Differentiation Questions and Answers

Write the first five terms of the sequence defined recursively a 27 ak 1 ak 4 a1 a a3 34 a5 Use the pattern to write the nth term of the sequence as a function of n Assume that n begins with 1 an

In this question we will use Newton s Method to approximate the x value of the intersection between f x and g x closest to 2 Start your approximation with co 2 and apply two iterations of Newton s Method to the function F x f x g x Give your answers correct to at least 4 decimal places Remember that WeBWork can understand calculator ready numbers and make liberal use of copy paste 21

Let z x 4y Then The rate of change in zat 3 1 as we change x but hold y fixed is The rate of change in zat 3 1 as we change y but hold a fixed is

Find all the first and second order partial derivatives of f x y 8 sin 2x y 10 cos x y A 1 B Sv c fen D ffwr 0 f FEEL

Find the sine and cosine of the angle 8 Calculater from the Pythagorean Theorem sin 0 cos A 11 8 10 24 y X i

Find the sine and cosine of the angle 0 sin 0 cos 0 0 y 90 54 72

Find the sine and cosine of the angle 0 sin 0 y 0 10 8 6 X

For an average person the rate of change of weight W in pounds with respect to height h in inches is given approximately by the following formula dW 0 0015h dh Find W h if W 80 256 pounds Also find the weight of a person who is 5 feet 6 inches tall W h

Problem 9 Evaluate the following integrals 3 i xe So 0 a b xe4x dx In x Se inta dx 1 11 e 12 1 16 Problem 9 a 11e 2 1 Enter your a as in these

Find the derivative of the function y y x2 1 1 5 ques

Find the derivative of the function y y 5 2 1

Find an equation of the tangent line to the curve at the given point y sin sin x 7 0 y

If h x 4 3f x where f 5 4 and f 5 2 find h 5 h 5

Find the derivative of the function y sin tan 9x y

Match the equation with its graph x 16 1 15 10 5 15 10 5 y 15 10 5 10 15 10 10 5 5 10 15 10 15 X 15 10 15 10 5 5 15 10 0 5 10 15 y y 15 10 5 10 15 5 10 15 5 10 15 X

Find the vertex focus and directrix of the parabola and sketch its graph Use a graphing utility to verify your graph x 20 y 5 vertex focus directrix No Solution x y 10 323 40 la 4 7 9 10 Fill Graph Layers After you add an object to the graph you can use Graph Layers to view and edit its properties

Find the x and y intercepts of the graph of the circle If an answer does not exist enter DNE x 10x y 2y 19 0 STEP 1 Consider the equation for the circle x2 10x y2 2y 19 0 Let y 0 Give the resulting equation below STEP 2 Solve your equation from Step 1 for x If one or more solutions exist these values are the x coordinates of the x intercepts of the graph of the circle Enter your answers as a comma separated list X STEP 3 Consider once again the equation for the circle x2 10x y2 2y 190 Now let x 0 Give the resulting equation below STEP 4 Solve your equation from Step 3 for y If one or more solutions exist these values are the y coordinates of the y intercepts of the graph of the circle Enter your answers as a comma separated list y STEP 5 Complete the problem by listing the x and y intercepts x y x intercepts y intercept x y x y smaller x value larger x value

K Prove the following identity and give the values of x for which it is true COS os sinx 2 1 08 2 c 2x Simplity sin sin x s n nh n sin de TR

Prove the following identity and give the values of x for which it is true cos sin x 1 x Take the derivative of each side of the equation cos sin x 1 x dx dx COC

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 a Use a trigonometric equation to find an expression for the angle e in terms of x Type an equation 8 tan 1 150 X de What is the rate of change of the angular size when the boat is x 400 m from the building dx rad m Round to five decimal places as needed 150 m

A boat is towed toward a dock by a cable attached to a winch that stands 10 ft above the water level see figure Let 6 be the angle of elevation of the winch and let L be the length of the cable as the boat is towed toward the dock Complete parts a to c below de OC dL b Compute LVL 10 10L 100 de dL When L 50 ft When L 20 ft When L 11 ft when L 50 20 and 11 ft de dL de dL de dL 0 0041 Round to four decimal places as needed 0 0289 Round to four decimal places as needed Round to four decimal places as needed GREY de D dL VL TU 10 LL 100 10 ft

K Consider f x 2 x 1 sinx on the interval 1 1 a Graph f with a graphing utility b Compute and graph f c Verify that the zeros of f correspond to points at which f has a horizontal tangent line b Compute f 2 x 1 4 SETTO Q OU DU

The function V t 3500t 18000 where Vis value and t is time in years can be used to find the value of a large copy machine during the first 5 years of use a What is the value of the copier after 2 years and 3 months After 2 years and 3 months the copier is worth 10 125 OF b What is the salvage value of the copier if it is replaced after 5 years After 5 years the salvage value of the copier is 500 c State the domain of this function o O OF d Sketch the graph of this function 1 X 18000 16000 14000 12000 10000 8000 6000 4000 2000 Clear All Draw 2 3 t 5 e What is the range of the function 500 V t 1800

A fence is to be built to enclose a rectangular area of 450 square feet The fence along three sides is to be made of material that costs 3 per foot The material for the fourth side costs 9 per foot Find the dimensions of the rectangl that will allow for the most economical fence to be built The short side is ft and the long side is ft

Find the gradient field F Vop of the potential function p x y z x y 2z 1 2

Calculate the Reynolds number for human blood flowing through a 2mm radius vessel at average velocity of 0 2 m s Charactize the flow i e turbulent laminar or transitional

et g x 3 1 5x and f x 3 1 5 Explore the rates of change constant average for th near function g and the exponential function f a Determine the value of the ratios i From x 0 to x 1 5 Ag x Ax Af x Ax ii From x Ag x Ax Af x Ax Ag x Ax iii From x Af x Ax 1 5 to x 3 3 to x 5 Ag x Ax and Af x Ax Preview Preview X Preview X Preview Preview Preview over the following intervals of x

By inspecting the graph of the function find the absolute maximum and absolute minimum on the given interval g x x on 1 64 10 2 20 10 OA The absolute minimum is at x U Use a comma to separate answers as needed B There is no absolute minimum U 10 Find the absolute minimum Select the correct choice below and if necessary fill in the answer boxes to complete your choice 10

Find the derivative of the following function and evaluate the derivative at the given value of x In 8x f x 8x f x 16 X e 180

Find the derivative of the following function and evaluate the derivative at the given value of x f x 8x 8x f x

Calculate the derivative y sin 2x in sin 2x y 4 cos 2x In sin 2x 1 Use parentheses to clearly denote the argument of each function

Check by differentiation that y 5 cos 2t 5 sin 2t is a solution to y 4y 0 by finding the terms in the sum So 1 411

Which of the following functions are solutions of the differential equation y 3y 18y 0 Select all that apply 3x A y x B y x e C y x er D y x e 3r E y x e F y x 6x G y x 0

Let y 16y 0 Find all values of r such that y separated list ker satisfies the differential equation If there is more than one correct answer enter your answers as a comma help numbers

A drug that stimulates reproduction is introduced into a colony of bacteria After t minutes the number of bacteria is given approximately by the following equation Use the equation to answer parts A through D N t 1000 27t t Osts 18 Dox to complete your choice A The rate of growth is increasing on 0 9 Type your answer in interval notation Use a comma to separate answer as needed B The rate of growth is never increasing When is the rate of growth decreasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The rate of growth is decreasing on Type your answer in interval notation Use a comma to separate answer as needed OB The rate of growth is never decreasing

Find the higher order derivative dx 6log x 4 6log x 4 Question 19

OA If b 0 and b 1 then c If b 0 and b 1 then d X OB If b 0 and b 1 thenx logbx Inb OD If b 0 and b 1 then d d dx logbx x in b for x 0 and dx logplx x Inb forx 0 dx d dx OB log bx OA 108x x Inx b dx x 109x x Inx dx OC log x In bx dx dx log bx hic function f x log x How does it differ from the derivative formula for Inx In b for x 0 and In Inb 1 x In b In bi d In b for x 0 and log x for x 0 dx X How does the rule of differentiation for the function f x log x differ from the derivative formula for Inx dx logolx X for x 0 and d dx In b BETER for x 0 1 logxxlnb for x 0

Find the higher order derivative 3x in 3x 3x in 3x 0 Question 20 7

K Use the definition of the derivative to evaluate the limit lim Inx 6 x X 2 6 Type the exact value of the limit lim Inx 6 Question 23 7 3 1

vative y sin 2x In sin 2x Y 0 Use parentheses to clearly denote the argument of each function www 17 7 3 65

K Evaluate the derivative of the following function H x 2x 5 H Question 16 7 3 61

K Find the derivative of the following function yain x 8 dy dx Type an exact answer using it as needed Question 12 7

Evaluate the derivative y 3x y Type an exact answer Question 15 7

of the function and evaluate the derivative at the given value of a f x sin x a Find the derivative of the function sin x x dx Use parentheses to clearly denote the argument of each function

V Find the derivative of the following function y 22 sinx dx 22 s n x Question 5 7 3

rentiation for the logarithmic function f x log x How does it differ from the derivative formula for Inx Which of the following is the rule of differentiation for the logarithmic function f x log x OA If b 0 and b 1 then d dx log bx x In b for x 0 and d OB If b 0 and b 1 then dx C If b 0 and b 1 then d dx log bx In b for x 0 and 1 log bx x In b d In b OD If b 0 and b 1 then log x dx X for x 0 and for x 0 and d dx log blx x Inb for x 0 d logb x Inb d d dx logb x logx xInb for x 0 1 In b X for x 0 for x 0

e g The area between the curve y Part 5 of 7 Determine whether the following statements are true and give an explanation or counterexample Assume x 0 and 0 a In xy Inx Iny b In 0 1 c in x y Inx Iny 7 2 71 e Choose the comect answer below f 3 for any integer m2 1 1 and the x axis on the interval 1 is 1 UD Ine statement is true because the derivative of an exponentia sunction is defined asjae Cerra OA The statement is true because the derivative of an exponerial function is defined as OB The statement is false because the derivative of an exponential function is defined as 0 OC The statement is false because the derivative of an exponential function is defined as OD The statement is true because the derivative of an exponential function is defined as dx HW Score 79 27 19 Points 0 43 of 1 woo

Part 1 of 3 Find the width function w y for each value of y on the face of the building w y Use integers or decimals for any numbers in the equation HW Score 48 8 11 71 of 24 points O Points 0 of 1 A large building shaped like a box is 60 m high with a face that is 70 m wide A strong wind blows directly at the face of the building exerting a pressure of 160 N m at the ground a increasing with height according to P y 160 2y where y is the height above the ground Calculate the total force on the building which is a measure of the resistance that mat be included in the design of the building CHOCO Save

K Determine whether the following statements are true and give an explanation or counterexample Assume x 0 and y 0 a In xy Inx Iny b In 0 1 c In x y In x Iny e xelx 1 d f d On dx 3x 33x for any integer n 2 1 1 g The area between the curve y and the x axis on the interval 1 e is 1 X OCED OD The statement is true The Sum Rule of Logarithms states that In x y Inx Iny d Choose the correct answer below d dx A The statement is false because the derivative of an exponential function is defined as x 1 d OB The statement is false because e is a constant and the derivative of a constant cis O C The statement is true because e is a constant and the derivative of a constant c is dx c 0 dx d c c d e dx The statement is true because the derivative of an exponential function is defined as

Cande derivative of the following function f x 3x x f x 3x 3x 3 3x log 3x Simplify your answer