## Description

10] 1. (a) Sketch a graph of the function

f(x) = (

−

√

4 − x

2 −2 ≤ x ≤ 0

2 − 2|x − 2| 0 < x ≤ 4

on the interval −2 ≤ x ≤ 4 and calculate the definite integral R

4

−2

f(x) dx

in terms of area (do not antidifferentiate).

(b) Use the Fundamental Theorem of Calculus to calculate the derivative of

F(x) =

x

2

R

x

e

sin(π t) dt,

and determine whether F is increasing or decreasing at x = 1.

[6] 2. Find F(x) such that F

0

(x) = x

2 + 2x

x

2 + 4

and F(0) = 0.

[11] 3. Find the following indefinite integrals:

(a)

Z

13 − x

x

2 − x − 6

dx (b)

Z

x

3/2

ln2

(x) dx

[12] 4. Evaluate the following definite integrals (give the exact answers):

(a)

Z

1

0

2

x

4

x + 1

dx (b)

Z

2

1

√

4 − x

2 dx .

[8] 5. Evaluate the given improper integral or show that it diverges:

(a)

Z∞

e

dx

x ln(x

2

)

(b)

Z

1

0

dx

(1 − x)

3/4

MATH 205 Final Examination

[18] 6. (a) Sketch the curves y = x (3 − x

2

) and y = −x, and find the area enclosed by

the two curves. (HINT: find first the points of intersection of the curves.)

(b) Sketch the region enclosed by y = cos(2x) and the x-axis on the interval

[0,

π

2

], and find the volume of revolution of this region about the axis y = −1.

(c) Find the average value of the function f(x) = sec4

(x) on the interval

−

π

4

,

π

4

.

[9] 7. Find the limit of the sequence {an} as → ∞ or prove that it does not exist:

(a) an =

e

n − n

3

3

n

(b) an =

(−1)nn

√

1 + 4n2

(c) an = ln(n + 2n

2

) − ln(2n + n

2

).

[12] 8. Determine whether the series is divergent or convergent, and if convergent,

then whether absolutely or conditionally convergent:

(a)

X∞

n=1

n

2/3

1 + 2n

(b)

X∞

n=1

(−1)n+1 sin

1

n

(c)

X∞

n=1

(−1)n+1 2

n+1

n!

[6] 9. Find (a) the radius and (b) the interval of convergence of the series P∞

n=1

(x + 2)3n

n2 8

n

[8] 10. (a) Derive the Maclaurin series of f(x) = x

2

e

3x

.

(HINT: start with the series for e

z and then let z = 3x).

(b) Find the values of x for which the following series converges

X∞

n=0

(x

2 + 1)n

2

n+1

and, for these values of x, find the sum of the series as a function of x.

Bonus question [5]. Find the values of p (if any) for which the series

P∞

n=5

1

n ln n(ln (ln n))p is convergent