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 +
=1
Intercept form
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
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xy = 3
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2x = 4y + 2
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x 2 + 3y = 2
I
I
x + 3 = y3
√
x +3=y
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x + y = 3x + 2
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x(3 + x) = y
I
y = 3x
I
3(xy + y 2 ) = 4y
I x + y =1
2
4
I 4a + 3b =
6
Answers
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4y = 3x + 2 Linear Function
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xy = 3 Not
I
2x = 4y + 2 Linear Function
I
x 2 + 3y = 2 Not
I
I
x + 3 = y 3 Not
√
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 + y 2 ) = 4y Linear Function
I x + y =1
2
4
I 4a + 3b =
Linear Function
6 Linear Function
Reasoning for the Nonlinear Functions
I
xy = 3 Not: Because the independent varialbe is multiplied to
the dependent variable.
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x 2 + 3y = 2 Not: Because the independent variable is raised to a
power other than 1.
I
x + 3 = y 3 Not: Because the dependent variable is raised to a
power other than 1.
I
I
√
x + 3 = y Not: Because the independent variable is raised to a
√
1
power other than 1. (i.e. x = x 2 )
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
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Demand Curves (economic analysis)
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Interest Rates and Investments (finance industry)
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Foreign Currency
Jobs
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Managers
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Financial Occupations
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Computer Programmers
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Scientists
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Engineers
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Administrators
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Construction
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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.
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Calculating travel times
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Converting hours to minutes
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Weights and measures (Doubling a recipe)
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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
1
slope must be f 0 (100) = 20
.
Now our equation is:
y=
1
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
x +b
20
1
10 =
(100) + b
20
10 = 5 + b
y
=
5 = b
Now we have our linear approximation of f (x) =
in and will use it to approximate f (99).
y=
1
x +5
20
√
x about x = 100
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
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http://en.wikipedia.org/wiki/Linear_approximation
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http://www.ehow.com/facts_6027891_
examples-equations-used-real-life.html