Integrals

∫dx

∫dx

∫uⁿ du

∫e^u du

∫b^u*ln(b) du

∫b^u du

∫cosu du

∫-sinu du

∫sinu du

∫sec²u du

∫-cscu*cotu du

∫cscu*cotu du

∫secu*tanu du

∫-csc²u du

∫csc²u du

∫du/√(1-u²)

∫-du/√(1-u²)

∫du/(1+u²)

∫-du/u√(1-u²)

∫du/u√(1-u²)

∫-du/(1+u²)

∫du/u

∫tan u du

-ln|cos u|+C

∫cot u du

ln|sin u|+C

∫sec u du

ln|sec u+tan u|+C

∫csc u du

-ln|csc u+ cot u|+C

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

derivative of the multiplaction of two functions

d/dx [f(x)/g(x)] =

quotient rule

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

chain rule

d/dx [ln|x|] =

d/dx [a^x] =

d/dx [log_a X] =

∫- sinx

∫secxtanx

∫- cscxcotx

d/dx [tanx]

d/dx [cotx]

∫cosx

quotient rule

derivative of a power

chain rule

derivative of an exponential function

derivative of a natural log

d/dx [a^x]

derivative of a log base a

d/dx [sinx]

d/dx [cscx]

d/dx [cosx]

d/dx [secx]

d/dx [tanx]

d/dx [cotx]

d/dx [arcsinx]

d/dx [arccscx]

d/dx [arccosx]

d/dx [arcsecx]

d/dx [arctanx]

d/dx [arccotx]

product rule

f is continuous at x=c if...

Intermediate Value Theorem

If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k

Global Definition of a Derivative

Alternative Definition of a Derivative

f '(x) is the limit of the following difference quotient as x approaches c

∫-sin(x)

∫sec²(x)

∫-csc²(x)

∫sec(x)tan(x)

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.

Rolle's Theorem

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

Mean Value Theorem

The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.

First Derivative Test for local extrema

Point of inflection at x=k

Combo Test for local extrema

If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c.
If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.

Horizontal Asymptote

L'Hopital's Rule

x+c

sin(x)+C

-cos(x)+C

tan(x)+C

-cot(x)+C

sec(x)+C

-csc(x)+C

Fundamental Theorem of Calculus #1

The definite integral of a rate of change is the total change in the original function.

Fundamental Theorem of Calculus #2

Mean Value Theorem for integrals or the average value of a functions

ln(x)+C

-ln(cosx)+C = ln(secx)+C

hint: tanu = sinu/cosu

ln(sinx)+C = -ln(cscx)+C

ln(secx+tanx)+C = -ln(secx-tanx)+C

ln(cscx+cotx)+C = -ln(cscx-cotx)+C

If f and g are inverses of each other, g'(x)

Exponential growth (use N= )

Formula for Disk Method

Axis of rotation is a boundary of the region.

Formula for Washer Method

Axis of rotation is not a boundary of the region.

Inverse Secant Antiderivative

Inverse Sine Antiderivative

Derivative of eⁿ

ln(a)*aⁿ+C

Derivative of ln(u)

Antiderivative of f(x) from [a,b]