AP Calculus BC

All memorized derivatives, integrals, and formulas

d/dx ln x

1/x

d/dx logbx

1/xlnb

d/dx e^x

e^x

d/dx b^x

b^x lnb

d/dx sinx

cosx

d/dx cos

-sinx

d/dx tanx

sec²x

d/dx secx

secxtanx

d/dx arcsinx

1/(1-x²)^1/2

d/dx arccosx

-1/(1-x²)^1/2

d/dx arctanx

1/1+x²

integration by parts

∫u dv = uv - ∫vdu

Partial Fractions

\frac{1}{\left(cx+d\right)\left(hx+k\right)}=\frac{A}{\left(cx+d\right)}+\frac{B}{\left(hx+k\right)}

Volume of a disc

\pi\int_{a}^{b}\!r^2\,dx

Volume of a washer

\pi\int_{a}^{b}\!R^2-r^2\,dx

total distance travelled

\int_{a}^{b}\!\left|v\left(t\right)\right|\,dx

displacement

∫v(t)dt

speed

|v(t)|

d/dx f(g(x))

f’(g(x))g’(x)

d/dx (f(x))/(g(x))

\frac{f^{\prime}\left(x\right)g\left(x\right)-f\left(x\right)g^{\prime}\left(x\right)}{g\left(x\right)^2}

d/dx f(x) g(x)

f^{\prime}\left(x\right)g\left(x\right)+f\left(x\right)g^{\prime}\left(x\right)

Alt Series Error

\le\left|a_{n+1}\right| (the next term)

Lagrange Error

\le\left|\frac{f^{\left(n+1\right)}\left(c\right)\left(b-a\right)^{n+1}}{\left(n+1\right)!}\right|

McLaurin Series : e^x

1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots+\frac{x^{n}}{n!}

McLaurin Series: sinx

x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+\frac{\left(-1\right)^{n}x^2n+1}{\left(2n+1\right)!}

McLaurin Series: cosx

1-\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots+\frac{\left(-1\right)^{n}x^{2n}}{\left(2n\right)!}

Point Slope

y-y_1=m\left(x-x_1\right)

Logistic Growth dP/dt =

\frac{k}{M}P\left(M-P\right)

Logistic Growth: M=

carrying capacity

AROC

\frac{f\left(b\right)-f\left(a\right)}{b-a}

IROC

f^{\prime}\left(c\right)

Average Value

\frac{1}{b-a}\int_{a}^{b}\!f\left(x\right)\,dx

Intermediate Value Theorem

f(x) is cont on [a,b] , then there is a y value between f(a) and f(b)

Extreme Value Theorem

f(x) is cont of [a,b] , then f(x) must have an abs min and max on [a,b]

Arc Length (Rectangular)

\int_{a}^{b}\!\sqrt{1+f^{\prime}\left(x\right)^2}\,dx

Arc Length (Parametric)

\int_{a}^{b}\!\sqrt{x^{\prime}\left(t\right)^2+y^{\prime}\left(t\right)^2}\,dx

Speed (Parametric)

\sqrt{x^{\prime}\left(t\right)^2+y^{\prime}\left(t\right)^2}

Polar Area

\frac12\int_{a}^{b}\!r^2\,dx

r²

x²+y²

x (polar conversion)

r\cos\theta

y (polar conversion)

r\sin\theta

\theta (polar conversion)

\arctan\frac{y}{x}

Area of a trapezoid

\frac12h\left(b_1+b_2\right)

nth term test

\lim_{n\rightarrow\infty}\ne0 then diverges

Geometric Series

\sum ar^{n} |r|<1 converges |r|>= 1 diverges

p-series

\sum\frac{1}{n^{p}} p>1 converges, p<=1 diverges

Ratio Test

\lim_{n\rightarrow\infty}\left\vert\frac{a_{n+1}}{a_{n}}\right\vert<1converges,\lim_{n\rightarrow\infty}\left\vert\frac{a_{n+1}}{a_{n}}\right\vert>1diverges

Integral Test

continuous, decreasing, and positive

Which test for testing absolute convergence

Ratio, geometric, nth term