Calculus 2 Exam

L'Hopitals Rule

100

What are the Justifications Needed for L'H?

1. Labeling the Indeterminate Form

2. Label the step that uses L'H

3. Label Which steps uses by continuity

200

By Continuity can happen in 2 different ways, what are the ways?

1. Plugging in a value and it not being indeterminate (infinity doesn't count)

2. Moving a limit inside of a function (like natural log)

300

limit x-> infinity 

(1+1/x)^x

e

400

limit x-> 0+ 

xlnx

0

500

limit x-> (pi/2)- 

tan(x)^cos(x)

1

Integration By Parts

100

What is LIATE, and what is it an acronym for?

LIATE (How to choose the best U for integration by parts)

Logrimithic 

Inverse Trigonometry

Algebra

Trigonometry

Exponentials 

200

What is the Justification for integration by parts

State what u,du,v,dv

Work with Limits of Integration if necessary

300

Integral:

x²e^-3xdx

-1/3e^-3x(x²+2/3x+2/9)+C

400

Integral: Bounds: (1,2)

ln(x)/x^2

-ln(1/2)+1/2

500

Integral:

xsec²3xdx

1/3(x)tan(3x)-1/9ln(sec(x))+C

Integrals of Powers

100

What is the Justification for Integrals with Powers?

1. Stating u and du

2. If definite integral, change the limits of integration

200

T/F if sine has an odd power and cosine has an even power it is better to have u = cosine and du = sine

True

300

Integral 

sin⁶xcos³x

1/7sin⁷x-1/9sin⁹x+C

400

Integral:

sec⁹xtan⁵x

1/13sec¹³x-2/11sec¹¹x+1/9sec⁹x

500

Integral:(0,pi/4)

sin⁵xdx

−(43√2−64)/120

Trig Substitution

100

What is the Justification for Trig sub?

1. State what range of values theta may take 

2. Draw a Triangle 

3. If usub you must change the limits of integration

200

T/F When using trig sub we can ignore the properties of inverse trig functions

False

300

Integral:

√9-4x²)/x

3ln((3-√9-4x²)/x)+√9-4x²)+C

400

Integral:

1/x⁴√9-x²

−(9−x2)^3/2/243x³ −√9−x²/81x + C

500

Integral: Bounds (0,3)

x³/√x²+9

18-9√2