Functions

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1

What Are Functions in Math?

2

Why Are Functions Important in Real Life?

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What Is the Definition of a Function?

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What Is a Coordinate System?

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What Are Function Tables?

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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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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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Logarithmic Functions

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Rational functions

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Calculate and Interpret the Average Rate of Change

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What Is the Point-Slope Equation?

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Interpret and Find Instantaneous Rate of Change in Math

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How to Approximate Instantaneous Rate of Change in Math

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How to Interpret and Calculate Limits of a Function

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How Do You Determine Continuity of a Function

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How to Interpret and Calculate Asymptotes of a Function

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What Does the Definition of the Derivative Mean?

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How Do You Determine if a Function Is Differentiable?

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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

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How to Differentiate Functions Using the Quotient Rule

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Make Sign Charts of the Derivatives of a Function

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Derivation of Distance to Get Velocity and Acceleration

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What Are Stationary Points of a Function?

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How to Calculate the Curvature of a Function

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What Are Inflection Points of a Function?

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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

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What Are Riemann Sums Used for?

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What Are the Important Integration Rules?

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Which Basic Relations in Integration Should You Know?

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How to Interpret and Calculate the Definite Integral

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How to Interpret and Calculate the Indefinite Integral

Integrals are mainly split into two categories: Definite and indefinite integrals. The indefinite integral is the same as the anti-derivative.

A challenge in working with integrals is having to find the expression without determining the constant term. The symbol $C$ that you add to the end of integrals represents this unknown constant of integration. To find the constant, you are dependent on having constraints, such that you can find $C$ by substitution and equation solving.

Theory

The Indefinite Integral

\int f (x) d x = F (x) + C, F^{'} (x) = f (x)

Here, $f (x)$ is called the integrand and $C$ is called the constant of integration.

Example 1

Compute the integral $\int \ln x + e^{x} + x^{3} d x$

$\begin{array}{l} \int \ln x + e^{x} + x^{3} d x & = x \ln x - x + e^{x} \\ + \frac{1}{4} x^{4} + C \end{array}$

$\begin{array}{l} \int \ln x + e^{x} + x^{3} d x = x \ln x - x + e^{x} + \frac{1}{4} x^{4} + C \end{array}$

Example 2

Compute the integral

$\int 3 \cos (3 x) - 4 \sin (2 x) d x$

$\int 3 \cos (3 x) - 4 \sin (2 x) d x$

$\begin{array}{l} \int 3 \cos (3 x) - 4 \sin (2 x) d x \\ = 3 \cdot \frac{1}{3} \sin (3 x) + 4 \cdot \frac{1}{2} \cos (2 x) + C \\ = \sin (3 x) + 2 \cos (2 x) + C \end{array}$

$\begin{array}{l} \int 3 \cos (3 x) - 4 \sin (2 x) d x & = 3 \cdot \frac{1}{3} \sin (3 x) + 4 \cdot \frac{1}{2} \cos (2 x) + C \\ = \sin (3 x) + 2 \cos (2 x) + C \end{array}$

Example 3

Compute the integral

$\int \sin (2 x) + 3 \cos (x) - e^{5 x} d x$

$\int \sin (2 x) + 3 \cos (x) - e^{5 x} d x$

$\begin{array}{l} \int \sin (2 x) + 3 \cos (x) - e^{5 x} d x \\ = - \frac{1}{2} \cos (2 x) + 3 \sin (x) - \frac{1}{5} e^{5 x} + C \end{array}$

$\int \sin (2 x) + 3 \cos (x) - e^{5 x} d x = - \frac{1}{2} \cos (2 x) + 3 \sin (x) - \frac{1}{5} e^{5 x} + C$

Done

49

How to Use the Formula for Integration by Parts

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How to Do Integration by Substitution

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Use Partial Fraction Decomposition for Integration

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How to Find the Area Between Two Graphs by Integration

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How to Use Solids of Revolution to Find Volumes

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What Is a Differential Equation?

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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?

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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

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Integrating Factor Method for Differential Equations

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Logistic Growth and Carrying Capacity of a Population

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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

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Find the Equation of a Tangent Line of a Function

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The Differential Equation for Harmonic Oscillators

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Reduction of Order Method for Differential Equations

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Mathematical Modeling

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Mathematical Regression

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The Difference Between Interpolation and Extrapolation

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Quadratic Models

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Exponential Models

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What Are Power Models in Math?

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What Are Logistic Models Used For?

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What Are Cubic Models in Math?

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What Are Trigonometric Models?

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What Does the Feasible Reagion in Optimization Mean?

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How to Use the Graphical Method for Linear Optimization

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How to Use the Ruler Method for Linear Optimization