5 Reflect the graph of f about the x axis H Vertically stretch the graph of f by a factor of 3 Shift the graph of f 3 units up J Shift the graph of f 3 units down

1 Use the parent function f x logx and transformation techniques to graph g x log x 5 indicate which transformation s is are applied Determine the domain range and vertical asymptote of g Select all transformations that apply A Horizontally stretch the graph of f by a factor of 5 B Reflect the graph of f about the x axis C Shift the graph of f 5 units down D Vertically compress the graph of f by a factor of 5 E Vertically stretch the graph of f by a factor of 5 F Reflect the graph of f about the y axis G Shift the graph of f 5 units to the left H Horizontally compress the graph of f by a factor of 5 1 Shift the graph of f 5 units to the right Shift the graph of f 5 units up

Solve the following basic calculus problems d a dx e cos x

Match the equation with its graph y x 36 25 1 5 y 10 12 5 10 X y 10 5 10 5 5 10 10 X O 10 5 10 10 5 5 15 y 5 y 10 10 5 5 X 10 X

4 Find L 12 x dx Jinlord de 5x dx Hints set u hu 2 pts dv dx and use I by P 3 times

Find suike dx Find sin e dx Hinto satu e G N 2 pts

Find the linearization L x of the function at a f x x 5 L x a 32 Question Help Message instructor

Use a linear approximation or differentials to estimate the given number Round your answer to two decimal places 32 05 4 5 Enter an integer or decimal number more Question Help Marcago in

a Find the differential dy y tan x dy b Evaluate dy for the given values of x and da x 4 and dx 0 01 dy Quartier

If F x f g x where f 2 3 f 2 5 f 4 1 g 4 2 and g 4 9 find F 4 F 4

Find the derivative of the function y x sin 3 8

Find the first and second derivatives of the function y sin x y y Questi 0 0

Write the composite function in the form f g x Identify the inner function u function y f u g x and the outer y 1 7x g x f u Find the derivative dy dx dy da Question Help Message

Find the derivative of the function y cos a x y x

Find the derivative of the function F z z 32 3 F z