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

Continuity Rule

A function is continuous at x = c if:

(1) f(c) is defined

(2) lim f(x) (x goes to c) exists

(3) lim f(x) (x goes to c) = f(c)

d/dx (x^n)

nx^n-1 (power rule)

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

n(f(x)^(n-1)) * 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] such that f(c) is between f(a) and f(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) 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 POSSIBLE max or min on f(x) [number line test of f'(x)]

If f'(x)>0

f(x) is increasing

If f'(x)<0

f(x) is decreasing

If x=a is a critical value

f'(a)=0 or f'(a)=DNE

If f''(x)>0

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

If f''(x)<0

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

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

means position function

p'(t) or x'(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)<0

p(t) is moving left

a(t)=0

v(t) not changing

a(t)>0

v(t) increasing

a(t)<0

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

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(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

End Behavior Model

g(x) is a right EBM for f(x) if lim (x goes to pos infinity) f(x)/g(x) = 1

g(x) is a left EBM for f(x) if lim (x goes to neg infinity) f(x)/g(x) = 1

Horizontal Asymptotes

If lim f(x) (x goes to pos/neg infinity) = L then y = L is a HA, describes the end behavior of a function

(a HA can be crossed)

Rules for lim f(x) (x goes to pos/neg infinity)

(1) when lim f(x) (x goes to pos infinity) and degree of x is the same, the ratio of the coefficients is the limit

(2) "" when degree in denominator is greater than the degree in numerator, limit = 0

(3) "" when degree in numerator is greater than the degree in denominator, limit DNE but can find end behavior model

Vertical Asymptotes

y = f(x) has a VA at x = c if lim f(x) (x goes to c from the right) = pos/neg infinity or lim f(x) (x goes to c from the left) = pos/neg infinity

Types of Discontinuity

(1) Removable discontinuity

(2) Jump discontinuity (piecewise or absolute value function)

(3) Infinite discontinuity (asymptote)