Application of derivatives Questions and Answers

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

54 57 Applying reduction formulas Use the reduction formulas Exercises 50 53 to evaluate the following integrals 54 55 x cos 5x dx 56 x sin x dx 57 In x dx 58 Two methods Evaluate sin x In cos x dx in the following two ways a Use integration by parts 59 Two methods a Evaluate f b Evaluate X x 1 X b Use substitution dx using integration by parts dx using substitution c Verify that your answers to parts a and b are consistent 60 Two methods a Evaluate fx ln x dx using the substitution u x and evaluating In u du b Evaluate xln x dx using integration by parts c Verify that your answers to parts a and b are consistent

Helium is pumped into a spherical balloon at a constant rate of 2 cubic feet per second How fast is the radius increasing after 3 minutes At what time if any is the radius increasing at a rate of 140 feet per second The radius is increasing at a rate of 0 0082 ft sec Type an integer or a decimal Do not round until the final answer Then round to four decimal places as needed Select the correct choice below and if necessary fill in the answer box to complete your choice A The time at which the radius increases at a rate of 140 feet per second is sec Type an integer or a decimal Do not round until the final answer Then round to five decimal places as needed There is no solution

Problem 7 A chain lying on the ground is 14 ft long and weighs 90 lbs How much work is required to raise one end of the chain to a height of 8ft 4s

Problem 5 Find an integral which represents the volume of the solid obtained by rotating the region enclosed by the curves y x 2 and y 2x 8 about the x axis A E 7 x H x 2x 8 dx B 7 2x 8 x2 2 dx C Problem 5 Select 2x 8 dx F 2x 8 dx F x 2x x4 2x 8 dix 4 2 2 2x 8 x D 10 S x 2x 8 x G 2x 8 x dx 2x 8 x dx

Problem 3 Which of the following represents the area of the region enclosed by the curves y x and y 2x 8 for 3 x 4 A 3 x 2x 8 dx C 2 3 x2 2x 8 dx H F x 2x 8 dx E x2 2x 8 dx x 2x x 2x 8 dx dx 2 x 2x 8 B x 2x 8 2 2 2 1 x 2x 8 dx D 2 x 2x 8 dx 3 x 2x 8 dx x 2x 8 dx G G x 2x 8 dx 3 x x 2x 8 dx 2 2x 8 dx x 2x 8 dx

Problem 8 Suppose that the temperature of a cup of coffee t minutes after it is poured is given by T t 24 77e kt where k 4 Find the average temperature of the coffee during the first 37 minutes after it is poured Enter YOUT

Problem 1 Evaluate the following integrals 1 S 5x 177 2 dbx a b je Sesved 4

Write an equation for the function graphed here 5 4 3 2 f x 3 2 1 2 3 3

The top and bottom margins of a poster are 2 cm and the side margins are each 4 cm If the area of printed material on the poster is fixed at 390 square centimeters find the dimensions of the poster with the smallest area Width Height

cos arcsin u CHE cos arcsin u Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Rationa all denominators

sin sin 1 sin sin 1 Type an exact answer using radicals as needed Rationalize all denominators

y sec 2 HECH y Type an exact answer using as needed Use integers or fractions for any numbers in the expression

xact value of the expression sin 2 sin 2 Type an exact answer using as needed

expression cos m2 2 Type an exact answer using x as

If a function f has an inverse and f x 1 then f 1 If a function f has an inverse and f x 1 then f 1

Complete the statement y cos 1 x means that x y cos x means that x for 0 y t for 0 syst

Watch the video that describes finding inverse function values Click here to watch the video Find the exact value in radians of the expression cot 1 3 3 cot 1 Type an exact answer using as n

K Watch the video that describes finding inverse cosine values Click here to watch the video Find the exact value of the expression arccos arccos Type an exact answer using a

A boat sails directly toward a 150 m skyscraper that stands on the edge of a harbor The angular size of the building is the angle formed by lines from the top and bottom of the building to the observer see figure Complete parts a and b below Round to five decimal places as needed de b Graph as a function of x and determine the point at which the angular size changes most rapidly Choose the correct graph below dx OA Q OB 0 1000 by 0 0 008 REISSE 0 1000 by 0 0 008 For what value of x is the angular size changing most rapidly x Round to the nearest integer as needed M 0 1000 by 0 008 01 OD 150 m S 0 1000 by 0 008 01

Find two numbers A and B with AB whose difference is 44 and whose product is minimizec A B

Find the eccentricity of the ellipse 2 1 x 25 36

wy oc cos 0 Yo Find the derivative of the inverse cosine using the theorem of the derivative of the inverse function The theorem states let f be differentiable and have an inverse on an interval 1 If x a point of I at which f x 0 then f isdifferentiable at yo Rx and Y x sin xo 1 sin xo Part 2 of 6 R 00 cos 0 How can the expression on the right side be rewritten in terms of f x OA Substitute using xocos Yo OB Substitute using Yo cos xo OC Rewrite the expression using cac Xosin xo OD Substitute using the identity sinx cos 1 sin xo HW Score 60 58 13 93 of 23 points O Points 0 of 1 CITES Save

A parcel delivery service will deliver a package only if the length plus the girth distance around does not exceed 120 inches Find the maximum volume of a rectangular box with square ends that satisfies the delivery company s requirements Maximum Volume

Given the following information evaluate the other five functions RECU and 0 so cos sin 0 35 tan 8 45 4 3 csc 8

Find the standard form of the equation of the parabola with the given graph 6 5 4 2 6 3 2 1 T y 11 9 7 5 3 1 3 5 7 9 11 0 0 1 2