Calculus Questions

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

A graphing calculator is recommended Determine the end behavior of P P x x y y x x 13x as x as x

A company has a price demand function given by p x 30e 0 000 dollars per item when z items are sold Determine how many items the company must sell in order to maximize revenue items NOTE Round to 3 decimal places if needed

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

Intervals of Increase and Decrease and Local Extrema Answer the following questions using the function f x The derivative of the given function is f x The critical point s of f are B In x x2 NOTE Enter values as a comma separated list if needed If you believe the answer is none type none that occurs at z f has a relative local maximum value of f has a relative local minimum value of NOTE Enter values as a comma separated list if needed If you believe the answer is none type none that occurs at z f is increasing on and decreasing on NOTE State the open intervals of increase and or decrease as a comma separated list if needed

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

The graph of f a is given below Use this graph to determine the intervals where f a is increasing or decreasing click on graph to enlarge a Interval s where f a is increasing b Interval s where f a is decreasing help intervals help intervals

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

Product Rule d dx x g x f x g x g x f x Quotient Rule d f x dx g x g x x f x g x g x 24 f x 1 xex X

2 Find the Taylor series generated by fat x a for f x 3x x 2x x 2 at a 1

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

1 Use the Ratio Test to determine if 2n 1 n3 1 converges or diverges

The price of a condominium is 109 000 The bank requires a 5 down payment and one point at the time of closing The cost of the condominium is financed with a 30 year fixed rate mortgage at 7 Use the following formula to determine the regular payment amount Complete parts a through e below PMT P nt a Find the required down payment b Find the amount of the mortgage c How much must be paid for the one point at closing his Round to the nearest dollar as needed d Find the monthly payment excluding escrowed taxes and insurance Round to the nearest dollar as needed e Find the total cost of interest over 30 years Dound to the nearest dollar as needed

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

Product Rule dx f x g x f x g x g x f x Quotient Rule d f x g x f x f x g x dx 19 y v 20 v v 2t

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

20 w 2 w ce 22 g t t t 11 3

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

19 y 21 f t v 2v v 2t

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

Write an algebraic expression that is equivalent to the expression Hint Sketch a right triangle as demonstrated in Example 8 sec arcsin x 1

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

Using the First Derivative Test what are the local extrema for the function h x 3x 9x Select the correct answer below 4 O There is a local minimum at x O There are no local extrema There is a local maximum at x There is a local minimum at x There is a local maximum at 9x 72x 9 2 and a local minimum at x 4 2 and a local maximum at x 4 2 and a local maximum at 4

Use an inverse trigonometric function to write as a function of x 0 0 X 9

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

27 28 Find the antiderivative F of f that satisfies the given condition Check your answer by comparing the graphs of f and F 27 f x 2e 6x F 0 1

3 The first tem fourth tom the number is 2 of of an 23 The terms arithmetic 2 sum of the series 28 0 A series is 7 and the the series 689 Find 13

24 g v 2 cos v 3 1

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

f t 2t 4 3 t F

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

Consider the function f t 1 2t 41 What is the smallest value of such that the relative rate of change is given by R 2 Give your answer as a decimal

Consider the price demand equation given by 8 1 5 2 500 Compute E 1 where E is the elasticity of demand Give your answer to two decimal places Provide your answer below P X

Find the derivative of the function f t 4 tant f t