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AP®MathsCollege BoardCalculus ABStudy GuidesUnit 4: Contextual Applications of DifferentiationLinearizationApproximating Values of a Function

Approximating Values of a Function (College Board AP® Calculus AB)

Study Guide

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Local linearity of a function

What does local linearity mean?

  • If you 'zoom in' far enough on the graph of a function at a point, a curve can look more like a straight line

  • This means the tangent to a graph of a function at a point, can act as an approximation for the function at that point

  • This linear approximation of a function is only appropriate very close to the point

    • Hence the term "local linearity"

Graphs of y = x^3 - 4x + 3 (black curve) and y = 8x - 13 (red line), intersecting at point (2, 3), shown in blue, with zoomed-in view on the right.
The graph of a cubic (black) and the graph of the tangent (red) to the curve at (2,3). The view on the right is zoomed in. You can see that close to the point, the curve is approximated by the straight line.

How do I use a tangent to approximate a function?

  • The equation of the tangent to f open parentheses x close parentheses at x equals a is given by

    • y minus f open parentheses a close parentheses equals f to the power of apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses or y equals f open parentheses a close parentheses plus f to the power of apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses

    • Provided that f open parentheses x close parentheses is differentiable at a

  • Due to the local linearity of a function this can be a linear approximation for f open parentheses x close parentheses at points close to open parentheses a comma space f open parentheses a close parentheses close parentheses

  • Using the example in the above image

    • For the graph of y equals x cubed minus 4 x plus 3

    • The tangent at (2, 3) is y equals 8 x minus 13

    • The tangent will be an approximation for the curve close to (2,3)

    • Substitute x values close to 2 into the equation of the tangent to find an approximation for the function (curve) at that point

    • See the table below for the approximated values compared to the real values

x

y equals 8 x minus 13 (Tangent)

y equals x cubed minus 4 x plus 3 (Curve)

2.3

5.4

5.967

2.2

4.6

4.848

2.1

3.8

3.861

2.01

3.08

3.0806

2

3

3

1.99

2.92

2.9205

1.9

2.2

2.259

1.8

1.4

1.632

1.7

0.6

1.113

  • The table shows how the approximation is more accurate closer to the point where the tangent intersects the curve

    • It will be less accurate further away from the point of intersection

  • Using a tangent to approximate a curve (within a small interval) can make calculations or computational processes easier to handle,

    • This is because a linear function is simpler than most other functions

    • However, this comes with a trade-off in accuracy

How do I know if the approximation is an overestimate or underestimate?

  • The values of the function approximated by the tangent will be either an overestimate or underestimate of the real value

  • Which one it is, depends on the concavity of the function at the point where the tangent intersects the curve

    • You can find out more about concavity in the 'Concavity of Functions' study guide

  • In general,

    • If the graph of the function is concave up open parentheses f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses greater than 0 close parentheses at the point where the tangent intersects it, the tangent will give an underestimate

    • If the graph of the function is concave down open parentheses f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses less than 0 close parentheses at the point where the tangent intersects it, the tangent will give an overestimate

Graph showing y=f(x), its tangents at x=-2 and x=2 in red, and its second derivative y=f''(x) in blue, with labeled axes and points.
The graph of f(x) is concave down for negative values of x, and concave up for positive values of x. This is also shown by the graph of the second derivative of f(x).

  • In the graph above, the tangent at x equals negative 2 will give an overestimate, as the function is concave down at this point

    • The tangent at x equals plus 2 will give an underestimate, as the function is concave up at this point

Worked Example

(a) Find an approximation to the value of square root of 65 using a linear approximation of y equals square root of x.

Answer:

We can use a linear approximation by finding a tangent to the graph of y equals square root of x at a point we know the coordinates of, and which is close to x equals 65

Use the point (64, 8) as this is close to 65, and has integer coordinates to make working easier

Tangent to f open parentheses x close parentheses equals square root of x equals x to the power of 1 half end exponent

at (64, 8)

For the equation of the tangent, use the form y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses

y minus f open parentheses 64 close parentheses equals f to the power of apostrophe open parentheses 64 close parentheses open parentheses x minus 64 close parentheses

Find f to the power of apostrophe open parentheses 64 close parentheses

f to the power of apostrophe open parentheses x close parentheses equals 1 half x to the power of negative 1 half end exponent f to the power of apostrophe open parentheses 64 close parentheses equals 1 half times fraction numerator 1 over denominator square root of 64 end fraction equals 1 over 16

Find the equation of the tangent

table row cell y minus f open parentheses 64 close parentheses end cell equals cell f to the power of apostrophe open parentheses 64 close parentheses open parentheses x minus 64 close parentheses end cell row cell y minus 8 end cell equals cell 1 over 16 open parentheses x minus 64 close parentheses end cell end table

Use the tangent to estimate the value of square root of 65 by substituting in x equals 65

table row cell y minus 8 end cell equals cell 1 over 16 open parentheses 65 minus 64 close parentheses end cell row y equals cell 8 plus 1 over 16 equals 8.0625 end cell end table

An approximation of square root of 65 is 8.0625

(b) Without calculating the real value of square root of 65, explain whether your approximation will be an overestimate or underestimate.

Answer:

Consider the concavity of f open parentheses x close parentheses equals square root of x to decide if the tangent at x equals 64 will be an over- or underestimate

f to the power of apostrophe open parentheses x close parentheses equals 1 half x to the power of negative 1 half end exponent f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses equals negative 1 fourth x to the power of negative 3 over 2 end exponent

Note that x to the power of negative 3 over 2 end exponent equals fraction numerator 1 over denominator x square root of x end fraction is only defined for x greater than 0, and for those values of x it is always positive

The second derivative is always negative, so the graph of y equals square root of x is always concave down. Therefore the approximation using a tangent will be an overestimate.

You can also see this when sketching a graph of y equals square root of x and the tangent at (64, 8)

The tangent is always above the curve, so will be an overestimate

Graph of root x and a tangent to it at (64, 8). The tangent is always above the curve.

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    Author
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    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.