CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 1.
Due January 15.
1.1. Formulate and prove a statement similar to Lemma 1.8 for the case when f (x0 ) < 0. (See Lecture 1 of the Course Notes). 1.2. Give an example of a function which is continuous on the in
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 10.
Due April 5.
10.1. Find the Maclaurin series for f (x). Find the radius of convergence of the series, and show, using Lagranges remainder theorem that the series converges to f (x). (i) f (x) = xe2x . (ii)
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 9.
Due March 26.
9.1. Find the radius and the interval of convergence of the following power series
(i)
n=0
5n x3n . 2n + 1 (x 1)3n . 3n2 + 2 3n xn .
n=0
n 2 2
(ii)
n=0
(iii)
9.2. Prove that if lim equals 1/
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 8.
Due March 12.
8.1. Determine whether the series converges absolutely, conditionally, or diverges. n (i) (1)n1 n + 10
n=1
(ii)
n=1
n5
2 3n 4n + 3
n
(iii)
n=1
sin 5n . n5
8.2. Let cfw_fn be the Fibonacci s
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 7.
Due March 5.
7.1. Find the values of p for which the series is convergent: 1 . n(ln n)p n=2 7.2. Determine whether the series converges or diverges. n+3 (i) 3 n7 + n2 + 1 n=1
(ii)
n=1
e1/n n
(iii)
n! nn
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 6.
Due February 26.
6.1. Determine whether the series
n=1
n2
1 + 5n + 6
is convergent or divergent. If it is convergent, nd its sum. 6.2. Find the value of c such that
n=1
2nc = 2010. an is Sn = 3 n2n , nd a
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 5.
Due February 12.
5.1. Using only the -N denition of convergence of a sequence prove 2n + 1 2 lim =. n 3n + 2 3 5.2. Determine without proof sup S , the supremum of the set S given by n S= , where n, m N . n
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 4.
Due February 5.
4.1. Evaluate
0
1
4.2. Evaluate
x dx 2 x4
y3 + 1 dy y3 y2
4.3. Determine whether the following improper integrals converge or diverge. Evaluate the integral if it converges. dx (i) 21 x 2
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 3.
Due January 29.
3.1. Evaluate x2
dx 6x + 13 dx 1 e2x
3.2. Evaluate ex 3.3. Evaluate
2x4 + 5 x2 2 dx 2x3 x 1
3.4. Write out the form of the partial fraction decomposition of the function t P (t) = 2 2 (t2 +
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 2.
Due January 22.
2.1. First use a substitution, then integration by parts to evaluate sin(ln x)dx. 2.2. Evaluate
0
ecos t sin 2t dt. 2.3. Use integration by parts to prove the reduction formula secn x dx =
CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 11.
Due April 9.
11.1. Find the equation of the tangent line to the parametric curve given by x = 3t2 + 1 y = 2t3 + 1, that passes through the point (4, 3). 11.2. Find the length of the loop of the curve x = 3