Math 115

Question 1

What makes Rn an underestimate? overestimate?

Answer 1

f’(x) 0 over

Question 2

What makes Ln and underestimate? overestimate?

Answer 2

f’(x) > 0 under

f’(x)

Question 3

What makes Tn an underestimate? overestimate?

Answer 3

f’‘(x) over

Question 4

What makes Mn an underestimate? overestimate?

Answer 4

f’‘(x) > 0 under

f’‘(x)

Question 5

Arclength

Answer 5

L = a-b | (sqrt(1+(dy/dx)^2))dx

Question 6

x (polar)

Answer 6

rcostheta

Question 7

y (polar)

Answer 7

rsintheta

Question 8

diffeq with carrying capacity

Answer 8

dP/dt = kP(1-P/M)

Question 9

diffeq exponential

Answer 9

dP/dt = kP

Question 10

Area (polar)

Answer 10

A = a–b | 1/2 r^2 dtheta

Question 11

Arclength (polar)

Answer 11

L = a–b | (sqrt(r^2 + (dr/dtheta)^2)) dtheta

Question 12

How does a geometric series converge?

Answer 12

if |x|

Question 13

Test for Divergence

Answer 13

if lim an != 0, then an is divergent

Question 14

Absolute convergence?

Answer 14

sig |an| is convergent

Question 15

Conditional convergence?

Answer 15

sigan converges but sig|an| does not

Question 16

Ratio test inconclusive?

Answer 16

lim = 1

Question 17

Power Series equality (3)

Answer 17

1/1-x =
1+x+x^2+…=
sig x^n
x

Question 18

Integral of Cn(x-a)^n

Answer 18

(Cn(x-a)^n+1)/n+1

Question 19

Derivative of Cn(x-a)^n

Answer 19

n*Cn(x-a)^n-1

Question 20

Cn in Taylor/Mclaurin Series

Answer 20

f^n(a)/n!

Question 21

Taylor inequality

Answer 21

M satisfying |f^(n+1)(x)|

Question 22

lim of x^n/n!

what is the sum of this

Answer 22

0

e^x

Question 23

sinx series and expansion

Answer 23

Sin[x] == Sum[((-1)^n x^(1 + 2n))/(1 + 2n)!, {n, 0, Infinity}]
=x-x^3/3! + x^5/5!….

Question 24

cosx series and expansion

Answer 24

Cos[x] == Sum[((-1)^n x^(2n))/(2n)!, {n, 0, Infinity}]

=1-x^2/2 + x^4/4! …

Question 25

tan-1x

Answer 25

Sum[((-1)^n x^(2n+1))/(2n+1), {n, 0, Infinity}]

= x - x^3/3 + x^5/5-…

Question 26

ln(1+x)

Answer 26

Sum[((-1)^n-1 x^n)/n, {n, 1, Infinity}]

x - x^2/2 + x^3/3 - x^4/4…

Question 27

Solution to a logistic equation

Answer 27

P(t) = M/1+Ae^-kt where A = (M-P0)/P0

Question 28

Solution to initial value problem

Answer 28

P(t) = P(0)e^kt