Math 20B Final Review

Grab Bag!

100

Integral formula for total distance traveled

What is integral of a to b of |r(t)|dt?

200

The approach for a trig integral with sec^mx and tanⁿx where the power of m is even and at least 2

What is converting all but 2 powers of sec to tan using Pythagorean identities and using u-sub with u=tan?

300

The amount of water that flows into a tank between t=1 minute and t = 5 minutes if the rate of flow is r(t)=t² ft³/min

What is 124/3 ft³?

400

The decomposition of 8/(3x³+7x²+4x)

What is [2/x] + [18/(3x+4)] - [8/(x+1)]?

500

The interval and radius of convergence for the series of [(n+1)/(2n+1)!](x-2)ⁿ

What is (-infinity, infinity) and R=infinity?

Complex Numbers

100

The standard form for complex numbers

What is a + bi?

200

i¹⁹

What is -i?

300

the polar form of z = 9 (as a complex number!)

What is z = 9(cos(pi)+isin(pi))?

400

(3+3i)⁵

What is -972-972i?

500

All roots of (2i)^1/2

Whate are 1+i and -1-i?

Math in the Arctic (Polar!)

100

The variables used as coordinates for polar coordinates

What is r and theta?

200

cos(4pi/3)

What is -1/2?

300

The formula for polar area

What is A = integral of (1/2)r²d(theta)?

400

The values of theta for which r = -1 (within the first 2pi) for the function r(theta)=2cos(theta)-1

What are pi/2 and 3pi/2?

500

The points of intersection of r = 1-sin(theta) and r=2+sin(theta)

What are -pi/6 (or 11pi/6) and 7pi/6?

Integrals All the Way Down

100

The antiderivative of 1/(1+x²)

What is arctanx +C?

200

The new bounds for the integral from -2 to 3 of 1/x³​​​​

What is -2 to 0 and 0 to 3?

300

The trig substitution for x⁵/(36x²+1)^3/2

What is x = (1/6)tany and dx = (1/6)sec²y dy?
(note y is used in place of theta)

400

An appropriate comparison for the integral from 3 to infinity of 1/(x-e^-x) 

(Specify the name of the test we are doing, whether the comparison is larger or smaller, and what this means for the test!)

What is 1/x < 1/(x-e^-x) where the integral diverges by the Direct Comparison Test?

500

The dv, u, du, and v for the integral of (lnx)dx

What is dv=dx, u=lnx, du= 1/x dx, v=x?

Series, Tests, and Convergence, Oh My!

100

The expression which governs the terms of a sequence/series

What is the general term?

200

A Taylor Series centered at c = 0

What is a Maclaurin Series?

300

A series with a common ratio larger than or equal to 1

What is a diverging geometric series?

400

A series of a_n for which the series of |a_n| also converges

What is an absolutely convergent series?

500

A convergence test with three rules for interpreting the results (for full points you must also identify the results!):

1) L = 0

2) L = C >0

3) L = infinity

What is the limit comparison test?

1) If sum b_n converges, sum a_n also converges

2) sum a_n and sum b_n both converge or both diverge

3) if sum b_n diverges, sum a_n also diverges