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Calculus - Assignment Questions with Solutions | MATH 1B, Assignments of Calculus

University of California - BerkeleyCalculus

Material Type: Assignment; Class: Calculus; Subject: Mathematics; University: University of California - Berkeley; Term: Summer 2009;

Typology: Assignments

Pre 2010

Uploaded on 10/01/2009

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Math 1B: Discussion Exercises
GSI: Theo Johnson-Freyd
http://math.berkeley.edu/~theojf/09Summer1B/
Find two or three classmates and a few feet of chalkboard. As a group, try your hand at the
following exercises. Be sure to discuss how to solve the exercises — how you get the solution is
much more important than whether you get the solution. If as a group you agree that you all
understand a certain type of exercise, move on to later problems. You are not expected to solve all
the exercises: some are very hard.
Exercises marked with an §are from Single Variable Calculus: Early Transcendentals for UC
Berkeley by James Stewart. Others are my own or are independently marked.
Taylor Series
If f(x) can be represented by a power series f(x) = P∞
n=0 cnxn, then fis ∞-times differentiable,
and the coefficients are given by
cn=f(n)(0)
n!
These are called the Taylor coefficients for fat 0, and the series P∞
n=0 f(n)(0) xn/n! is called the
Taylor series for fcentered at 0, or the Maclaurin series for f. By translating, we can also define
the Taylor series centered at a:P∞
n=0 f(n)(a) (x−a)n/n!.
Some infinitely-differentiable functions cannot be represented by power series. But: the Taylor
series P∞
n=0 f(n)(0) xn/n! converges to f(x) for |x|< R if fsatisfies that
lim
n→∞ Rn
n!sup
|x|<R f(n)(x)!= 0
Most functions that are actually used — e.g. rational functions, exponential and trigonometric
functions and their inverses, etc. — are equal to their Taylor series inside the radius of convergence.
Functions that equal their Taylor series expansion inside the interval of convergence are called
analytic. Here are a few analytic functions and their Taylor series:
ex=
∞
X
n=0
xn
n!
1
1−x=
∞
X
n=0
xn
cos x=
∞
X
k=0
(−1)kx2k
(2k)! sin x=
∞
X
k=0
(−1)kx2k+1
(2k+ 1)!
ln(1 + x) =
∞
X
n=1
(−1)n−1xn
narctan x=
∞
X
k=0
(−1)kx2k+1
2k+ 1
(1 + x)k=
∞
X
n=0 k
nxn
The last equation is the binomial theorem. The symbol k
nis defined as (k)(k−1) . . . (k−n+
1)/n!, for any number kand integer n. (The numerator is a product of knumbers, just like the
denominator.)
1. Find the Taylor series expansion centered at 0 for 1/(1 −x), by computing derivatives. Is
this what you expect?
1
pf2

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Download Calculus - Assignment Questions with Solutions | MATH 1B and more Assignments Calculus in PDF only on Docsity!

Math 1B: Discussion Exercises

GSI: Theo Johnson-Freyd http://math.berkeley.edu/~theojf/09Summer1B/

Find two or three classmates and a few feet of chalkboard. As a group, try your hand at the following exercises. Be sure to discuss how to solve the exercises — how you get the solution is much more important than whether you get the solution. If as a group you agree that you all understand a certain type of exercise, move on to later problems. You are not expected to solve all the exercises: some are very hard. Exercises marked with an § are from Single Variable Calculus: Early Transcendentals for UC Berkeley by James Stewart. Others are my own or are independently marked.

Taylor Series

If f (x) can be represented by a power series f (x) =

n=0 cnx n, then f is ∞-times differentiable,

and the coefficients are given by

cn = f (n)(0) n!

These are called the Taylor coefficients for f at 0, and the series

n=0 f^ (n)(0) xn/n! is called the

Taylor series for f centered at 0, or the Maclaurin series for f. By translating, we can also define the Taylor series centered at a:

n=0 f^ (n)(a) (x − a)n/n!. Some infinitely-differentiable functions cannot be represented by power series. But: the Taylor series

n=0 f^ (n)(0) xn/n! converges to f (x) for |x| < R if f satisfies that

nlim→∞

Rn n! sup |x|<R

∣∣f (n)(x)

Most functions that are actually used — e.g. rational functions, exponential and trigonometric functions and their inverses, etc. — are equal to their Taylor series inside the radius of convergence. Functions that equal their Taylor series expansion inside the interval of convergence are called analytic. Here are a few analytic functions and their Taylor series:

ex^ =

∑^ ∞

n=

xn n!

1 − x

∑^ ∞

n=

xn

cos x =

∑^ ∞

k=

(−1)k^ x^2 k (2k)! sin x =

∑^ ∞

k=

(−1)k^ x^2 k+ (2k + 1)!

ln(1 + x) =

∑^ ∞

n=

(−1)n−^1 xn n arctan x =

∑^ ∞

k=

(−1)k^ x^2 k+ 2 k + 1

(1 + x)k^ =

∑^ ∞

n=

k n

xn

The last equation is the binomial theorem. The symbol

(k n

is defined as (k)(k − 1)... (k − n + 1)/n!, for any number k and integer n. (The numerator is a product of k numbers, just like the denominator.)

  1. Find the Taylor series expansion centered at 0 for 1/(1 − x), by computing derivatives. Is this what you expect?
  1. Write out the Taylor series expansion centered at 0 of the following functions. You should use a combination of direct differentiation and manipulation of power series. For each function, determine the radius of convergence. (a) sin 2x (b) xex^ (c)

1 + x (d) cosh x (e) ln(5 + x) (f) 1 /(2 − x) (g) sin(2x − π/4) (h) (3 + x)^1 /^3

  1. § Let f (x) = ex 2 . Prove that f (2n)(0) = (2n)!/n!. What is f (2n+1)(0)?
  2. (a) Fill in the following outline to prove that

lim x→ 0 p(1/x)e−^1 /x 2 = 0

for any polynomial p(x):

  • e−^1 /x 2 < e−^1 /x^ for 0 < x < 1;
  • limx→∞ xne−x^ = limx→ 0 nxn−^1 e−x^ by L’Hospital’s Rule;
  • limx→∞ xne−x^ = 0 by induction;
  • limx→∞ p(x)e−x^ = 0 by the sum rule;
  • limx→ 0 p(1/x)e−^1 /|x|^ = 0 by a substitution;
  • limx→ 0 p(1/x)e−^1 /x 2 = 0 by the Squeeze Theorem. (b) Let f (x) = e−x 2 . Prove that any derivative of f (x) is of the form p(1/x)e−x 2 for some polynomial p. (c) It is a fact that the derivative of any function does not have a removable discontinuity. Use this fact to show that f (n)(0) = 0 for every n. (d) What is the Taylor series expansion of f (x) = e−^1 /x^2 centered at 0? What is the radius of convergence of this series? Why should this make you troubled?
  1. § By using the Taylor series expansion for sin(x) only up to the cubic term, approximate the non-zero solution for x^2 = sin(x)
  2. § Prove that (^) ∫ 1 0

1 + x^30 1 + x^60 dx = 1 +

c 31 where 0 < c < 1.

  1. Let fn be the Fibonacci sequence, given by the rules f 0 = 0, f 1 = 1, and fn = fn− 1 + fn− 2 for n ≥ 2. Let F (x) =

n=0 fnx n. You may ignore questions of convergence for this exercise.

(a) Explain why the equation ∑^ ∞

n=

fnxn^ =

∑^ ∞

n=

fn− 1 xn^ +

∑^ ∞

n=

fn− 2 xn

follows from the definition of the Fibonacci sequence. (b) Rewrite each side of the above equation in terms of x and F (x). (c) Use your answer from part (b) to find an elementary expression for F (x). (d) Use partial fractions to find an explicit power series for F (x). (e) Thus find an explicit formula for fn, the nth Fibonacci number.