Precalculus/Calculus 1 Formula Review

Pythagorean Identity (sin)

sin²(x)+cos²(x)=1

Pythagorean Identity (sec)

1+tan²(x)=sec²(x)

Pythagorean Identity (csc)

1+cot²(x)=csc²(x)

sin(-x)=

-sin(x)

cos(-x)=

cos(x)

tan(-x)=

-tan(x)

sin(A+B)=

sin(A)cos(B)+cos(A)sin(B)

sin(A-B)=

sin(A)cos(B)-sin(B)cos(A)

cos(A+B)=

cos(A)cos(B)-sin(A)sin(B)

cos(A-B)=

cos(A)cos(B)+sin(A)sin(B)

sin(2x)=

2sin(x)cos(x)

cos(2x)=

cos²(x)-sin²(x)=2cos²(x)-1=1-2sin²(x)

sin²(x)=

1/2(1-cos(2x)

cos²(x)=

1/2(1+cos(2x))

A function is even if…

f(x)=f(-x) for every x in the function’s domain.

Every even function is symmetric about the…

y-axis

A function is odd if…

-f(x)=f(-x)  for every x in the function’s domain

Every odd function is symmetric about the…

origin

A function f(x) is periodic with period p if…

f(x+p)=f(x) for every value of x.

The period of the function y=Asin(Bx+C) or y=Acos(Bx+C) is…

2pi/abs(B)

The amplitude of the function y=Asin(Bx+C) or y=Acos(Bx+C) is…

abs(A)

The period of y=tan(x) is…

pi

e as a limit of n as n approaches 0

lim n→0(1+n/1)^(1/n)=e

e as a limit of n as n approaches infinity

lim n→infinity(1+1/n)^n=e

Rolle’s Theorem - If f is continuous on [a, b] and differentiable on (a, b) such that f(a)=f(b), then…

there is at least one number c in the open interval (a, b) such that f’(c)=0.

Where are the only candidates for an absolute min or max on an interval?

Where f’(x)=0 or does not exist and the endpoints of the interval

If f(x) is an odd function, then the integral from -a to a equals…

0

If f(x) is an even function, then the integral from -a to a equals…

2*the integral from 0 to a

If g(x) is greater than or equal to f(x) on [a, b], then the integral from a to b of g(x) is…

greater than or equal to the integral from a to b of f(x)

Fundamental Theorem of Calculus

The integral from a to b of f(x)dx equals F(b)-F(a), where F’(x)=f(x)

d/dx(x^n)=

nx^(n-1)

d/dx(fg)=

f’g+fg’

d/dx(f/g)=

(f’g-fg’)/g²

d/dx(f(g(x))

f’(g(x))*g’(x)

d/dx(sin(x))

cos(x)

d/dx(cos(x))

-sin(x)

d/dx(tan(x))

sec²(x)

d/dx(cot(x))

-csc²(x)

d/dx(sec(x))

sec(x)tan(x)

d/dx(csc(x))

-csc(x)cot(x)

d/dx(e^x)

e^x

d/dx(a^x)

a^x*ln(a)

d/dx(ln(x))

1/x

d/dx(sin^-1(x))

1/sqrt(1-x²)

d/dx(tan^-1(x))

1/1+x²

d/dx(sec^-1(x))

1/(abs(x)sqrt(x²-1))

Chain Rule: dy/dx=

dy/du*du/dx

Integral of a dx where a is a constant

ax+C

Integral of xⁿ dx where n is a constant not equal to -1

x^(n+1)/(n+1)+C

Integral of x^-1

ln|x|+C

Integral of e^x dx

e^x+C

Integral of a^x dx where a is a constant

a^x/ln(a)+C

Integral of ln(x) dx

x*ln(x)-x+C

Integral of sin(x) dx

-cos(x)+C

Integral of cos(x) dx

sin(x)+C

Integral of tan(x) dx

ln|sec(x)|+C or -ln|cos(x)|+C

Integral of cot(x) dx

ln|sin(x)|+C

Integral of sec(x) dx

ln|sec(x)+tan(x)|+C

Integral of csc(x) dx

-ln|csc(x)+cot(x)|+C or ln|csc(x)-cot(x)|+C

Integral of sec²(x) dx

tan(x)+C

Integral of sec(x)tan(x) dx

sec(x)+C

Integral of csc²(x) dx

-cot(x)+C

Integral of csc(x)cot(x) dx

-csc(x)+C

Integral of tan²(x) dx

tan(x)-x+C