Calculus Questions

Calculate the following integrals 4x 5 2x dx 4x 5 2x dx mencerit sulky nes srltyd in A B C f 5 cos x sin x dx 4 x e 6x 2 x 1 2 dx

1 Given a function of f x 2x 8x 3 on 1 5 answer the following questions You cannot use the graphic calculator to find the answers ing que A Find the critical point s within the given interval mo 8 B Determine the absolute max and absolute min of f x within the given intenol

4 Given a function p x x 1 on 1 2 Find the point s guaranteed to exist by the Mean Value Theorem

ven a function f x 6x 3x find the antiderivative F x that satisfies F 1 3 7 10t 5 and g 1 10 find g t

hips at an engineering firm As part of their work they get to assist in the planning of a brand new roller coaster For this assignment you help Jim and Michelle as they tackle the math behind some simple curves in the coaster s track Part A The first part of Jim and Michelle s roller coaster is a curved pattern that can be represented by a polynomial function 1 Michelle has a list of possible functions Pick one of the g x functions below and then describe to Michelle the key features of g x including the end behavior y intercept and zeros a g x x x 4x 4 Pick ONE b g x x 2x 9x 18 c g x x 3x 4x 12 d g x x3 2x 25x 50 Le g x x 7x x 7 2 Create a graph of the polynomial function you selected X6 from Question 1 Please label all intercepts and critical points art B he second part of the new coaster is a parabola 3 Jim needs help creating the second part of the coaster Create a unique parabola in the pattern f x ax bx c Describe the edirection of the parabola and determine the y intercept and zeroes Make sure this parabola will connect to the equation you selected in e Part A You may arrange the two functions in any order you choose as long as they connect and graph 4 The safety inspector notes that Jim also needs to plan for a vertical ladder through the center of the coaster s parabolic shape for access to the coaster to perform safety repairs Find the vertex and the equation for the axis of symmetry of the parabola showing your work so Jim can include it in his coaster plan PA My Drive Kneeskerm Roller Coaster Project pdf 5 Create a piecewise graph of the polynomial function you created in Question 1 and the quadratic equation from Question 4 You will need to restrict the domain on both your g x function in Part A and your f x function from Part B Please label all intercepts critical points domain restrictions and the range for each piece of the graph Part C 6 Now that the curve pieces are determined use those pieces as sections of a complete coaster By hand or by using a drawing program sketch a design of Jim and Michelle s coaster that includes the shape of the g x and f x functions that you chose in the Parts A and B You do not have to include the coordinate plane in the drawing Be sure to label each section of the graph with the corresponding function for your instructor to view Part D 7 Create an ad campaign to promote Jim and Michelle s roller coaster It can be a 15 second advertisement for television or radio an interview for a magazine or news report or a song poem or slideshow presentation for a company These are just examples you are not limited to how you prepare your advertisement so be creative Make sure to include a script of what you will say if you are preparing an interview or a report The purpose of this ad is to get everyone excited about the roller coaster 8 On a half sheet of poster board build your presentation that would be used to win the award of getting your roller coaster built The poster board should include a The first polynomial equation and graph g x with its key characteristics explained and labeled b The second quadratic equation and graph f x with its key motni I

What two nonnegative real numbers with a sum of 23 have the largest possible product

Use n 6 and p 0 2 to complete parts a through d below a Construct a binomial probability distribution with the given parameters P x 4 5 6 Round to four decimal places as needed b Compute the mean and standard deviation of the random variable using ux x P x and ox x P x Hx x Round to two decimal places as needed Round to two decimal places as needed c Compute the mean and standard deviation using Hx np and ox np 1 p Round to two decimal places as needed X 0 1 2 3

Determine if the following probability experiment represents a binomial experiment If not explain why If the probability experiment is a binomial experiment state the number of trials n and probability of success p An investor randomly purchases 13 stocks listed on a stock exchange Historically the probability that a stock listed on this exchange will increase in value over the course of a year is 42 The number of stocks that increase in value is recorded S www w Select the correct choice below and if necessary fill in the answer boxes to complete your answer OA Yes because the experiment satisfies all the criteria for a binomial experiment and p Type integers or decimals Do not round n OB No because the experiment is not performed a fixed number of times OC No because there are more than two mutually exclusive outcomes for each trial D No because the trials of the experiment are not independent because the PATE differs from trial to trial

56 59 Functions from derivatives Use the derivative f to determine the x coordinates of the local maxima and minima of f and the inter vals on which f is increasing or decreasing Sketch a possible graph of f f is not unique 56 f x x 1 x 2 x 4 10 sin 2x on 2m 2m 57 f x 58 f x x 1 x 2 x 3 59 f x x x 2 x 1

5 f x x tan x on TT TT 22

Refer to the functions f and g and evaluate the function for the given value of x Select Undefined If applicable f 4 2 1 1 3 3 3 1 2 1 and g 0 3 5 8 1 1 4 6 g f 3 is Undefined

Given m x x 2 find mm x and write the domain in Interval notat Part 0 2 Part 1 of 2 mm x

Refer to functions p and q Evaluate q p x and write the domain in interval notation Write your answers as integers or simplified fractions p x x 3x Part 0 2 Part 1 of 2 q p x 1 q x 2 x 2 0 0

2 Use the parent function f x bx and transformation techniques to graph h x 3 x 5 Also indicate which transformation s is are applied Determine the domain range and horizontal asymptote of h Select all transformations that apply A Reflect the graph of f about the x axis B Horizontally stretch the graph of f by a factor of 5 C Shift the graph of f 5 units to the right D Shift the graph of f 5 units to the left E Vertically stretch the graph of f by a factor of 5 F Vertically compress the graph of f by a factor of 5 G Reflect the graph of f about the y axis H Horizontally compress the graph of f by a factor of 5 1 Shift the graph of f 5 units down J Shift the graph of f 5 units up

Test the series for convergence or divergence Use the Select and evaluate lim 7140 Since the limit is Select Note Use INF for an infinite limit 7th n 10 2n 8 Select

Test the series for convergence or divergence Part 1 Divergence Test Evaluate the limit lim bn 71 00 Identify the corresponding positive terms b Gipss li t 00 Calea 1 n 0 n21 8 then the Divers

4 7 Optimization Problems 1 A closed box top bottom and all four sides needs to be constructed to have a volume of 9 m and a base whose width is twice its length See Figure 1 2x T Y Figure 1 A box Use calculus to determine the dimensions length width height of such a box that uses the least amount of material Justify why your answer corresponds to a minimum not a maximum

Use the Root Test to determine the convergence or divergence of the given series or state that the Root Test is inconclusive L lim Van Enter inf for o 11 700 is A convergent B divergent C The Root Test is inconclusive n 6n

Apply the Ratio Test to determine convergence or divergence or state that the Ratio Test is inconclusive n 2n p lim Enter inf for o 71 00 00 n 1 n 2n is A convergent B divergent C The Ratio Test is inconclusive n 1

2 We want to find the points on y x that are closest to 0 3 0 3 Y D x x x X Figure 2 Distance to 0 3 on y x a For each point x x on the parabola find a formula for its distance to 0 3 Call this distance D x See Figure 2 b Let f x D x 2 which is the squared distance between x z2 and 0 3 Finding where D x is minimal is the same as finding where f x is minimal Determine all x where f x has an absolute minimum The points x z2 for such a are the dlogget points to 03 on

3 Three line segments of length 1 are joined together at endpoints to form a base and the legs of an isosceles trapezoid as in Figure 3 Let 0 in 0 2 be the common angle measurement between the legs and the line passing through the base of length 1 We want to find the angle that maximizes the area of the trapezoid 1 1 Figure 3 An isosceles trapezoid with base and legs of length 1 a Compute the area A 0 of the trapezoid The general area formula for a trapezoid ish b b where h is the height and b and b2 are the lengths of the bases Hint Break up the trapezoid into a rectangle with two right triangles at both ends Use trigonometry to compute the height and the length of the longer base in terms of 0 b Find all solutions to A 0 0 with 0 0 2 Hint Write A 0 entirely in terms of cos 0 The answer is not 0 4 45 c Verify that the area A 0 is a maximum not a minimum at the angle found in part b and compute this maximum area

5 Consider the boundary value problem as in Q3 b y 4x 2x2y 2r2ln x ze 1 2 y 1 1 2 y 2 ln 2 1 a Write the system of first order IVP regarding y and y2 that we need to compute if we would like to apply the linear shooting method to solve the above BVP b Computational exercise Write a script to solve this problem Let h 0 1 using Euler s method to solve the system of IVP in a and compute the approximate solution to y in 0 1 DI 11 91

1 Consider the nonlinear shooting method with Newton s algorithm We are aiming to approximate the solution to the boundary value problem y y yy 1 x 2 y 1 1 2 y 2 1 3 with h 0 5 and to 1 6 a Write down the system of first order IVP that we need to solve for each tk b Compute the value of y 2 to or y 2 to with Euler s method that you introduced in a You may round up the solution up to two decimal places c Compute t by using Newton s method to solve y b t 8 0

2 Consider the following initial value problem of the second order y 3y 2y 6e t te 0 1 y 0 y 0 2 h 0 5 a Rewrite the above IVP into a system of first order IVPs by letting u y and u y b Compute an approximate solution to y 1 by using Euler s method to solve the system of firs order IVPs that you get in a

Given the following graph what factor would appear in the denominator O x 1 O x 1 O x 0

5 Find where f x 0 Give your answer in interval notation O 2 infinity O 1 infinity O 2 1 O 2 1 Search PRE IC O

What is the multiplicity of the x 3 factor O Even O Not Enough Information Odd

Use the graph to find f x 0 Assume the function extends from negative infinity to infinity 2 O infinity 0 O infinity 0 u 0 3 O 3 infinity 3 infinity 10

10 The graph of a function f is shown where f x a z p z q for some constants a p and 9 Which of the following is true about the function f A a p q 0 B a 0 p q C D P 9 0

8 1 8e x 1 8e x dx

Let z 8 cos isin and W 3 3 cos Which statement describes the geometric construction of the product zw on the complex plane O Stretch z by a factor of 3 and rotateradians counterclockwise Stretch z by a factor of 3 and rotate radians counterclockwise O Stretch z by a factor of 24 and rotate radians counterclockwise O Stretch z by a factor of 24 and rotate radians counterclockwise sin 7 TE i sin

Find the Vertical Asymptote O x 1 O y 1 O x 1 O y 1 2 6

Given f x 2x 1 if x 0 find f 3 06 07 01 O 10 O x 2x x 1ifx 0

Use the end behavior to determine that as xx f x infinity 0 Onegative infinity 3

A given function has 3 complex zeros Two of the zeros are 3 and 2 4i What is the missing zero O 2 4i O 3 O 2 4i O 2 4i

Question 10 If 6000 dollars is invested in a bank account at an interest rate of 7 per cent per year Find the amount in the bank after 6 years if interest is compounded annually Find the amount in the bank after 6 years if interest is compounded quarterly Find the amount in the bank after 6 years if interest is compounded monthly Finally find the amount in the bank after 6 years if interest is compounded continuously Question Help Video Message instructor

Question 11 Find the horizontal asymptote of f x y 4x 3x 1 4x32x 3

Question 7 Divide and simplify 126 41i 9 4i Question Help Video Messago inc

Simplify 562 5 6i

Question 8 2 Find a formula for the exponential function passing through the points 3 27 and 1 6 y

An oil well is expected to produce oll at the rate of 46e 0 05t thousand barrels per month indefinitely where it is the number months that the well has been in operation Find the total output in barrels over the lifetime of the well by integrating this r from 0 to Note The owner will shut down the well when production falls too low but it is convenient to estimate the total output as if production continued forever Thousand barrels barrald 46 production rate total production Months

R G B K Match each equation with a graph above 3 1 43 3 0 8 4 1 22 3 1 22 3 0 68 a red R b blue B c green G d orange 0 e black K Question Help Video Message instructor Submit Question

Find the size of the permanent endowment needed to generate an annual 25 000 forever at a continuous interest rate of 5

x 2 x 7 0 Give your answer in interval notation

X Given f x 3x 2x 1 What is the degree of the function Good job 4

x 2 x 7 Try again 2 7 0 Give your answer in interval notation

Question 9 Given that a cubic function has the zeros 4 and 8 4i what is the missing zero

5 10 TT 1 5 VE 20 15 10 5 0 5 10 is 5 10 tion 5 Use the graph to determine the denominator of the rational function Write the denominator as a factor Submit

TT VE vo 4x 5 3X 8 x 2 x 2 Use the piecewise function to ans Find f 2 Submit

10 sing the Fundamental Theorem of Calculus Part 2 Interpret each result geometrically Evaluating definite integrals Evaluate the following definite 1 t 1 6x dx 1 16 So 60x Ass b eg 2T 530 3 sin x dx