AP Calculus AB - Review

Limit Definition of Derivative

limit (as h approaches 0)= F(x+h)-F(x)/h

Alternate Definition of Derivative

limit (as x approaches a number c)=
f(x)-f(c)/x-c x≠c

limit as x approaches 0: sinx/x

1

limit as x approaches 0:
1-cosx/x

0

Continuity Rule

If the limit exists (aka left limit and right limit are equal), and the limit equals the function at that point.

Basic Derivative

f(x^n)= nX^(n-1)

d/dx(sinx)

cosx

d/dx(cosx)

-sinx

d/dx(tanx)

sec²x

d/dx(cotx)

-csc²x

d/dx(secx)

secxtanx

d/dx(cscx)

-cscxcotx

d/dx(lnu)

u'/u

d/dx(e^u)

e^u(u')

d/dx(a^u)

a^u(lna)(u')

Chain rule of f(x)^n

nf(x)f'(x)

Product rule of f(x)g(x)

f'(x)g(x)+g'(x)f(x)

Quotient rule of f(x)/g(x)

g(x)f'(x)-f(x)g'(x)/g(x)²

Intermediate Value Theorem

if f(x) is continuous on [a,b], then there will be a point x=c that lies in between [a,b]

Extreme Value Theorem

if f(x) is continuous on [a,b], then f(x) has an absolute max or min on the interval

Rolle's Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), and if f(a)=f(b), then there is at least one point (x=c) on (a,b) [DON'T INCLUDE END POINTS] where f'(c)=0

Mean Value Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), there is at least one point (x=c) where f'(c)= F(b)-F(a)/b-a

If f'(x)=0

there is a max or min on f(x) [number line test]

If f'(x)>0

f(x) is increasing

If f'(x)

f(x) is decreasing

If f''(x)=0

f(x) has a point of inflection & f'(x) has a max or min

If f''(x)>0

f(x) is concave up & f'(x) is increasing

If f''(x)

f(x) is concave down & f'(x) is decreasing

p(t), x(t), s(t)

means position function

p'(t)

v(t)= velocity

p''(t) or v'(t)

a(t)= acceleration

v(t)=0

p(t) is at rest or changing direction

v(t)>0

p(t) is moving right

v(t)

p(t) is moving left

a(t)=0

v(t) not changing

a(t)>0

v(t) increasing

a(t)

v(t) decreasing

v(t) and a(t) has same signs

speed of particle increasing

v(t) and a(t) has different signs

speed of particle decreasing

∫(x^n)dx

x^(n+1)∕(n+1) +C

∫(1/x)dx

ln|x|+C

∫(e^kx)dx

ekx/k +C

∫sinx dx

-cosx+C

∫cosx dx

sinx+C

∫sec²x dx

tanx+C

∫csc²x dx

-cotx+C

∫secxtanx dx

secx+C

∫cscxcotx

-cscx+C

∫k dx [k IS A CONSTANT]

kx+C

1st fundamental theorem of calculus

(bounded by a to b) ∫f(x)dx= F(b)-F(a)

2nd fundamental theorem

(bounded by 1 to x)
d/dx[∫f(t)dt]= f(x)(x')

average value

(1/(b-a))[∫f(x)dx] [BOUNDED BY A TO B]

Area between curves

A=∫f(x)-g(x) dx

Volume (DISK)

V=π∫f(x)²dx

Volume (WASHER)

V=π∫f(x)²-g(x)²dx

∫f(x)dx [BOUNDS ARE SAME]

0

Displacement of particle

∫v(t)dt

total distance of particle

∫|v(t)|dt

position of particle at specific point

p(x)= initial condition + ∫v(t)dt (bounds are initial condition and p(x))

derivative of exponential growth equation:
P(t)=Pe^kt

dP/dt=kP

Cross section for volume: square [A=s²]

v=∫[f(x)-g(x)]²dx

Cross section for volume:
isosceles triangle [A=1/2s²]

v= 1/2∫[f(x)-g(x)]²dx

Cross section for volume:
equilateral triangle [A=√3/4s²]

v= √3/4∫[f(x)-g(x)]²dx

Cross section for volume:
semicircle [A=1/2πs²]

v= 1/2π∫[f(x)-g(x)]²dx

d/dx(sin⁻¹u)

u'/√(1-u²)

d/dx(cos⁻¹u)

-u'/√(1-u²)

d/dx(tan⁻¹u)

u'/(1+u²)

d/dx(cot⁻¹u)

-u'/(1+u²)

d/dx(sec⁻¹u)

u'/|u|√(u²-1)

d/dx(csc⁻¹u)

u'/|u|√(u²-1)

∫du/√(a²-u²)

(sin⁻¹u/a)+C

∫du/(a²+u²)

(1/a)(tan⁻¹u/a)+C

∫du/|u|√(u²-a²)

(1/a)(sec⁻¹u/a)+C