Chapter 3 & 5

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Derivatives

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Area beneath the curve

splitting up the changes in x into rectangles that you then find the area of. The sum is multiplied by the change of x value over the number of total rectangles.

Optimization

Using a function to find the max or min values of a given example. Similar to the first derivative test.

Chapter 3 & 5

Antiderivatives

Derivatives that are directly reversable with exception to constants.

Logs

Polynomials or Exponentials

x^n= (x^n+1)/n+1+C

Trig

Inverse trig

Used to revert function from its derivative state to its original state or close to it.

Constants do not have derivatives so functions will sometimes differ while finding the antiderivative. Hence the (+C) behind some functions.

Substitution Rule

Substitution rule simplifies integrand

For definite integrals

int/ [a,b] f(g(x))g'(x) dx = int/ [g(a),g(b)] f(u) du

int/ f(g(x))g'(x) dx = int/ f(u) du

du=g'(x) dx

u=g(x)

Integrals

Net change theorem

int/ [a,b] F'(x) dx = F(b)-F(a)

The integral of a rate of change is the net change.

An integral relies on antiderivatives for finding solutions

Definite

A definite integral can be found geometrically via area beneath the curve.

Fundamental Theorem of Calculus

Part 2) int/ [a,b] f(x) dx = F(b)-F(a), where F is any antiderivative of f, that is F' = f

Part 1) If g(x) = int/ [a,x] f(t) dt, then g'(x) = f(x)

Indefinite

int/ b^x dx = (b^x)/ln(b) + C

int/ 1/x dx = ln|x| + C

int/ 1/x^2+1 dx = arctan(x) + C

int/ sin(x) dx = -cos(x) + c

int/ e^x dx = e^x + C

int/ x^n dx = (x^n+1)/(n+1) + C

int/ k dx = kx + C

Derivatives

The derivative of a function is used to find the velocity or slope of a tangent line at a given point.

Derivatives of inverse trig

dy/dx arccsc(x)= -1/x*sqrt(x^2-1)

dy/dx arccot(x)= -1/1+x^2

dy/dx arccos(x)= -1/sqrt(1-x^2)

dy/dx arcsec(x)= 1/x*sqrt(x^2-1)

dy/dx arctan(x)= 1/1+x^2

dy/dx arcsin(x)= 1/sqrt(1-x^2)

Derivatives of trig

dy/dx csc(x)= -csc(x)cot(x)

dy/dx cot(x)= -csc^2(x)

dy/dx cos(x)= -sin(x)

dy/dx sec(x)= sec(x)tan(x)

dy/dx tan(x)= sec^2(x)

dy/dx sin(x)= cos(x)

Finding derivatives

Derivatives of exponential functions

dy/dx f(x)^n= (n*f(x)^n-1) * f'(x)

Derivatives of polynomials

dy/dx f(x) + g(x) + h(x)= f'(x) + g'(x) + h'(x)

Derivatives of logs

dy/dx ln(x)= 1/x

Implicit differentiation

Quotient rule

dy/dx f(x)/g(x)= f'(x)g(x) - g'(x)f(x)/g(x)^2

Product rule

dy/dx f(x)g(x)= f'(x)g(x) + g'(x)f(x)

Chain rule

dy/dx f(g(x))= f'(g(x)) * g'(x)

Curve Sketching

Extrema

Extrema can be found while using first derivative test and mapping out a number line to find positive and negative values.

Derivative goes from negative to positive = min

Derivative goes from positive to negative = max

Graphing Guidelines

Concavity

If F''(x) < 0 then f(x) is concave down

If F''(x) > 0 then f(x) is concave up

Increasing or decreasing

If F'(x) < 0 then f(x) is decreasing

If F'(x) > 0 then f(x) is increasing

Critical Numbers

Use derivative of f(x) and set equal to zero

Critical numbers are either extrema or discontinuity points

Asymptotes

Vertical- when denominator=0

Horizontal- use limits

Intercepts

Where f(x) meets the x-axis and y-axis

Domain

check for all values of x of f(x) that make the function continuous.

Second Derivative Test

When you use the second derivative of a function and set = 0 to find inflection points or concavity

First Derivative Test

When you use the first derivative of a function and set = 0 to find critical numbers or intervals of increasing and decreasing