Midterm Memorization

Pythagorean Identities

\sin^2x+\cos^2x=1

1+\cot^2x=\csc^2x

\tan^2x+1=\sec^2x

Sum and Difference Identities

\cos\left(A\pm B\right)=\cos A\cos B\mp\sin A\sin B

\sin\left(A\pm B\right)=\sin A\cos B\pm\cos A\sin B

Double Angle

\sin2x=2\sin x\cos x

\cos2x=1-2\sin^2x

            =2\cos^2x-1

            =\cos^2x-\sin^2x

Degree Reducing Identities

\cos^2x=\frac12\left(1+\cos2x\right)

\sin^2x=\frac12\left(1-\cos2x\right)

f\left(x\right)=\sin x

f^{\prime}\left(x\right)=\cos x

f\left(x\right)=\cos x

f^{\prime}\left(x\right)=-\sin x

f\left(x\right)=\tan x

f^{\prime}\left(x\right)=\sec^2x

f\left(x\right)=\cot x

f^{\prime}\left(x\right)=-\csc^2x

f\left(x\right)=\sec x

f^{\prime}\left(x\right)=\sec x\tan x

f\left(x\right)=\csc x

f^{\prime}\left(x\right)=-\csc x\cot x

f\left(x\right)=\ln x

f^{\prime}\left(x\right)=\frac{1}{x}

f\left(x\right)=e^{x}

f^{\prime}\left(x\right)=e^{x}

f\left(x\right)=a^{x}

f^{\prime}\left(x\right)=a^{x}\ln\left(a\right)

f\left(x\right)=\log_{a}\left(x\right)

f^{\prime}\left(x\right)=\frac{1}{x\ln\left(a\right)}

\int\tan x

\ln\left|\sec x\right|+C

-\ln\left|\cos x\right|+C

\int\cot x

\ln\left|\sin x\right|+C

\int\csc x

\ln\left|\csc x-\cot x\right|+C

-\ln\left|\csc x+\cot x\right|+C

\int\sec x

\ln\left|\sec x+\tan x\right|+C

Arc Length

\int_{a}^{b}\sqrt{1+\left(\frac{\differentialD y}{\differentialD x}\right)^2}

Surface Area

2\pi\int_{a}^{b}r\sqrt{1+\left(\frac{\differentialD y}{\differentialD x}\right)^2}

Mean Value Theorem

f^{\prime}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}

Average Value Theorem

f^{\prime}\left(c\right)=\frac{1}{b-a}\int_{a}^{b}f\left(x\right)

Integration By Parts

\int udv=vu-\int vdu

Find the Derivative Equation

f^{\prime}\left(x\right)=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}