AP Calc Exam formulas and definitions

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.

Derivative of x raised to a power

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')

Deriv of (f(g(x)))^n using the chain rule

nf(g(x))g'(x)

Deriv of f(x)g(x)

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

Deriv 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 and f' changes sign at x

there is a relative 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 and f" changes sign at x

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

particle is at rest or changing direction

v(t)\>0

particle is moving right

v(t)

particle 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) have same signs

speed of particle increasing (particle is speeding up)

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

speed of particle decreasing (particle is slowing down)

∫(x^n)dx

(x^(n+1))∕(n+1) +C

∫(1/x)dx

ln|x|+C

∫(e^(kx))dx

(e^(kx))/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(0))

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\=(1/2)s²]

v\= (1/2)∫[f(x)-g(x)]²dx

Cross section for volume:

equilateral triangle [A\=√(3/4)s²)]

v\= √3/4∫[f(x)-g(x)]²dx

Cross section for volume:

semicircle [A\=(1/2)πs²]

v\= (1/2)π∫[(f(x)-g(x))/2]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