AB Calculus Review

Derivartives

100

f(x) = secx + cscx then, f'(x) =

f'(x) = secxtanx + cscxcotx 

200

y = ln (6x³ - 2x²), then f'(x) = 

f'(x) = (9x + 2)/(3x² - x)

300

f(x) = 6x²/ x- 2 

f'(x) = 24x - 6x²/ (2-x)²

400

f(x) = tan(x)sec(x)

f'(x) = sec3(x)+sec(x)tan2(x)

500

Find the second derivative of x²y = 2

f"(x) = 6y/x²

Limits

100

lim _{x -> 0}   3x - 3sinx/ x²

0 

200

lim _{x ->0}  sin2x/2sinx 

2

300

lim _{h-> 0}  5(½ + h)⁴ - 5(½)⁴/ h  

⁵⁄₂

400

lim _x->∞ 3xe^-3x

0

500

lim _{x-> π⁄2}  1 - sinx/1+ cos2x

¼

Antiderivatives

100

∫18x² sec²x (3x³) dx

2tan(3x³) + C 

200

∫ln³x/x dx

ln⁴x/4 + C

300

_-1¹∫ 4/ 1+x² dx

2π 

400

∫(x²+2x)cos(x³ + 3x²) dx

⅓ sin (x³+ 3x²) + C

500

∫x² sin(3x³+2) dx

- cos(3x³+2)/9 + C

Theroems

100

This Theorem States that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b]. 

Mean Value Theorem

200

This theorem states that if a function f(x) is continuous over an interval [a, b], then the function takes on every value between f(a) and f(b). 

Intermediate Value Theorem 

300

This Theorem states that if a function f has an antiderivative F, then the definite integral of f from a to b is equal to F(b)-F(a)

Fundamental Theorem of Calculus

400

If a function f(x) is continuous on a closed interval [ a, b], then f(x) has both a maximum and minimum value on [ a, b].

Extreme Value Theorem

500

if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them.

Squeeze Theorem 

Equations

100

Average Value 

∫1/b-a f(x) dx

200

Area of a Triangle 

1/2bh

300

∫a^x

a^x/ln(a) + C

400

if dy/dx = Ky then

y = Ce^kx

500

Special Trig Limits

limθ->0 cosθ-1/θ=0

limθ->0 sinθ/θ=1