Section 1
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f'(x)=lim(h→0)
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Date created
Mar 1, 2020
Cards (68)
Section 1
(50 cards)
f'(x)=lim(h→0)
[f(x+h)-f(x)]/h
a²-u²
u=asinΘ
volume by disks for rotation about the y-axis
Integration by Parts
∫udv=uv-∫vdu
Mean Value Theorem for Derivatives
f'(c)=(f(b)-f(a))/(b-a)
∫du/(√(1-u²))
sin⁻¹u+C
Geometric form to maclaurin series
a/(1-r)=∑arⁿ
∫du/(1+u²)
tan⁻¹u+C
Arc Length of a Polar Equation
L=∫√(r^2+(dr/dθ)^2) dθ
∫sin u du
-cos u + C
∫cot u du
ln|sin u| + C
Definition of a definite integral
∫(a→b)f(x)dx=lim(n→∞) ∑(i=1 to n) f(xi)Δx, where Δx=(b-a)/n and xi=a+iΔx
Arc Length
∫tan u du
-ln|cos u| + C
x definition Polar
x= r cos(θ)
y definition Polar
y= r sin(θ)
cos(x) as a maclaurin series
∫(a^u)du
(a^u)/(ln a) + C
∫sec^2 u du
tan u + C
Trapezoid Rule
∫f(x)dx= (b-a)/ 2n [f(x₀) + 2f(x₂) + 2f(x₃)..... f(x)]
u²-a²
u=asecΘ
Taylor Polynomial
=f(0)+f'(0)x+(f"(0)x²)/2!+ (f"'(0)x³)/3!+...+ (fⁿ(0)xⁿ)/n!
∫e^u du
e^u + C
∫du/(u√(u^2-1))
sec⁻¹u+C
volume by washer around the x-axis
e^x as a maclaurin series
f'(x)=lim(x→a)
[f(x)-f(a)]/(x-a)
∫csc^2 u du
-cot(u)+C
∫csc u du
-ln lcsc u + cot ul + C
Average value of a function on the interval [a,b]
average value=f(c)=∫f(x)dx/(b-a) with a and b as integral bounds
Derivative for Parametric Equations
(dy/dx)= (dy/dt)/(dx/dt) dx/dt≠0
a²+u²
u=atanΘ
Odd- symmetry with respect to origin
f(-x)=-f(x)
Arc Length of a Parametric Curve
L= ∫√( (dx/dt)^2 + (dy/dt)^2) dt
Even- symmetry with respect to y-axis
f(-x)=f(x)
∫u^n
(u^(n+1))/(n+1) + C, where u≠-1
shell formula for revolution about the vertical line
volume of a solid of integrable cross-sectional area
∫cos u du
sin u + C
volume by disks for rotation about the x-axis
Cartesian to Polar conversion, Angle
θ=tan⁻¹(y/x)
Lagrange Remainder Formula
∫csc(u)cot(u) du
-csc u + C
Second derivative for Parametric Equations
(d^2y)/(dx^2)=((d/dt)(dy/dx))/(dx/dt)
∫sec(u)tan(u) du
sec u + C
Euler's Method
y1 = y0 + f (x0,y0) (dx)
Area enclosed by a Polar equation
A=∫ 1/2 r∧₂ dθ
sin(x) as a maclaruin series
∫sec u du
ln lsec u + tan ul + C
∫du/u
ln|u|+C
Section 2
(18 cards)