Calculus Questions

Find a power series solution of the form y x 0 Cnx of the given differential equation x y y 0 First compute y x Use cn for c in your answer Then x y x y Co C1 Cn n 1 Use c0 for co c1 for c cn for c and cnm1 for C 1 in your answer Requiring that the terms of this series for x y y vanish gives X7 1 Cn 1 for n 2 100 x 2 The series solution converges to the familiar function y f x x

Find the power series solution of the initial value problem Assume the form y x 0 Cnx n 0 y x n 1 Then compute n C x 1 0 Cn First compute y x 1 n 1 n n Cn 1 The recurrence relation implies Cn 0 n 1 y 4y 0 Then y 4y n 2 n 1 Requiring that the terms of this series for y 4y vanish gives the recurrence relation Cn 2 Cn for n 0 1 2 The initial conditions imply Co C for odd indices n Cn 2 0 Cn 1X Cn 1x 1 n 2 n 1 2m 1 y 0 1 Cn 1 4 for even indices n 2m The series 0 Cnx is familiar as the series for the elementary function in 0 y f x m 0 1 2 m 0 1 2 y 0 0 Cn x Cn 2x

3 Consider the function f x 1 x4 5x 4 Use graphical methods to discuss the differentiability of the function In particular find all values of x for which the function appears non differentiable and explain why Include at least one graph illustrating each point of non differentiability

Find a power series solution Cnx of the given differential equation Determine the radius of convergence of the resulting series and by comparing with a table of well known series identify the series solution in terms of familiar elementary functions y 4y y x 0 The radius of convergence is p The series gives the familiar function Use c0 for co in your answer Use c0 for co in your answer

6 Below is a graph of the curve y 10 1 50sinx C x 3 10 Looking at the graph it s clear that the function has three zeros in the domain 5 x 5 the three zeros and find the slopes of the tangent lines to the curve at those x values

1 Below is a graph of the function y f x 5 From your observations of this graph state with reasons the values of x at which fis not differentiable 2 Consider the function f x x x Use graphical methods to discuss the differentiability of the function In particular find all values of x for which the function appears non difforontin

Find the recurrence relation for the coefficients of the power series solution of the differential equation y y 2y 0 Assume the form y x 0 Cnx in y x n 1 Then compute First compute Cnx 1 0 y x 1 Then y y 2y Cn 2 Cn 1 Requiring that the terms of this series for y y 2y vanish gives the recurrence relation Cn 2 Cn for n 0 1 2 Cn 1 Cn 1X Cn 1X 1 0 Cn 2X Cn x

6 Below is the graph of y g x 17 0 4 I Note The open circles at x 2 are to imply that the function g is not defined there Answer the following questions about the function g x A Is is possible impossible or certain that g is continuous at x 2 Explain B Is it possible impossible or certain that g 2 0 Explain C Is it possible impossible or certain that g has a vertical asymptote at x 2 Explain D Is it possible impossible or certain that g has a jump discontinuity at x 2 E Is it possible impossible or certain that g has a removable discontinuity at x 2 F Make a sketch of what the graph of y g x could look like

x2 2x 1 2x3 3x 3x 1x4 x 9x 24 if 4 x Discuss the continuity and differentiability of the function at x 1 and x 4 Prove your assertions mathematically a graphic proof will NOT suffice 4 Consider the function f x W 5 Below is the graph of y g x 101 0 if x 1 if 1 x 4 Is it possible impossible or certain that the function is continuous atv 22 Explain

1 point For the system of differential equations x t x y 2xy 20 y t 8x 2y xy the critical point xo yo with xo 0 yo 0 is xo Change variables in the system by letting x t xo u t y t yo v t The system for u u is u v Use u and u for the two functions rather than u t and u t For the u v system the Jacobian matrix at the origin is A You should note that this matrix in 1 Yo

1 point Find the eigenvalues and corresponding eigenvectors for the matrix P A V saddle center point ellipses spiral source spiral sink none of these 28 12 6 12 and For the linear system of differential equations Px the critical point is best described as source unstable node sink stable node U

a Find the eigenvalues and corresponding eigenvectors for the matrix P A x t x 1 V 17 20 15 18 and A 11 E b Find the most general solution to the linear system of differential equations Px Use c and c to denote arbitrary constants and enter t as c1 and c2

5 Below is a graph of the curve y 4x cosx 10 10T SIX 10 Your friend claims that the slope of the tangent line to the curve at x 4 is the negative of th slope of the tangent line to the curve at x 4 Prove mathematically whether your friend is right or wrong Is the slope of the tangent line to the curve at x a the negative of the slope of the tangent to the curve at x a for any other values of a Explain

13 Find the limit of the sequence if it converges otherwise indicate divergence 2 2 n an 14 Find the limit of the sequence if it converges otherwise indicate divergence an n n 20n

Now take a look at the following three functions f x x 2x 1 500 g x x 0 X k x x 21 For each function do the following 1 Compute the derivative that is the derivative function by evaluating the limit for f x h f x example limo h Show your work Notice that for the function k x you ll have to compute the limit differently depending on whether x 2 or x 2 and you ll end up with a piecewise function for the derivative Since the graph of k x is made of straight lines you can also find its derivative by looking at the actual slopes 2 Graph the derivative f x on your calculator by assigning it to the function Y Draw the graphs that you create on your calculator Make sure to label and submit them alon with the rest of the assignment 3 Graph the function Y nDeriv original function X X By doing this you re having the calculator evaluate the derivative numerically at each point and graph the results 4 Discuss how the two graphs are the same or different Zoom in and out to find places where the graphs differ A good idea is to set Y3 Y2 Y deselect but don t erase the functions Y and Y2 and look at the graph of Y3 Discuss what this illustrates and see if it helps you in your comparison Discuss the things you did in order to compare the two graphs After you ve done these things for each of the three functions make a final comment about when the graph made by nDeriv was most or least accurate Try to come up with a general comment such as Dori

1 Find the slope of the tangent line to the curve y x3 2x 1 at the point 1 0 Find the equation of that tangent line On your calculator graph both the curve and the tangent line on the same set of axes with viewing window x 2 3 y 2 2 Show all work except for anything concerning the graphs you made on your calculator they were for your observation only I 2 Find the slope of the tangent line to the curve y at the point 1 1 Find the equation of that tangent line On your calculator graph both the curve and the tangent line on the same set of axes with 11 523 Show all work oxcont for anythin ho aranhe

3 Below is a graph of the curve y 10 5 15 10 5 0 10 cosx sinx 1 2x2 I 3 10

4 Here is a graph of the curve y x3 5x 4x 101 I Your friend tells you that the slope of the tangent line to the curve at x 1 is the same as the slope of the tangent line to the curve at x 1 Prove mathematically whether your friend is right or wrong Is the slope of the tangent line to the curve at x a the same as the slope of the tangent line the curve at x a for any other values of a Explain

Earth s distance from the center of the sun as a function of time 1 Point

7 Use a half angle formula to find the exact value of tan 8 tan 7z 8 2 16

Find the amplitude phase shift and period of the function y 1 2 cos 2x 2 Give the exact values not decimal approximations Amplitude 1 Phase shift 3 X B

37 If f x sec x then lim X A 0 B sec C sec tan 5 D nonexistent 15 00 f x f 5 T X 3 is

4 Suppose f x if x 2 x 3 if 2 x Identify any points of discontinuity and determine giving reasons if they are removab infinite essential or jump discontinuities 6 Points

A First class postage as a function of weight 1 Point

B Atmospheric pressure as a function of altitude 1 Point

1 Given f x x 17 identify any points of discontinuity and determine giving reasons if 6x x 1 they are removable infinite essential or jump discontinuities 2 Points

x3 2x 2 x ax b 4x4 3x 5x 2 if 1 x find values for a and b so that the function is continuous for all x 4 Points 6 Given g x if x 2 if 2 x 1

C Average daily temperature as a function of the day of the year 1 Point

5 Find all the constants a that make the function f x 3 a x if x 3 a x if x 3 continuous for all x 2 Points

3 Suppose f x sin x2 4 1 Identify any points of discontinuity and determine giving reasons if they are removable infinite essential or jump discontinuities 3 Points

Consider the system 8 238 E nd the solution curve passing through x y 1 0 and write it in th m F x y 0

Sally s anime club is selling brownies to raise money Instead of just aiming blindly Sally does some market research to try and maximize the club s prom E 4p2 10P 125 Where is the amount the club will earn from the bake sale and P is the price they charge for each brownie both measured in dollars A What price should the club charge for their brownies to maximize their profits B How much money should the club expect to make if they sell their brownies at this price

C Write an equation for C then state the slope of the graph and determine its interpretation in the context of the problem Total Cost for Electrician Services 330 900 850 Boo 730 00 6gn ban 49 300 450 400 350 300 230 ann 200 54 o 0 15 Number of Hours Working The slope of the function is which represents the number of hours the electrician spends at the house the set fee for the house call the hourly rate charged H MET CERE S O O d U

Khalil Problem 4 3 Consider the system x x 1 x x x x 1 x x Use a quadratic Lyapunov function to show that the origin is asymptotically sta

Given z 2 xy x uv w y u ve z du then find 20 043 when u 1 v 3 w 0

The weight in pounds of a newborn baby t months after birth can be modeled by W A graph of W is shown below Write an equation for W then state the slope of the graph and determine its interpretation in the context of the problem Weight of the Baby Pounds 26 24 13 20 18 9 14 12 10 CO 4 W O 3 8 Number of Months Since Birth 10 11 ha

a c Use the criteria indicated to create a pair of 3 x 3 matrices D D where the entries of the matrix D are all integers 20 marks i d 1 5 d3 2 2 Create your own substitution cipher Then use it and the matrix D you created in a to encrypt the title of your favourite movie 20 marks ii Use the matrix D and the appropriate substitution cipher to decrypt the message from b 20 marks

f x 20x3 12x2 6 f 0 3 f 1 0 find f

b Let f R R be a function and suppose ff t dt a Show that x y z b y 2 where D is the unit ball x y z x y z 1 dv 4

Solve the following boundary value problem Ytt 25yxx 0 x 4 t 0 y 0 t y 4 t 0 y x 0 0 y x 0 x 4 x y x t Type a series using n as the index variable and 1 as the starting index

3 The int operator in your calculator is the greatest integer function It returns the greatest integer less than or equal to the operand and is written using square brackets x or x For example it would give back 82 for 82 976 In Smalltown AZ the students receive grade points as follows 4 0 for any grade 90 or higher 90 100 3 0 for any grade 80 or higher 80 90 2 0 for any grade 70 or higher 70 80 1 0 for any grade 60 or higher 60 70 Notice students NEVER fail And students never get 100 A Write a function that correctly returns grade points for gradesfrom 60 to 100 78 45 should give a 2 0 Is your function even odd or neither I

The number of times a new pop song has been downloaded t weeks after its initial release is given by f t a After how many weeks is the rate of change of the number of downloads maximized b What is the rate of change of the number of downloads at the time found in part a c How many times has the song been downloaded at the time found in part a a The rate of change of the number of downloads is maximized after weeks Do not round until the final answer Then round to two decimal places as needed 3 000 000 1 300 0 85

x b Given the function g x cos x that will make the function differentiable A 0 B 1 C 2 D No such value exists x 0 x 0 at x 0 Find the value of b if any E There is not enough information to determine the value

4 The domain of f x is 0 and 1 2 3 4 1 17 X f x 1 12 1 2 5 10 10 A Find an equation for f x that satisfies the domain requirements and the data table Demonstrate that your function satisfies the data points in the table Is your function even odd or neither I

If f x sin 7 5x find f 7 O A 0 657 OB 0 754 O C 0 OD 0 657 OE 3 770

Which of the following functions is continuous but not differentiable at x 1 x x 1 2x x 1 I y x 1 II y III y x x 1 1 x 1 X A I only B II only C I and III only D II and III only E All of these functions are continuous but not differentiable at x 1

The derivative of y 1 sin x COS X is O A tan x O B O C O D O E 1 1 sin x sin x COS X sin x Cos x cos x sin x 1 sin x 2

The slope of the tangent line to the graph y x at the point 1 is A 1 B 0 C 2 D 1 3 E Undefined

Suppose the functions fand g and their derivatives have the following values at x 1 and x 2 Let h x f g x Evaluate 1 A 6T B 24 C 10 D 1 E 15 X 1 2 f x g x f x g x 8 2 3 3 4 113 2TT LO 5 K

The inflection point of the curve y 8x 24x 7x 3 is O A 1 23 OB 0 3 O C 1 21 OD 2 59 OE There is no point of inflection