Math 21: Spring 2014Final Exam

NAME:

LECTURE:

Time: 180 minutes

For each problem, you should write down all of your work carefully and legibly to receive fullcredit. When asked to justify your answer, you should use theorems and/or mathematicalreasoning to support your answer, as appropriate.Failure to follow these instructions will constitute a breach of the Stanford Honor Code:

• You may not use a calculator or any notes or book during the exam.

• You may not access your cell phone during the exam for any reason.

• You are required to sit in your assigned seat.

• You are bound by the Stanford Honor Code, which stipulates among other things thatyou may not communicate with anyone other than the instructor during the exam, orlook at anyone else’s solutions.

I understand and accept these instructions.

Signature:

Problem Value Score

1 7

2 6

3 6

4 12

5 8

6 11

7 9

8 6

9 11

10 16

11 8

TOTAL 100

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Be of good cheer. [...] You have set yourselves a difficult task, but you will succeed if youpersevere; and you will find a joy in overcoming obstacles. Remember, no effort that wemake to attain something beautiful is ever lost. – Helen Keller

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Problem 1 : (7 points) For this question, suppose that f is a smooth function at x = 0.In other words, all of the derivatives of f exist at x = 0.

a) (3 points) Write down the definition of the Maclaurin series of f (aka the Taylor seriescentered at a = 0).

b) (2 points) Consider the function f : R → R given by the rule f(x) = 5x2 + 3x − 4.Compute the Maclaurin series of f .

c) (2 points) Consider now the function g : R→ R given by the rule g(x) = 4x76− 2x53 +32x17 + 5x12 − 3x2 + 5x+ 1. What is the Maclaurin series of g?

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Problem 2 : (6 points) For each of the following statement, decide if it is TRUE orFALSE. You do not need to show your work.

a) If f is a solution to the differential equation

y′′ − 3y = 0

then 3f is also a solution to this differential equation.

b) Every solution to the differential equation

y′′ + 4y = 0

is of the formy = C1 cos 2x+ C2 sin 2x

for some choice of the constants C1 and C2.

c) If f and g are solutions to the differential equation

y′′′ + 2y′ + y = sinx

then f + g is also a solution.

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Problem 3 : (6 points) Find the sum of the following series, if it exists. Justify youranswer.

a)∞∑

n=0

(−2)n+2

32n

b) 2π − 23π

3!+

25π

5!− 27π

7!+ . . .

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Problem 4 : (12 points) Decide whether the following series converge or diverge. Justifyyour answer.

a)∞∑

n=1

(−1)n

(−4)nn

b)∞∑

n=0

n− 1

3n− 1

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c)∞∑

n=0

ne−n2

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Problem 5 : (8 points) Solve the following differential equations and initial value prob-lems.

a) xy′ = y + x2 sinx, y(π) = 0.

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b)dy

dx= 2 + 2y + x+ xy, y > −1.

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Problem 6 : (11 points) Consider the differential equation

y′′′ + 2y′′ + y′ + 2y = 0.

a) (3 points) Of the following functions, circle all of the ones that satisfy the differentialequation. Show your work.

i. f1 : R→ R given by f1(x) = ex.

ii. f2 : R→ R given by f2(x) = e−2x.

iii. f3 : R→ R given by f3(x) = sin(2x).

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b) (3 points) There are exactly two numbers b such that the function f : R→ R given bythe rule

f(x) = cos(bx)

is a solution of the differential equation

y′′′ + 2y′′ + y′ + 2y = 0.

What are these two numbers?

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c) (3 points) Of the two numbers that you obtained in part b), one should be positive,and one should be negative. Keep the positive one. Assume that for this positive valueof b, the function g : R→ R given by the rule

g(x) = sin bx

is also solution of the differential equation

y′′′ + 2y′′ + y′ + 2y = 0.

You may further assume that f , g, f1, f2, and f3 (these are all of the functions thathave appeared in this problem) are all linearly independent. Using you work so far,write down the general solution of the differential equation

y′′′ + 2y′′ + y′ + 2y = 0.

d) (2 points) True, False, or Maybe true: Any solution of the differential equation is givenby your solution in c), when a choice is made for the arbitrary constant(s).

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Problem 7 : (9 points) Use a power series to solve the differential equation

(x− 3)y′ + 2y = 0.

Simplify your answer.

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Problem 8 : (6 points) You arrive to tutoring shortly after Aparna has finished solvinga problem on Euler’s method. On the board you see the differential equation y′ = y + 2xand the table below:

n xn yn

0 1 0

1 1.5 1

2 2 3

3 2.5 6.5

4 3 12.25

5 3.5 21.3750

6 4 35.5625

a) (2 points) Write down the initial value problem that Aparna was working on.

b) (2 points) What is the step size h in this application of Euler’s method?

c) (2 points) Use Aparna’s work to find an estimate for y(3) where y is a solution to theinitial value problem you wrote in part a).

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Problem 9 : (11 points) An unknown function f : [−1, 1]→ R and several of its deriva-tives are pictured below.

Graph of f :

f(x)

Graph of f ′′:

f’’(x)

Graph of f (4):

f (x)(4)80

-80

Graph of f ′:

f’(x)

Graph of f ′′′:

f’’’(x)

a) (2 points) Find the following:

i. f(0) =

ii. f ′(0) =

iii. f ′′(0) =

iv. f ′′′(0) =

v. f (4)(0) =

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b) (3 points) Write down the Taylor polynomial of degree 3 of f .

c) (3 points) Use your answer from part b) to approximate f(0.1).

d) (3 points) Use Taylor’s inequality to find an upper bound for the error in your approx-imation in part c).Hint: Use the pictures on the previous page to help you find K.

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Problem 10 : (16 points) In this problem, we will compute some digits of the numberπ = 3.14159265 . . ..

a) (2 points) Compute the Maclaurin series of the function f : (−1, 1) → R where f is

given by the rule f(x) =1

1 + x2. Simplify your answer.

b) (2 points) What is the interval of convergence the Maclaurin series of f , where f(x) =1

1 + x2?

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c) (3 points) Use the fact that ∫1

1 + x2dx = arctanx+ C

and the fact that arctan 0 = 0 to compute the Maclaurin series of the function whoserule is g(x) = arctan x.

d) (2 points) What is the radius of convergence of the Maclaurin series of g, where g(x) =arctanx?

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e) (2 points) What is the interval of convergence the Maclaurin series of g, where g(x) =arctanx?

f) (2 points) Write down the Taylor polynomial of degree 3 of the function g.

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g) (3 points) It is a fact that

arctan1

2+ arctan

1

3=π

4.

Therefore,

4

(arctan

1

2+ arctan

1

3

)= π.

For the following questions, you may use the following values:

1

2= 0.5

1

3≈ 0.333

1

24≈ 0.042

1

81≈ 0.012

Hint: If you get any denominator that is not listed here, then you are doing the problemwrong.

i. Use the Taylor polynomial of degree 3 of g to approximate arctan 12.

ii. Use the Taylor polynomial of degree 3 of g to approximate arctan 13.

iii. Add these two numbers together and multiply by 4 to get an approximation forπ.

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Problem 11 : (8 points) The simple gravity pendulum is an idealized model of a pendulumin which a weight (also called a bob in this context) swings on the end of a massless cordfrom a pivot, without any effects from friction or air drag, as in the picture below.

In this context, the angle θ between the cord’s actual position and its equilibrium positionis a function of t satisfying the following initial value problem:

θ′′ = −k sin θ, θ(0) = a, θ′(0) = 0.

for some positive constants a and k. For simplicity, let us use the values a = 1/4 and k = 1in this problem, so that θ satisfies the initial value problem

θ′′ = − sin θ, θ(0) = 1/4, θ′(0) = 0.

a) (1 point) Is the differential equation θ′′ = − sin θ, where θ is a function of t, a lineardifferential equation?

b) (1 point) For small values of θ, we have that sin θ ≈ θ. Making this substitution wenow have the differential equation

θ′′ = −θ.

Is this a linear differential equation?

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c) (4 points) Use a power series to solve the initial value problem

θ′′ = −θ, θ(0) = 1/4, θ′(0) = 0.

Please note that you must use a power series to solve this problem to obtain full credit.

Hint: You should be able to recognize your solution; if you can recognize your solutionyou can check your work.

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d) (2 points) With a set-up as above, we will have that

−1/4 ≤ θ ≤ 1/4

for all values of t. Use Taylor’s Inequality to compute how much of an error in thevalue of θ is introduced by making the substitution sin θ ≈ θ.

Hint: There is a function here which is approximated by its Taylor polynomial. Whatis the function? What is the independent variable? What is the degree of this Taylorpolynomial?

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