Finalprep Flashcards
Differential equation rate equation
P(t) =de^(kt) + C
What’s the Cooling formula
T - Temp
t - time
T_a is room temp
dT/dt = K(T-T_a)
General Equation: For Cross sections
b
∫ (Area Function in terms of the axis perpendicular to)^2 d(axis perpendicular)
a
What are the area functions?
Squares: S^2
Semi-cricles: (pi)*r^2/(2)
Equilateral triangle: sqrt(3)/4 * (s)^2
Leaving Rate Formula
ds/dt = Rate of Entry 1 * Rate of Entry 2 - Rate of Draining * Function (S(t))/Initial Containment
How do you know if it’s a washer or disk problem?
You washer’s method if there’s a the hole in the middle of the rotation and disk if there’s no hole.
Gen Eq for washers method
b
pi ∫ (outer radius)^2 - (inner radius)^2 d(axis of rotation)
a
Gen Eq for disk method
b
pi ∫ (dist. from axis of rotation to curve)^2 d(axis of rotation)
a
Gen Eq for integrating polar curves
b
∫ 1/2 (r)^2 dθ
a
how do you know you’re graphing a rose?
There is a number inside the r= a sin or cos θ
how you know if you’re graphing a circle left or right of y axis?
Use cos for the simple polar equation r=acosθ. -cos means left and +cos means right.
how do you know if you’re graphing a circle above or below
Use sin for simple polar equation r=asinθ. -sin means down and +sin means up
Limacon graph equation
r= a +or- b sinθ or cosθ. cos for left or right. sin for up or down
how do you know if the limacon has an inner loop?
a/b < 1
how do you know if the limacon is a cardiod
a/b =1
arclength for parameteric curve
b
∫sqrt((dy/dt)^2 + (dx/dt)^2) dt
a
what makes a sequence montonic
if it is always increasing or decreasing
when does a sequence converge or diverge?
if the limit as n goes to infinity exists then it converges, but diverges if it doesn’t,
how do you find error for alternating series?
basically plug in K+1 into the n of the absolute value of the series
how do you find error for a p series?
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.
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?
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.
equation for area of parametric curve
bounded ∫y*x’ dt
convert (x,y) to r, theta
r = sqrt(x^2 + y^2)
theta= arctan (y/x)
how to find concaveity
(dy/dx)/dx. <- quotient rule this
