Linear Approximations

ACADEMIC RESOURCE CENTER Table of Contents

Linear Function

Linear Function or Not

Real World Uses for Linear Equations

Why Do We Use Linear Equations?

Estimation with Linear Approximations

References Table of Contents

Linear Function

Linear Function or Not

Real World Uses for Linear Equations

Why Do We Use Linear Equations?

Estimation with Linear Approximations

References Linear Function

Definition: A mathematical equation in which no independent-variable ”x” is raised to a power greater than one. A simple linear function with only one independent variable ”y” (y = ax + b) traces a straight line when plotted on a graph. Also known as a linear equation.

Famous Forms: Y-axis form y = mx + b

Point-slope form (y − y1) = m(x − x1) x  y  Intercept form + = 1 c b Table of Contents

Linear Function

Linear Function or Not

Real World Uses for Linear Equations

Why Do We Use Linear Equations?

Estimation with Linear Approximations

References Linear Function or Not

I 4y = 3x + 2

I xy = 3

I 2x = 4y + 2 2 I x + 3y = 2 3 I x + 3 = y √ I x + 3 = y

I x + y = 3x + 2

I x(3 + x) = y

I y = 3x 2 I 3(xy + y ) = 4y x y I 2 + 4 = 1 I 4a + 3b = 6 Answers

I 4y = 3x + 2 Linear Function

I xy = 3 Not

I 2x = 4y + 2 Linear Function 2 I x + 3y = 2 Not 3 I x + 3 = y Not √ I x + 3 = y Not

I x + y = 3x + 2 Linear Function

I x(3 + x) = y Not

I y = 3x Linear Function 2 I 3(xy + y ) = 4y Linear Function x y I 2 + 4 = 1 Linear Function I 4a + 3b = 6 Linear Function Reasoning for the Nonlinear Functions

I xy = 3 Not: Because the independent varialbe is multiplied to the dependent variable.

2 I x + 3y = 2 Not: Because the independent variable is raised to a power other than 1.

3 I x + 3 = y Not: Because the dependent variable is raised to a power other than 1.

√ I x + 3 = y Not: Because the independent variable is raised to a √ 1 power other than 1. (i.e. x = x 2 )

I x(3 + x) = y Not: Because after distribution, the indenpendent variable is raised to a power other than 1. Table of Contents

Linear Function

Linear Function or Not

Real World Uses for Linear Equations

Why Do We Use Linear Equations?

Estimation with Linear Approximations

References Real World Uses for Linear Equations

Popular Uses

I Demand Curves (economic analysis)

I Interest Rates and Investments (finance industry)

I Foreign Currency Jobs

I Managers

I Financial Occupations

I Computer Programmers

I Scientists

I Engineers

I Administrators

I Construction

I Health Care Table of Contents

Linear Function

Linear Function or Not

Real World Uses for Linear Equations

Why Do We Use Linear Equations?

Estimation with Linear Approximations

References Why Do We Use Linear Equations?

Linear Equations are used in everyday life.

I Calculating travel times

I Converting hours to minutes

I Weights and measures (Doubling a recipe)

I Estimation Table of Contents

Linear Function

Linear Function or Not

Real World Uses for Linear Equations

Why Do We Use Linear Equations?

Estimation with Linear Approximations

References Estimation with Linear Approximations

√ √Suppose√ we wanted to approximate 99. We could say that 99 ≈ 100 = 10. However, using linear approximations, we can obtain a better approximation than 10. Let us take a look at the √ non-linear function f (x) =√ x. This function represents all of the square roots. i.e. f (3) = 3.

Now using Mathematica to visualize. Estimation with Linear Approximations Estimation with Linear Approximations

Now that we have motivation, we should find a linear approximation around the point x = 100. Our reasoning is simply because we know the function value at that point and it is near 99. i.e. f (100) = 10. √ So we wish to find a line that passes through the function x at the point x = 100, then we will use that line to approximate the point x = 99. To start, let us take the form

y = mx + b

, where m is the slope and b is the y-intercept. Estimation with Linear Approximations

In order to determine the linear equation, we must determine what the slope of the line is. Since m = f 0(x),

1 m = f 0(x) = √ 2 x

And we wish to know the slope of a line at the point x = 100, so the 0 1 slope must be f (100) = 20 .

Now our equation is: 1 y = x + b 20 Estimation with Linear Approximations

Next we must determine b. We can use the point at which we are making this linear approximation, x = 100. By plugging in 10 for y and 100 for x, we get: 1 y = x + b 20 1 10 = (100) + b 20 10 = 5 + b 5 = b √ Now we have our linear approximation of f (x) = x about x = 100 in and will use it to approximate f (99).

1 y = x + 5 20 Estimation with Linear Approximations Using Mathematica, we can plot the function and the linear approximation together. Estimation with Linear Approximations Zooming in near the point x = 100 we have: Estimation with Linear Approximations Plotting the error of the two functions, we can clearly see that the linear approximation will be a good approximation for f (99). Estimation with Linear Approximations

We can see that the error for the linear approximation at√x = 99 will be small. So then we can obtain the estimation for the 99. The actual value is given by Mathematica.

This concludes the example for linear approximations. Hopefully you find more uses in everyday life for linear approximations. Table of Contents

Linear Function

Linear Function or Not

Real World Uses for Linear Equations

Why Do We Use Linear Equations?

Estimation with Linear Approximations

References References

I http://en.wikipedia.org/wiki/Linear_approximation

I http://www.ehow.com/facts_6027891_ examples-equations-used-real-life.html