Functions

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1

What Are Functions in Math?

2

Why Are Functions Important in Real Life?

3

What Is the Definition of a Function?

4

What Is a Coordinate System?

5

What Are Function Tables?

6

How to Find x and y Values of a Function

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How to Calculate Domain and Range in Math

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How to Calculate and Interpret Intersection Points

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How to Calculate Zeros or Roots of a Function

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What Are Linear Functions?

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How Are Linear Functions Used in Real Life?

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What Makes a Function Proportional?

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What Are Quadratic Functions?

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What Are Polynomial Functions?

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What Are Exponential Functions?

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Exponential Functions with Euler's Number

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What Does Inversely Proportional Mean in Math?

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What Are Power and Root Functions?

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What Are Logarithmic Functions?

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What Are Rational Functions?

21

Calculate and Interpret Average Rate of Change in Math

22

What Is the Point-Slope Equation?

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Interpret and Find Instantaneous Rate of Change in Math

24

How to Approximate Instantaneous Rate of Change in Math

25

How to Interpret and Calculate Limits of a Function

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How Do You Determine Continuity of a Function?

27

How to Interpret and Calculate Asymptotes of a Function

28

What Does the Definition of the Derivative Mean?

29

How Do You Determine if a Function Is Differentiable?

30

Rules of Differentiation for Finding Derivatives

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How Does the Chain Rule for Differentiation Work?

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How to Find Derivatives using the Product Rule

33

How to Differentiate Functions Using the Quotient Rule

34

Make Sign Charts of the Derivatives of a Function

35

Derivation of Distance to Get Velocity and Acceleration

36

What Are Stationary Points of a Function?

37

How to Calculate the Curvature of a Function

38

What Are Inflection Points of a Function?

39

How to Analyze Polynomial Functions

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How to Analyze Logarithmic Functions

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How to Analyze Power Functions

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How to Analyze the Cosine Function

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Analyzing Exponential Functions

44

What Are Riemann Sums Used for?

45

What Are the Important Integration Rules?

46

Which Basic Relations in Integration Should You Know?

47

How to Interpret and Calculate the Definite Integral

48

How to Interpret and Calculate the Indefinite Integral

49

How to Use the Formula for Integration by Parts

50

How to Do Integration by Substitution

51

Use Partial Fraction Decomposition for Integration

52

How to Find the Area Between Two Graphs by Integration

53

How to Use Solids of Revolution to Find Volumes

54

What Is a Differential Equation?

55

What Are Integral Curves and Direction Fields Used for?

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What Can Differential Equations Be Used for?

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What Are First Order Differential Equations?

58

First Order Differential Equations with Constant Coefficients

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First Order Differential Equations with Initial Conditions

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How to Solve Exact First Order Differential Equations

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How to Write and Solve Separable Differential Equations

62

Integrating Factor Method for Differential Equations

63

Logistic Growth and Carrying Capacity of a Population

64

What Are Second Order Differential Equations?

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How to Solve Exact Second Order Differential Equations

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Homogeneous Second Order Differential Equations with Constant Coefficients

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Second Order Differential Equations with Initial Conditions

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Verify Solutions to Second Order Differential Equations

69

Find the Equation of a Tangent Line of a Function

The equation for an arbitrary tangent to the graph of a function is given by the point-slope equation:

Formula

Point-Slope Equation

y - y_{1} = f^{'} (x_{1}) (x - x_{1})

where $(x_{1}, y_{1})$ is the point of tangency and $f^{'} (x_{1})$ is the slope. After substituting into the formula, you always solve for $y$ .

When solving differential equations, you are often asked to find the tangent to a graph at a certain point. It is then useful to use the fact that

y^{'}

from the differential equation can be used to find

f^{'} (x_{1})

. You do as follows:

Rule

Find the Tangent at a Given Point Using Differential Equations

1.: Solve the differential equation with respect to $y^{'}$ .
2.: Insert the $x_{1}$ -value at the point of intersection with the tangent, and find the value of $y^{'} (x_{1})$ .
3.: Enter $x_{1}$ , $y_{1}$ and $y^{'} (x_{1})$ into the point-slope equation and find the equation for the tangent.

Example 1

Given the differential equation $x y^{'} + 3 y = - x$ , find the tangent at the point $(1, \frac{3}{4})$

1.

Solve for

y^{'}

:

\begin{array}{l} x y^{'} + 3 y & = - x \\ x y^{'} & = - 3 y - x \\ y^{'} & = \frac{- 3 y}{x} - 1 \end{array}

2.

Insert the point

(1, \frac{3}{4})

:

y^{'} = \frac{- 3 \cdot \frac{3}{4}}{1} - 1 = \frac{- 13}{4}

3.

Insert this into the point-slope equation:

\begin{array}{l} y - \frac{3}{4} & = \frac{- 13}{4} (x - 1) \\ y & = \frac{- 13 x}{4} + 4 \end{array}

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70

The Differential Equation for Harmonic Oscillators

71

Reduction of Order Method for Differential Equations

72

What Does Mathematical Modeling Mean?

73

What Is Mathematical Regression?

74

The Difference Between Interpolation and Extrapolation

75

What Are Linear Models?

76

What Are Quadratic Models?

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What Are Exponential Models?

78

What Are Power Models in Math?

79

What Are Logistic Models Used For?

80

What Are Cubic Models in Math?

81

What Are Trigonometric Models?

82

What Does the Feasible Region in Optimization Mean?

83

How to Use the Graphical Method for Linear Optimization

84

How to Use the Ruler Method for Linear Optimization