12-01: Rates of Change

Rates of Change & The Slope of a Curve

Rate of change: the change in one quantity (dependent variable) with respect to another quantity (independent variable)

Slope = ∆y/∆x = (y2 - y1)/(x2-x1)

Rates of change are obtained this way

  • Independent variables: are picked and can be controlled, e.g. time
  • Dependent variables: depend on the independent variables, are an outcome

\ \ AROC - average rate of change: rate of change over an interval, between 2 points → corresponds to the slope of the secant line

IROC - instantaneous rate of change: rate of change at an instance, at a single point → corresponds to the slope of the tangent line

\ Find IROC from a table of values

Proceeding and Following method:

  1. Look at the data that proceeds (comes before) and follows (comes after) the target data
  2. Find AROC: (y2-y1)/(x2-x1)

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Rates of Change Using Equations

Slope = ∆y/∆x = (y2 - y1)/(x2-x1)

can be written as

 

Difference Quotient - use for IROC

 

The closer h is to zero, the more accurate the estimate becomes

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Limits

Using limits, the interval can be made infinitely small, approaching 0. As this happens, the slope of the secant line approaches its limiting value: the slope of the tangent line

  • Gets closer to tangent as the secant numbers get smaller

\ Left hand limits: use values that are less than or to the left side of the value being approached

 

Right hand limits: use values that are greater than or to the right side of the right side of the value being approached

 

A limit is a boundary placed, saying that no matter how close it tries, it will never be equal to a

This is represented as

 

The limit exists if the following 3 criteria are met:

 

Continuous Functions

Continuous function: a function that is continuous at x for all values of xER

A function, f(x) is continuous at x=a if:

 

  • A function is continuous if you can draw its graph without lifting your pencil   * Holes, asymptotes, gaps mean that the it is discontinuous or has discontinuity at the point at which the gap occurs

\ \ Piecewise function: a function made up of 2 or more functions, each corresponding to a specific interval within the entire domain (each graph has its own domain). These are also discontinuous

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Types of Discontinuity

Removable discontinuity: Hole (when factor cancels in equation). Function is undefined at a particular point

 

Jump discontinuity: One sided limits exist but are not equal

 

Infinitely discontinuity: one sided limit is infinite so limit doesn’t approach a particular finite value and limit does not exist

 

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Properties of Limits: Evaluating Limits Algebraically

  • Any constants are equal to their limit
  • The limit of x as x approaches a is = to a
  • The limit of a sum is the sum of the limits
  • The limit of a difference is the difference of the limits
  • The limit of a constant times a function is the constant times the limit of the function
  • The limit of a product is the product of the limits
  • The limit of a quotient is the quotient of the limits, provided that the denominator ≠ 0
  • The limit of a power is the power of the limit, provided that the exponent is a rational number
  • The limit of a root is the root of the limit provided that the root exists

\ Direct substitution: When you are evaluating a limit, you can substitute your a value in for x within the function to find it

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  • This doesn’t always work though because it may result in 0/0 (intermediate form)

Intermediate form: 0/0 as a result, we cannot conclude that this limit DNE so we need more tools to determine the limit. The tools we use are factoring, rationalizing both numerator and denominator, and simplifying and expanding

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Derivatives

Differentiation: output is called the derivative which can be used to find the slope of the tangent at any point in the function’s domain

First Principles Defintion of the Derivative

  • The difference quotient with a swapped out for x
  • h is 0 because this is the smallest interval between 2 numbers

 

When this limit is simplified by letting h → 0, the resulting expression is expressed in terms of x

  • This expression can be used to determine the derivative of the function at any x-value in the function’s domain
  • In the end, you shouldn’t have any h’s left as they should all be swapped for 0 after simplifying

f’(x) is a way to communicate that it’s a derivative

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