formula reviews Flashcards
(47 cards)
Intermediate Value Theorem
f a function is continuous in a closed interval [π, π] β¦
β¦ then there is an π₯-value for every π¦-value in the range π(π) β€ π¦ β€ π(π)
Extreme Value Theorem
If a function is continuous in a closed interval [π, π] β¦
β¦ then the functionβs absolute extrema occur either at the endpoints, or where its derivative is zero or undefined
Mean Value Theorem
If a function is continuous in a closed interval [π, π] and is differentiable in the open interval (π, π) β¦
β¦ then there is at least one moment when its instantaneous slope equals is average slope
Average Value Theorem
n a closed interval [π, π] β¦
β¦ a functionβs average value equals the area it captures, divided by the intervalβs length.
(integral over interval)
Taylors Theorm
Construct a functionβs series from its behavior at a single point π₯ = π.
f(x) = f(a)+ fβ(a)(x-a) + (1/2!)fββ(a)(x-a)^2+(1/3!)fβββ(a)(x-a)^3
derivative of tan(x)
sec^2(x)
derivative of sec(x)
sec(x)*(tan(x)
derivative of cot(x)
-csc^2(x)
derivative of csc(x)
-csc(x)*cot(x)
derivative of b^x
(b^x)(lnb)
derivative of arcsin(x)
1/sqrt(1-x^2)
derivative of arccosx
-1/(sqrt(1-x^2))
derivative of arctanx
1/(1+x^2)
a/1-r series
a + ar + ar^2
series of e^x
1 + x +(1/2!)(x^2) + (1/3!)(x^3)
Series of sinx
x - (1/3!)(x^3) + (1/5!)(x^5)
series of cos x
1 - (1/2!)(x^2) + (1/4!)(x^4)
arctanx series
x - (1/3)(x^3) + (1/5)(x^5)
lnx series
(x-1) - (1/2)(x-1)^2 + (1/3)(x-1)^3
Eulers Method
4 column chart to approximate a solution
headings are x, y, dy/dx, delta y (which equals dy/dx * delta x)
exponential growth and decay
dy/dt = ky
y=ce^kt
logistic growth
dp/dt = kP(M-P)
Integrating techniques
integrate by parts, u-substitution, partial fractions, synthetic division, use series
area of 2d regions
A = integral from a to b (upper curve-lower curve)dx
