Calc Midterm 2

Question 1

When should you use substitution?

Answer 1

When you have a function inside a function in a product with the derivative of the inner function

Question 2

When should you use integration by parts?

Answer 2

when you have the product of two separate functions

Question 3

Formula of Integration by Parts

Answer 3

∫udv = uv - ∫vdu

Question 4

What should you remember about in the v symbol?

Answer 4

don’t add + c

Question 5

What are you NOT, ABSOLUTLEY UNDER NO CIRCUMSTANCES, GOING TO FORGET

Answer 5

+ c for the final answer

Question 6

In LIATE, _ should be earlier and _ should be later.

Answer 6

u’s earlier, du’s later

Question 7

What does LIATE stand for

Answer 7

Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential

Question 8

What will you not forget for the dv?

Answer 8

Adding the dx

Question 9

Antiderivative of e^-x

Answer 9

-e^-x

Question 10

When should you use tabular integration by parts?

Answer 10

If you know you’ll need to do regular more than once

Question 11

Steps of tabular integration by parts

Answer 11

1. Differentiate u until its zero
2. antidifferentiate dv accordingly
3. Draw the arrows
4. Label starting + then -
5. Make it all one addition phrase with the last product being and integral

If both don’t go to zero, go until dv matches the original dv

Question 12

Antiderivative of -e^-x

Answer 12

e^-x

Question 13

e^x should go

Answer 13

after the x expression

Question 14

Double angle identities
cos(2x) =

Answer 14

cos^2(x) - sin^2(x)
1 - 2sin^2(x)
2cos^2(x) -1

Question 15

Double angle identities
sin(2x) =

Answer 15

2sin(x)cos(x)

Question 16

Walk me through how FTC Part Two is made

Answer 16

A’(x) = f(x)
A(x) =∫(a->x)f(t) dt

A’(x) = d/dx A(x)
A’(x) = d/dx[∫(a->x f(t)dt]
A’(x) = f(x)

Question 17

Definition of FTC Part Two

Answer 17

If f is continuous on an interval, then f has an antiderivative on that interval.

If “a” is any point on the line, then the function is
F(x) = ∫(a->x)f(t)dt

Question 18

FTC Part Two
f(x) =

Answer 18

d/dx [∫(x->a)f(t)dt]

Question 19

FTC Part Two when x is a function

Answer 19

d/dx [∫(g(x)->a)f(t)dt] = f(g(x))g’(x)

Question 20

Factor
2x^2-5x-3

Answer 20

2(-3) = -6
-6 + 1 = -5
2x^2 -6x + x -3
2x(x-3) + 1(x-3)
(2x+1)(x-3)

Question 21

Restrictions of Heavyside Method

Answer 21

1. The denominator must factor into non-repeated linear factors
2. The degree of the numerator must be only one less than the degree of the denominator
3. Don’t use it for a quadratic denominator

Question 22

Improper Rational Fractions

Answer 22

If the degree of the numerator is greater than the denominator

Use long division. The answer is an integral, find the integral.

Question 23

Three types of improper integrals

Answer 23

1. Infinite intervals
2. Infinite discontinues
3. Infinite discontinuities and intervals

Question 24

If the limit exists it

Answer 24

Converges (find the value of the integral)

Question 25

If the limit doesn’t exist

Answer 25

Diverge, there is no value

Question 26

∫sin^m(x)cos^n(x) dx
n is odd

Answer 26

Split of a cos(x)
apply cos^2(x) = 1 - sin^2(x)
Make the substitution u = sin(x)

Question 27

∫sin^m(x)cos^n(x) dx
m is odd

Answer 27

Split of a sin(x)
apply sin^2(x) = 1 - cos^2(x)
Make the substitution u = cos(x)

Question 28

∫sin^m(x)cos^n(x) dx
m even
n even

Answer 28

Use relevant identities to reduce the powers on sin(x) and cos(x)

Question 29

Restrictions for Heavyside Method

Answer 29

1. The denominator must factor into non-repeated linear factors
2. The degree of the numerator must be only one less than the denominator
3. Don’t use this for quadratic functions (class rule)

Question 30

Improper Fractions

Answer 30

Degree of the numerator > or equal to degree of the denominator

Question 31

Improper integral for [a, +∞]

Answer 31

∫+∞-> a f(x)dx = lim∫[b->a]f(x)dx
b->∞

Limit exists? converge. solve the integral

Limit does not exist? diverge. no value

Question 32

Improper integral for [-∞, b]

Answer 32

∫b->-∞f(x)dx = lim∫[b->af(x)dx
a->-∞
Limit exists? converge. solve the integral

Limit does not exist? diverge. no value

Question 33

Improper integral for [-∞, +∞)

Answer 33

∫-∞->+∞f(x)dx = ∫(-∞->c)f(x)dx + ∫(c->+∞)f(x)dx

Use 0 for c
Both converge? converge
either diverge? diverge