Application of derivatives Questions and Answers

A cadet rappelling down a cliff on a rope needs help A cadet on the ground pulls tight on the end of the rope that hangs down from the rappelling cadet to lock the cadet in place The length of the rope between the t cadets is 143 feet and the angle of elevation of the rope is 70 How high above the ground is the cadet on the rope Round your answer to two decimal places ft

63 66 Solve each equation for x 63 a e7 4x 6 b

A graphing calculator is recommended A cardboard box has a square base with each edge of the base having length x inches as shown in the figure The total length of all 12 edges of the box is 144 in X a Express the volume V of the box as a function of x V x

Select all that apply 5 2 500 Calculate E 2 where E is the elasticity of demand and select the correct interpretations of the result When p When p P X When p 2 the elasticity of demand satisfies E 2 1 which is elastic 2 the elasticity of demand satisfies E 2 1 which is inelastic When p 2 an increase in price will result in increased revenue 2 an increase in price will result in decreased revenue

The graph of a polynomial function is given P x 12x 3x 1 6 4 y intercept x y y x y 6 4 2 2 4 6 2 4 a From the graph find the x and y intercepts If an answer does not exist enter DNE x intercepts x y smaller x value 6 X larger x value b Find the coordinates of all local extrema If an answer does not exist enter DNE

This exercise involves local maxima and minima of polynomial functions Consider the following YA AN 0 Explain X What is the smallest possible degree that the polynomial whose graph is shown can have

Describe the end behavior of each polynomial a y x 8x 4x 16 End behavior b End behavior y y y 4x4 14x 100 y y as X as x 00 as x co as x 8

Find Critical Points of a Function Find the critical points of the function on the interior its domain 11 f 2 49 2 2 then the critical points of fare Note Type answer in form z c Separate multiple answers with a comma such as z 1 2 1 or type None if there are no cri

Find the domain and all critical points in the domain of y 4 2 a Domain of y 4 2 is help help intervals b Critical point s of y 4 2 on its domain are 2 domain help numbers use a comma to separate answers enter NONE if there are no critical points

Use PMT PA to determine the regular payment amount rounded to the nearest dollar The price of a home is 206 000 The bank requires a 20 down payment and three points at the time of closing The cost of the home is financed with a 30 year fixed rate mortgage at 7 Complete parts a through e below a Find the required down payment 41200 b Find the amount of the mortgage 164800 c How much must be paid for the three points at closing Round to the nearest dollar as needed

1 Find the Taylor Polynomials of orders 0 1 2 and 3 generated by fat a for f x 1 x 2 a 0

C Use Euler s method to obtain a numerical solution of the differential equation dx 2 yx given the initial conditions that x 1 when y 1 in the range x 1 0 0 1 1 2 Work dx to an accuracy of 6 decimal places If the exact solution of the given differential equation is y 0 3 679 ex determine the percentage error at x 1 1 1 and 1 2 Do you think we need to use a smaller step size h in this problem Explain your answer 10 20 20 50 marks

Current Attempt in Progress An oil company discovered an oil reserve of 400 million barrels The company plans that q t the rate of extraction of oil in millions of barrels per year at time t will be How long does it take to exhaust the entire reserve Round your answer to one decimal place t i q t a bt years a 40 b 0 5

With an interest rate of 2 compounded continuously a Find the future value in 10 years of a payment of 14 000 made today Round your answer to two decimal places i b Find the future value of an income stream of 1400 per yea Round your answer to two decimal places i c Which is larger the future value from the lump sum in part a or from the income stream in part b Why does this make sens financially The future value from the This makes sense because er 10 years is larger

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Your company needs 600000 dollars in five years time for renovations and can earn 10 interest on investments a What is the present value of the renovations Present value i dollars b If your company deposits money continuously at a constant rate throughout the five year period at what rate should the money be deposited so that you have the 600000 dollars when you need it Round your answers to 2 decimal places Money deposited at a rate of i dollars per year

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15 22 Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f Use the graphs and transformations of Sections 1 2 autoade 1 2 15 f x 3x 1 x 3

Given the equation below find dy dx 216x8 120x23 72 6 y dy 336x 337 dx Now find the equation of the tangent line to the curve at 1 1 Write your answer in mx b forr 24x 5x2y y 30 X X

Write the function in terms of the sine function by using the identity A cos ut B sin wt A2 B2 sinut arctan an 4 f t 7 cos nt 24 sinnt f t Use a graphing utility to graph both forms of the function y 6 5 4 3 2 20 What does the graph imply 10 20 WWW 6 5 4 3 2 2 3 4 6 5 4 3 2 30 20 10 nf t 20 10 WWW 30 f t

dy Use implicit differentiation to determine given the equation e e x sin y dx dy dx ee 3x cos x ee Add Work X

Use implicit differentiation to find the equation of the tangent line to the curve cy xy 10 at th point 5 1 The equation of this tangent line can be written in the form y mx b where m is and where b is

Find the slope of the tangent line to the curve 1x 4xy 2y 36 at the point 2 2 DNE X

Find the value Enter your answer in radians As x 1 the value of arccos x

Complete the equation arcsin 16 x arccos 0 x 4

Use the properties of inverse functions to find the exact value of the expression if possible Enter your answer in radians If an answer is undefined enter UNDEFINED COS

For the function below find the local extrema by using the First Derivative Test p x x 22x 96x 7 4 23 3 Select the correct answer below There is a local minimum at x 3 O There is a local maximum at x 8 There is a local maximum at x 8 and a local minimum at x 3 There is a local minimum at x 8 and a local maximum at x 3 O There are no local extrema

Find the exact value of the expression Use a graphing utility to verify your result Hint Make a sketch of a right triangle Enter your answer in radians sec arcsin 49

Consider the following 3 Find the length of the third side of the triangle in terms of x e cos 1 Find in terms of x for all three inverse trigonometric functions sin 1 e tan 1 0 Need Help X 9 Read It

Find the exact value of each expression if possible without using a calculator If an answer is undefined enter UNDEFINED a arccos 3 in V2 b arcsin

Suppose that you decide to borrow 15 000 for a new car You can select one of the following loans each requiring regular monthly payments Installment Loan A three year loan at 5 5 Installment Loan B five year loan at 4 8 Use PMT LIC nt to complete parts a through c below The monthly payment for Loan A is Do not round until the final answer Then round to the nearest cent as needed The total interest for Loan A is Round to the nearest cent as needed b Find the monthly payments and the total interest for Loan B The monthly payment for Loan B is Do not round until the final answer Then ound to the nearest cent as needed 43 The total interest for Loan B is Round to the nearest cent as needed c Compare the monthly payments and the total interest for the two loans Determine which loan is more economical Choose the correct answer below A The five year loan at 4 8 is more economical B The three year loan at 5 5 is more economical The buyer will save approximately S in interest Round to the nearest cent as needed

5 26 Find the most general antiderivative of the function Check your answer by differentiation 5 f x 4x 7 6 f x x 3x 2 8 f x 6x5 8x4 9x 7 f x 2x 3x 5x 2 342

An element with mass 970 grams decays by 15 1 per minute How much of the element is remaining after 14 minutes to the nearest 10th of a gram

P 8 5 2 500 Calculate E 2 where E is the elasticity of demand

Consider the function f t 1 21 41 What is the relative rate of change when 1 2 Give your answer to one decimal place

Differentiate y y 1 seca tan a

59 A retailer has been selling 1200 tablet computers a week at 350 each The marketing department estimates that an addi tional 80 tablets will sell each week for every 10 that the price is lowered a Find the demand function b What should the price be set at in order to maximize revenue

29 Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r

A thermometer is taken from a room where the temperature is 21 C to the outdoors where the temperature is 13 C After one minute the thermometer reads 10 C a What will the reading on the thermometer be after 5 more minutes b When will the thermometer read 12 C minutes after it was taken to the outdoors

oblem 2 When a golfer strikes a golf ball and we observe the ball s trajectory we can represent its vertical height h measured in feet above the ground after t seconds using the following representation h t 96t 16t a Find the height of the ball after 4 seconds Is the golf ball moving upward or downward after 4 seconds b Evaluate the rate of change of h t on the time interval 3 3 5 Interpret the meaning of your answer and be sure to include appropriate units c Find the maximum height of the golf ball d State a realistic domain for the function h t e Sketch the function h t and label when the ball is moving upward moving downward and the point where the golf ball is at its highest point include the coordinates of the highest point

y f x 89876 32 10 5 y 10 9 8 7 6 5 4 3 2 1 2 234567O9O 3 4 5 6 7 8 9 10 Evaluate g f 2 1 2 3 4 5 6 8 9 10 y g x 10 8 7 6 5 43 2 10 999664321 36789 W Ni 2 3 4 5 6 8 9 10 1 2 3 4 5 6 7 8 9 10

2 For the quadratic function f x x 7x 10 a Find the vertex b Does the vertex represent a local maximum or a local minimum c State the axis of symmetry d Find the y intercept

3 Use your calculator to graph f x x x 4x x 5 Then find the following Round 10 pts answers to 2 decimal places as needed a Local maxima stated as ordered pairs b Local minima stated as ordered pairs c Increasing intervals d Decreasing intervals dy intercepts stated as ordered pairs

The price p in dollars and the quantity x sold of a certain product satisfy the demand equation x 5p 200 Answ parts a through g ww a Find a model that expresses the revenue R as a function of p Remember R xp R p Simplify your answer Use integers or decimals for any numbers in the expression

the graph of the quadratic function f to determine the solution a Solve f x 0 b Solve f x 0 a The solution to f x 0 is Type your answer in interval notation

A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R x 100x 0 2x where the revenue R x is measured in dollars What is the maximum revenue and how many units should be manufactured to obtain this maximum at units

Carol has 1100 ft of fencing to fence in a rectangular horse corral 550 x a Find a function that models the area A of the corral in terms of the width x of the corral A x b Find the dimensions of the rectangle that maximize the area of the corral width ft ft length

The effectiveness of a television commercial depends on how many times a viewer watches it After some experiments an advertising agency found that if the effectiveness E is measured on a scale of 0 to 10 the E n 3n 34 where n is the number of times a viewer watches a given commercial For a commercial to have maximum effectiveness how many times should a viewer watch it times

If a ball is thrown directly upward with a velocity of 24 ft s its height in feet after t seconds is given by y 24t 16t2 What is the maximum height attained by the ball Round your answer to the nearest whole number Need Help Read It Watch It Manteell

Find the maximum or minimum value of the function g x 100x 1500x Is this a maximum or minimum value O maximum value minimum value