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Notes, Lesson 3.8

Linear Approximations and Differentials

Linear Approximations | Differentials | Links to Other Explanations of Differentials | Check Concepts

Linear Approximations

Review: Linear Approximations were first experienced in Lesson 2.9. It would be healthy to go back and briefly reviewour first contact with this topic.

In the two graphs above, we are reminded of the principle that a tangent line to a curve at a certain point can be a good approximation ofthe value of a function if we are "close by" the point we are interested in. In the above graphs, we see that near the point (8,2) (the pointof tangency) the green tangent is extremely close to the red original curve. We also notice that the closer we get to the point of tangency,the more accurate our linear approximation is.

If you have theJourney Through Calculus CD, load and run MResources/Module 3/Linear

Approximations/Start of Linear Approximations.

Example problem: Find a linear approximation or linearization for . Use this approximation to estimate .

We first calculate the derivative of the function. This will allow us to know thegeneral formula for the slope of the tangent line at any point on the curve.

Next, we need to find the slope of the tangent line atx = 5.

Now we can begin to formulate the equation of the tangent line, because weknow its specific slope.

We now subsitute in the coordinates of the given point or (5,1.71) and solveforb.

We now have our linear approximation.

Now we can find our estimate for the cube root of 5.03

Differentials

If we take the two derivative notations that we have been using and set them equal, we have the equation: . If we then

multiply both sides of this equation by we get: . This equation shows that we can calculate dy as a dependent variable,

based on the inputs of dependent variablesx anddx.This means that we can calculate an estimated error (dy) in linear approximationusing such a formula. An example follows.

Calculus I Notes, Section 3-8 http://www.blc.edu/fac/rbuelow/calc/nt3-8.html

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7/27/2019 Calculus I Notes, Section 3-8

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Sample Problem:

The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere?

If the radius of the sphere is rthen its volume is . If the error in the measured value ofris denoted

by then the corresponding error in the calculated value ofVis , which can be approximated bythe differential .

When r = 21 and dr = 0.05, this becomes:

Therefore, the maximum error in the calculated volume is about 277 cubic centimeters.

Links to Other Explanations of Differentials:

Differentials

Mathematics Help Central

University of Kentucky Tutorial on Differentials

If you have theJourney Through Calculus CD, load and run MResources/Module 3/Linear

Approximation/Start of Linear Approximation.

Check Concepts

#1: True or False: Linear approximations use the derivative.

#2: True or False: When you zoom in enough on any function, it can beapproximated with a straight line.

#3: True or False. Differentials can be used to check on the accuracy of

linear approximations.

#4: True or False. A differential is a partial derivative

#5: Linear approximations are often used in ________________.

Calculus I Notes, Section 3-8 http://www.blc.edu/fac/rbuelow/calc/nt3-8.html

2 of 2 1/16/2011 7:31 PM