Linear Approximations

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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 functionwith only one independent variable ”y” (y = ax + b) traces a straightline 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)

Intercept form(xc

)+(yb

)= 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

Linear Function or Not

I 4y = 3x + 2

I xy = 3

I 2x = 4y + 2

I x2 + 3y = 2

I x + 3 = y3

I√x + 3 = y

I x + y = 3x + 2

I x(3 + x) = y

I y = 3x

I 3(xy + y2) = 4y

I x2 + y

4 = 1

I 4a + 3b = 6

Answers

I 4y = 3x + 2 Linear Function

I xy = 3 Not

I 2x = 4y + 2 Linear Function

I x2 + 3y = 2 Not

I x + 3 = y3 Not

I√x + 3 = y Not

I x + y = 3x + 2 Linear Function

I x(3 + x) = y Not

I y = 3x Linear Function

I 3(xy + y2) = 4y Linear Function

I x2 + y

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 tothe dependent variable.

I x2 + 3y = 2 Not: Because the independent variable is raised to apower other than 1.

I x + 3 = y3 Not: Because the dependent variable is raised to apower other than 1.

I√x + 3 = y Not: Because the independent variable is raised to a

power other than 1. (i.e.√x = x

12 )

I x(3 + x) = y Not: Because after distribution, the indenpendentvariable 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 thenon-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 approximationaround the point x = 100. Our reasoning is simply because we knowthe 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 pointx = 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 whatthe slope of the line is. Since m = f ′(x),

m = f ′(x) =1

2√x

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

20 .

Now our equation is:

y =1

20x + b

Estimation with Linear Approximations

Next we must determine b. We can use the point at which we aremaking this linear approximation, x = 100. By plugging in 10 for yand 100 for x , we get:

y =1

20x + b

10 =1

20(100) + b

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).

y =1

20x + 5

Estimation with Linear ApproximationsUsing Mathematica, we can plot the function and the linearapproximation together.

Estimation with Linear ApproximationsZooming in near the point x = 100 we have:

Estimation with Linear ApproximationsPlotting the error of the two functions, we can clearly see that thelinear 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 willbe 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 youfind 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