Differential Calculus 2

Differential Calculus

branch of mathematics which deals with derivatives and limits

One-Sided Limits

Suppose f is a function such that it is not defined for all values of x. Rather, it is defined in such a way that it “jumps” from one y value to the next instead of smoothly going from one y value to the next.

If f is defined in an open interval containing a, except possible at a, then limf(x) as x approaches a = L if and only if

lim f(x) as x approaches -a= lim f(x) as x approaches a

= 1

special limit

Domain

Set of values of x or inputs

Range/codomain

Set of values of y or outputs

Function f(x)

Relationship between inputs where each input is related to exactly one output

General representation of a function

y

dependent variable

x

independent variable

Limit of a function

value that the function approaches as its input approaches a particular value

Primary rule in limits of a function

Undefined is equivalent to

infinity

Calculus

Latin “pebble”
Branch of mathematics concerned on very, very small quantities or changes

Newton

Discovered calculus

Leibniz

German mathematician

First to publish works on Calculus

Velocity

First derivative of position function with respect to time

Instantaneous acceleration

2nd derivative of position function with respect to time

Jerk / Jounce

3rd derivative of position function with respect to time

Snap

4th derivative of position function with respect to time

Crackle

5th derivative of position function with respect to time

Pop

6th derivative of position function with respect to time

Derivative

• Rate of change of a fin relation to a variable.

• Its value of a function at a specific input value is the slope of the tangent line to the functions graph at that point

Line that intersects 2 points in a function

Secant line

definition of First derivative

differential charge =

very small change

Newton Notation

Leibniz notation

Lagrange

Composition of Functions

The composition operator takes 2 functions and returns a new function.

Forward Difference [First Derivative Approximations]

Backward Difference [First Derivative Approximations]

Central Difference [First Derivative Approximations]

Chain Rule

0

trick to memorize the 6 trigo derivatives

Implicit Differentiation

We differentiate each side of the equation to find the implicit derivative dy/dx but in the process, we write dy/dx wherever we are differentiating y

Partial Differentitation

The derivative of a function of several variable is its derivative with respect to one of those variables, with others held constant (treating the other variables as a constant)

Time Rates

Critical Points

Defined as the greatest and least values in the set, when they exist. They are found when a function starts to change in direction or when the slope is equal to zero.

Max point 1st derivative

0

Min point 1st derivative

0

Inflection point 1st derivative

0

Max point 2nd derivative

negative, concave downward

Min point 2nd derivative

positive, concave upward

Inflection point 2nd derivative

0

2nd derivative test

shows the concavity

1st derivative test

shows if the slope is increasing or decreasing

1st step in maxima/minima or optimization problems

2nd step in maxima/minima or optimization problems

3rd step in maxima/minima or optimization problems