Math 115 Flashcards
What makes Rn an underestimate? overestimate?
f’(x) 0 over
What makes Ln and underestimate? overestimate?
f’(x) > 0 under
f’(x)
What makes Tn an underestimate? overestimate?
f’‘(x) over
What makes Mn an underestimate? overestimate?
f’‘(x) > 0 under
f’‘(x)
Arclength
L = a-b | (sqrt(1+(dy/dx)^2))dx
x (polar)
rcostheta
y (polar)
rsintheta
diffeq with carrying capacity
dP/dt = kP(1-P/M)
diffeq exponential
dP/dt = kP
Area (polar)
A = a–b | 1/2 r^2 dtheta
Arclength (polar)
L = a–b | (sqrt(r^2 + (dr/dtheta)^2)) dtheta
How does a geometric series converge?
if |x|
Test for Divergence
if lim an != 0, then an is divergent
Absolute convergence?
sig |an| is convergent
Conditional convergence?
sigan converges but sig|an| does not
Ratio test inconclusive?
lim = 1
Power Series equality (3)
1/1-x =
1+x+x^2+…=
sig x^n
x
Integral of Cn(x-a)^n
(Cn(x-a)^n+1)/n+1
Derivative of Cn(x-a)^n
n*Cn(x-a)^n-1
Cn in Taylor/Mclaurin Series
f^n(a)/n!
Taylor inequality
M satisfying |f^(n+1)(x)|
lim of x^n/n!
what is the sum of this
0
e^x
sinx series and expansion
Sin[x] == Sum[((-1)^n x^(1 + 2n))/(1 + 2n)!, {n, 0, Infinity}]
=x-x^3/3! + x^5/5!….
cosx series and expansion
Cos[x] == Sum[((-1)^n x^(2n))/(2n)!, {n, 0, Infinity}]
=1-x^2/2 + x^4/4! …
tan-1x
Sum[((-1)^n x^(2n+1))/(2n+1), {n, 0, Infinity}]
= x - x^3/3 + x^5/5-…
ln(1+x)
Sum[((-1)^n-1 x^n)/n, {n, 1, Infinity}]
x - x^2/2 + x^3/3 - x^4/4…
Solution to a logistic equation
P(t) = M/1+Ae^-kt where A = (M-P0)/P0
Solution to initial value problem
P(t) = P(0)e^kt
