Exam 4

Question 1

Extended Ratio Test

Answer 1

the limit as n approaches infinity of the absolute value of a of n+1 over the absolute value of a of n. If it is less than one it converges, if it is greater the one it diverges.

Question 2

Extended Root Test

Answer 2

The limit as n approaches infinity of the nth root of the absolute value of a of n. If it is less than one it converges, if it is greater than one it diverges.

Question 3

Power Series

Answer 3

a series which has variable terms.

Question 4

Interval of Convergence

Answer 4

the set of all values of x for which a power series converges.

Question 5

Radius of Convergence

Answer 5

half the length of the interval of convergence.

Question 6

The variable x can be:

Answer 6

any real number

Question 7

Because x can be any real number we must use tests that:

Answer 7

Work for any type of series
-Extended Ratio Test
-Extended Root Test

Question 8

General form of Power Series Representation

Answer 8

For x is greater than negative one but less than one: 1/(1-x) = the series from n=0 to infinity of x^n = 1 + x + x^2 + x^3 + …

Question 9

Does differentiating of integrating a power series change its interval of convergence or radius of convergence?

Answer 9

No

Question 10

Taylor Series

Answer 10

f(x) = f(a) + (f’(a)/1!)(x-a) + (f’‘(a)/2!)(x-a)^2 +…

Question 11

Maclaurin Series (Taylor series for a=0)

Answer 11

f(x) = f(0) + (f’(0)/1!)x + (f’‘(0)/2!)(x^2) + …

Question 12

Power Series Expansion : f(x) = e^x

Answer 12

the series from n=0 to infinity of (x^n)/n! for x is greater than negative infinity but less than infinity.

Question 13

Power Series Expansion: f(x) = sinx

Answer 13

the series from n=0 to infinity of (-1)^n ((x^2n+1)/(2n+1)!) for x is greater than negative infinity but less than infinity.

Question 14

Power Series Expansion: f(x) = cosx

Answer 14

the series from n=0 to infinity of (-1)^n ((x^2n)/(2n)!) for x is greater than negative infinity but less than infinity.

Question 15

nth degree Taylor polynomial

Answer 15

Tn(x) = f(a) + (f’(a)/1!)(x-a) + (f’‘(a)/2!)(x-a)^2 + … + (f^(n) (a)/n!)(x-a)^n

Question 16

Taylor’s formula/Taylor’s Inequality

Answer 16

|Rn(x)| is less than or equal to (M/(n+1)) *(|x-a|^n+1) where |f^(n+1) (x)| is less than or equal to M.

Question 17

Arc length - function of y

Answer 17

the integral from c to d of the square root of 1 + (dx/dy)^2 dy

Question 18

Arc length - function of x

Answer 18

the integral from a to b of the square root of 1 + (dy/dx)^2 dx

Question 19

Area of a surface of revolution - function of x

Answer 19

the integral from a to b of 2pir times the square root of (1+ (dy/dx)^2) dx

Question 20

Area of a surface of revolution - function of y

Answer 20

the integral from c to d of 2pir times the square root of (1+ (dx/dy)^2) dy

Question 21

Parameter

Answer 21

variable t

Question 22

Cycloid

Answer 22

curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line.

Question 23

First derivative of parametric equations

Answer 23

(dy/dx) = (dy/dt)/(dx/dt)

Question 24

Second derivative of parametric equations

Answer 24

(d^2 y/dx^2) = (d/dt(dy/dx))/(dx/dt)

Question 25

Point-slope form

Answer 25

(y-y1) = m(x-x1)

Question 26

Area under a Curve - Parametric equations

Answer 26

the integral from alpha to theta of g(t) * f’(t) dt

Question 27

Arc Length - Parametric equations

Answer 27

the integral from alpha to theta of the square root of ((dx/dt)^2 + (dy/dt)^2) dt

Question 28

Surface Area - Parametric equations

Answer 28

the integral from alpha to theta of 2pir * square root of ((dx/dt)^2 + (dy/dt)^2) dt