AP Calc Gold Sheets

d/dx cos^-1(u)

-1/√(1-u^2) du/dx

d/dx tan^-1(u)

1/(1+u^2) du/dx

d/dx cos^-1(u)

-1/√(1-u^2) du/dx

d/dx tan^-1(u)

1/(1+u^2) du/dx

d/dx cot^-1 (u)

-1/(1+u^2) du/dx

d/dx sec^-1(u)

1/|u|√u^2-1 du/dx

d/dx csc^-1(u)

-1/|u|√u^2-1 du/dx

Derivatives of Inverse Functions (at inverse points)

The slopes are reciprocals at inverse point

d/dx(e^u) u is a function

e^u du/dx

d/dx (a^u) a is a constant

a^u du/dx ln(a)

d/dx(ln(u))

1/u du/dx

d/dx(loga(u)) (a is a constant)

1/ulna du/dx

critical points (if f(x) is continuous)

any interior points where F'(x)=0

max or min points

check endpoints F'(x) changes signs

Mean Value Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b) then there exists some value x=c in (a,b) where f'(c) = (f(b) - f(a))/ (b - a)

When finding antiderivatives

DON'T FORGET + C

inflection points (if f(x) is continuous)

where F''(x) changes signs

Solving related rates

difference implicitly with respect to time(t)

area by Riemann sums (rectangles)

integral notation

A = ∫(a-b) f(x)dx

Integrals

area under the curve * area between the curve a*

Integral of a constant function

(integral a-b) (c dx)

c(b-a)

(integral a-b) (kf(x)dx)

k ∫(a-b) (f(x)) dx

(integral a-b) ([f(x) + or - g(x)] dx)

∫ (a-b) (f(x)dx) + or - ∫ (a-b) (g(x)dx)

Average Value of a Function

1/b-a ∫(a-b) f(x)dx

Mean Value Theorem for Integrals

if f is countinuous on [a,]b] then at some point c in [a,b]

f(c)=1/(b-a) ∫(a-b) f(x)dx

antiderivative of integrals let f(x)= antiderivatives of f(x)

then ∫ (a-x) f(t)dt= F(x)-F(a)=[F(t)]up right x bottom right a

lim x->c f(x)

if f(x) is nice let x=c simplify

if lim x->a^+ f(x) = lim x->a^- f(x) then

both equal to lim x->a f(x)

Finding Horizontal Asymptotes

lim(x->+/- infinity) f(x)=L

vertical asymptote

set denominator equal to zero

f(x) is continuous at x=a if...

interior point

lim x->a f(x)=f(a)

Intermediate Value Theorem

A function f(x) must take on every y-value between f(b) and f(a) if it's continuous from a to b

Average Rate of Change of f(x) on [a,b]

f(b)-f(a)/b-a

instantaneous rate of change

slope of the tangent line

f(a+h)-f(a)/h

Definition of Derivative

lim h->0 f(x+h)-f(x)/h

f(x) is differentiable if

f(x) is continuous and limx->a^- f'(x) = lim x->a^+ f'(x) (slopes have to match)

Alternate definition of derivative

f'(x) = limit (as x approaches a number c)=

f(x)-f(c)/x-c

d/dx (c) , c is a constant

0

d/dx(x^n)

nx^n-1 (power rule)

d/dx (cf(x)) (Constant Multiple Rule)

cf'(x)

d/dx[u+/- v]

u'+/-v'

d/dx [uv]

uv' + vu'

d/dx[u/v]

(vu'-uv')/v^2

position, velocity, acceleration

f(x)

f'(x)

f''(x)

Speed

|v(t)|

d/dx tan(x)

sec^2x

d/dx sec(x)

sec(x)tan(x)

d/dx cot(x)

-csc^2x

d/dx csc(x)

-csc(x)cot(x)

d/dx sin(x)

cosx

d/dx cos(x)

-sinx

Chain Rule: d/dx f(g(x))

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

Slope of a parametric curve

dy/dx = dy/dt / dx/dt

Implicit Differentiation

1. d/dx everything

2. any time you take the derivative of y term tack on dy/dx

3. solve for dy/dx

d/dx sin^-1(x)

1/√(1-x^2)

Fundamental Theorem of Calculus part 1

1. derivative of an integral

2. derivative match upper limit variable

3. lower limit is a constant f(x)

Fundamental Theorem of Calculus part 2

F(b)=F(a) where F(x) is the antiderivative of f(x)

Trapezoid Rule (approx.)

T=h/2 (y + 2y + ... + 2y + y) when [a,b] is portioned into subintervals of equal lengths h=b-a/n

Simpson's Rule (Approximation)

s= h/3 (y+4y+2y.....+2y+4y+y) where h is the width of sunintervals h=b-a/n

Solving Differential Equations

1. Separate Variables

2. Integrate both sides

3. general solution y= antiderivative + c

4. particular solution Find c and plug into equation

Euler's Method

y_new = y_old + ∆x(dy/dx | x old, y old)

x new= x old + delta x

int du

u + C

int u^n du

(u^(n+1))/(n+1) + C

int cos(u) dx

sinu + c

int u^-1 du = int 1/u du

ln|u| + c

int sin(u) dx

-cos(u) + c

int sec^2 (u) du

tan u + C

int csc^2(u) dx

-cot(u) + c

int sec(u)tan(u) du

sec(u) + C

int csc(u)cot(u) du

-csc(u) + C

int e^u du

e^u + C

int a^u du

(a^u)/(ln a) + C

int ln(u) du

ulnu - u + C

int logu + udu

ulnu-u/lnu + C

int tan(u) du

ln|sec(u)| + C or -ln|cos(u) + C

Integration by parts

∫ u dv = uv - ∫ v du

Seperation of Variables

1. separate the variables

2. integrate

3. find c

4. solve for y

if dy/dx = k(y)

y= ye^kt

Law of exponential change

Partial Fractions if the quotient has a bigger power on the denominator

1. factor the denominator

2. A/( ) + B/( )

3.Find A and B

4. integrate