Unit 7

Question 1

POWER-REDUCING FORMULAS

Question 2

sin^2 θ =

Answer 2

1-cos(2θ) / 2

Question 3

cos^2 θ =

Answer 3

1+cos(2θ) / 2

Question 4

tan^2 θ

Answer 4

1-cos(2θ) / 1+cos(2θ)

Question 5

INVERSE TRIG INTEGRATION

Question 6

∫ du / (a^2 - u^2)^1/2 =

Answer 6

arcsin(u/a) + C

Question 7

∫ du / a^2 + u^2 =

Answer 7

1/a arctan(u/a) + C

Question 8

∫ du / u(u^2 - a^2)^1/2 =

Answer 8

1/a arcsec(|u|/a) + C

Question 9

TRIG. INTEGRATION

Question 10

∫ tanx dx =

Answer 10

ln|secx| + C

Question 11

∫ cotx dx =

Answer 11

ln|sinx| + C

Question 12

∫ secx dx =

Answer 12

ln|secx + tanx| + C

Question 13

∫ cscx dx =

Answer 13

ln|cscx - cotx| + C

Question 14

u-substitution

Answer 14

make sure to evaluate EVERYTHING in terms of u

Question 15

INTEGRATION BY PARTS
- ultraviolet voodoo / tabular method

Answer 15

∫ u dv = uv - ∫ v du

• use when it seems like the function can’t be integrated

Question 16

for ultraviolet voodoo, choose u in this order:

Answer 16

1. inverse trig.
2. polynomial
3. exponential
4. trig.

(make the part that’s easiest to integrate dv)

Question 16

complete the square

Answer 16

watch a video

Question 17

TRIG. SUBSTITUTION WITH RADICALS

Answer 17

either in the form
- (a^2 - u^2)^1/2 (adjacent side)
- (u^2 - a^2)^1/2 (opposite side)
- (u^2 + a^2)^1/2 (hypotenuse)

• find x, dx, and the radical by putting the other sides over the constant
• look up examples

Question 18

PARTIAL FRACTION DECOMPOSITION

Answer 18

• factor denominator, then split into parts
• make sure if part has a power higher than one, make numerator in terms of that
  - ex. (Cx+D) / (x^2 + 4)

Question 19

IMPROPER INTEGRALS

Answer 19

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

• if you get a value, it converges (reaches a distinct value)
• if you get DNE, it diverges (continues to grow without bounds)

Question 20

for a limit as x approaches infinity, if the higher power is in the _____

Answer 20

• denominator: limit = 0
• numerator: limit => DNE
• equal powers: limit = coefficients

Question 21

ln(∞) =>

Answer 21

∞

Question 22

ln(-∞) =>

Answer 22

DNE

Question 23

e^∞ =>

Answer 23

∞

Question 24

e^(-∞) =>

Answer 24

0

Question 25

NEWTON'S LAW OF COOLING

Answer 25

y - T = Ce^(Kt) C: initial condition T: ambient (room temperature)

Question 26

half-life

Answer 26

y = Ce^(Kt)

Question 27

K =

Answer 27

ln(0.5) / half-life

Question 28

SLOPE FIELDS

Question 29

solution curve

Answer 29

the graph of a solution of a differential equation

Question 30

differential equation

Answer 30

can be interpreted as a statement about the slopes of its solution curves

Question 31

EULER'S METHOD

Answer 31

- with slope fields we are always tangent to the solution curve - with Euler's we calculate the slope at that point

Question 32

Euler's Method Equation

Answer 32

y, = previous solution + (derivative at that previous point)(step size)

Question 33

step size =

Answer 33

final-initial / steps

Question 34

INTEGRATION BY SEPARATION OF VARIABLES

Answer 34

- push as much as possible on x side before integrating - e^C = C - dividing/multiplying by C = multiplying by C

Question 35

RELATED RATES INFO.

Question 36

volume of a cylinder =

Answer 36

hπr^2

Question 37

volume of a cone =

Answer 37

(h/3)πr^2

Question 38

volume of a cube =

Answer 38

a^3

Question 39

volume of a sphere =

Answer 39

(4/3)πr^3

Question 40

area of a circle =

Answer 40

πr^2

Question 40

area of a square =

Answer 40

a^2

Question 41

a^2 + b^2 =

Answer 41

c^2

Question 42

L'HOPITAL'S RULE

Answer 42

- use when you plug in the value the limit is approaching, and you get 0/0 or ∞/∞ - take the derivative of the numerator and denominator separately and re-evaluate - PROVE it can be applied by showing limit of numerator and denominator each = 0, or DNE