Calculus

Limits

100

lim(x->0)x^2-2

-2

200

lim(x->0)(-(x^2-3x)/x)

3

300

lim(x->0+)(x^3-4)/4x

-infinity

400

lim(x->Inf)sqrt(x)

1

500

lim(x->0)(-6(x^3-6)/(4x^2+4x)

undefined

Derivatives

100

x^2

2x

200

x^3+x^2+x

3x^2+2x+1

300

sin(x)+cos(x)

cos(x)-sin(x)

400

f(x)/g(x)

(f(x)g'(x)-g(x)f'(x))/g(x)^2

500

1/f(x)

-f'(x)/f(x)^2

Integrals

100

x^3

1/3x^2+c

200

ln(x^2)

1/(1/3x^3)+c

300

e^(12x)

12xe^12x+C

400

(ln(x)/12)x

((x^2(2ln(x)-1))/48)+C

500

x^2-e^(sqrt(5x))

(x^3/3)−2(5sqrt(x)−sqrt(5)esqrt(5)sqrt(x)/5^3/2+C

Trig Derivative Rules

100

cos(x)

-sin(x)

200

csc(x)

-csc(x)cot(x)

300

 cot(x)

-csc^2(x)

400

cot(x)csc(x)

-csc(x)(csc^2(x)+cot^2(x))

500

(cot(x)csc(x))/tan(x)

-csc(x)((csc^2(x)+cot^2(x))tan(x)+cot(x)sec^2(x))/tan^2(x)

Theorems

100

Chain Rule

f*g=(f'*g)g'

200

Product Rule

fg=g'f-f'g

300

FTC

integral from a to b of f(x) = F(b)-F(a)

400

Intermediate Value Theorem

A function that is continuous between two points takes on every value between those two points

500

Rolle's Theorem

When f(x) is both continuous and differentiable between a and b where f(a)=f(b) than at some point between a and b the derivative must be 0