Finalprep

Question 1

Differential equation rate equation

Answer 1

P(t) =de^(kt) + C

Question 2

What’s the Cooling formula

Answer 2

T - Temp
t - time
T_a is room temp

dT/dt = K(T-T_a)

Question 3

General Equation: For Cross sections

Answer 3

b
∫ (Area Function in terms of the axis perpendicular to)^2 d(axis perpendicular)
a

Question 4

What are the area functions?

Answer 4

Squares: S^2
Semi-cricles: (pi)*r^2/(2)
Equilateral triangle: sqrt(3)/4 * (s)^2

Question 5

Leaving Rate Formula

Answer 5

ds/dt = Rate of Entry 1 * Rate of Entry 2 - Rate of Draining * Function (S(t))/Initial Containment

Question 6

How do you know if it’s a washer or disk problem?

Answer 6

You washer’s method if there’s a the hole in the middle of the rotation and disk if there’s no hole.

Question 7

Gen Eq for washers method

Answer 7

b
pi ∫ (outer radius)^2 - (inner radius)^2 d(axis of rotation)
a

Question 8

Gen Eq for disk method

Answer 8

b
pi ∫ (dist. from axis of rotation to curve)^2 d(axis of rotation)
a

Question 9

Gen Eq for integrating polar curves

Answer 9

b
∫ 1/2 (r)^2 dθ
a

Question 10

how do you know you’re graphing a rose?

Answer 10

There is a number inside the r= a sin or cos θ

Question 11

how you know if you’re graphing a circle left or right of y axis?

Answer 11

Use cos for the simple polar equation r=acosθ. -cos means left and +cos means right.

Question 12

how do you know if you’re graphing a circle above or below

Answer 12

Use sin for simple polar equation r=asinθ. -sin means down and +sin means up

Question 13

Limacon graph equation

Answer 13

r= a +or- b sinθ or cosθ. cos for left or right. sin for up or down

Question 14

how do you know if the limacon has an inner loop?

Answer 14

a/b < 1

Question 15

how do you know if the limacon is a cardiod

Answer 15

a/b =1

Question 16

arclength for parameteric curve

Answer 16

b
∫sqrt((dy/dt)^2 + (dx/dt)^2) dt
a

Question 17

what makes a sequence montonic

Answer 17

if it is always increasing or decreasing

Question 18

when does a sequence converge or diverge?

Answer 18

if the limit as n goes to infinity exists then it converges, but diverges if it doesn’t,

Question 19

how do you find error for alternating series?

Answer 19

basically plug in K+1 into the n of the absolute value of the series

Question 20

how do you find error for a p series?

Answer 20

make it an integral with the upper bound as t and the lower bound as k. Integrate the definite integral with the limit of t as it goes to infinity than set your inequality.

Question 21

lets says you have a series that has cos(n) or sin(n), you want to find if it converges (conditionally or absolutely) or diverges. steps?

Answer 21

understand that sin and cos can only take up values from [-1, 1]. Manpliluate the interval like you would for an interval of convergence problem and take the absolute value of the highest value and compare the series to the other series.

Question 22

equation for area of parametric curve

Answer 22

bounded ∫y*x’ dt

Question 23

convert (x,y) to r, theta

Answer 23

r = sqrt(x^2 + y^2)
theta= arctan (y/x)

Question 24

how to find concaveity

Answer 24

(dy/dx)/dx. <- quotient rule this