Linear Approximation and Differentials
Site: | Saylor Academy |
Course: | MA005: Calculus I |
Book: | Linear Approximation and Differentials |
Printed by: | Guest user |
Date: | Saturday, May 18, 2024, 4:48 AM |
Description
Read this section to learn how linear approximation and differentials are connected. Work through practice problems 1-10.
Introduction
Newton's method used tangent lines to "point toward" a root of the function. In this section we examine and use
another geometric characteristic of tangent lines:
This idea is used to approximate the values of some commonly used functions and to predict the "error" or uncertainty in a final calculation if we know the "error" or uncertainty in our original data. Finally, we define and give some examples of a related concept called the differential of a function.
Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.9-Linear-Approximation.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.
Linear Approximation
Since this section uses tangent lines frequently, it is worthwhile to recall how we find the equation of the line tangent to at a point . The line tangent to at goes through the point and has slope so, using the point-slope form for linear equations, we have
This final result is the equation of the line tangent to at .
Example 1: Find the equation of the line which is tangent to the graph of at the point . Evaluate and to approximate and .
. If is close to , then the value of is a good approximation of the value of (Fig. 2). The number is close to 9 so
Similarly, In fact, , so our estimate, using , is within of the exact answer. (accurate to 6 decimal places) and our estimate is within of the exact answer.
In each example, we got a good estimate of a square root with very little work. The graph indicates the tangent line is slightly above , and each estimate is slightly larger than the exact value.
Practice 1: Find the equation of the line tangent to the graph of at the point (Fig. 3). Evaluate and to approximate and . Are your approximations using larger or smaller than the exact values of the square roots?
Practice 2: Find the equation of the line tangent to the graph of at the point and use to approximate and . Do you think your approximations using are larger or smaller than the exact values?
The process we have used to approximate square roots and cubics can be used to approximate any differentiable function, and the main result about the linear approximation follows from the two statements in the boxes. Putting these two statements together, we have the process for Linear Approximation.
If is differentiable at and is close to ,
(algebraically) the values of the tangent line function
Example 2: Use the linear approximation process to approximate the value of .
Solution: so We need to pick a value a near for which we know the exact value of and , and is the obvious choice. Then
This approximation is within of the exact value of
Practice 3: Approximate the value of , the amount becomes after 4 years in a bank which pays interest compounded annually. (Take and ).
Practice 4: Use the linear approximation process and the values in the table to estimate the value of when and .
We can also approximate functions as well as numbers.
Example 3: Find a linear approximation formula for when is small. Use your result to approximate and .
Solution: so . Since "x is small", we know that is close to , and we can pick Then and so
If is small, then and . Use your calculator to determine by how much each estimate differs from the true value.
Applications of Linear Approximation to Measurement "Error"
Most scientific experiments involve using instruments to take measurements, but the instruments are not perfect, and the measurements we get from them are only accurate up to a certain level. If we know the level of accuracy of our instruments and measurements, we can use the idea of linear approximation to estimate the level of accuracy of results we calculate from our measurements.
If we measure the side of a square to be 8 inches, then, of course, we would calculate its area to be square inches. Suppose, as is reasonable in a real measurement, that our measuring instrument could only measure or be read to the nearest inches. Then our measurement of 8 inches would really mean some number between inches and inches, and the true area of the square would be between and square inches. Our possible "error" or "uncertainty", because of the limitations of the instrument, could be as much as square inches so we could report the area of the square to be square inches. We can also use the linear approximation method to estimate the "error" or uncertainty of the area. (For a function as simple as the area of a square, this linear approximation method really isn't needed, but it is used to illustrate the idea).
For a square with side , the area is and If represents the "error" or uncertainty of our measurement of the side, then, using the linear approximation technique for ,
so the uncertainty of our calculated area is In this example, inches and inches so
and the uncertainty in our calculated area is approximately
This process can be summarized as:
Linear Approximation Error:
Practice 5: If we measure the side of a cube to be with an uncertainty of , what is the volume of the cube and the uncertainty of our calculation of the volume? (Use linear approximation).
Example 4: We are using a tracking telescope to follow a small rocket. Suppose we are 3000 meters from the launch point of the rocket, and, 2 seconds after the launch, we measure the angle of the inclination of the rocket to be with a possible "error" of (Fig. 5). How high is the rocket and what is the possible error in this calculated height?
Solution: Our measured angle is radians and radians (all of our trigonometric work is in radians), and the height of the rocket at an angle is so Our uncertainty in the height is
If our measured angle of can be in error by as much as , then our calculated height of can be in error by as much as . The height is meters.
In some scientific and engineering applications, the calculated result must be within some given specification. You might need to determine how accurate the initial measurements must be in order to guarantee the final calculation is within the specification. Added precision usually costs time and money, so it is important to choose a measuring instrument which is good enough for the job but not too good or too expensive.
Example 5: Your company produces ball bearings (spheres) with a volume of , and the volume must be accurate to within . What radius should the bearings have and what error can you tolerate in the radius measurement to meet the accuracy specification for the volume?
Solution: Since we want , we can solve for to get . and so . In this case we have been given that , and we have calculated so '(1.3365 .
Solving for , we get . To meet the specifications for the allowable error in the volume, we must allow no more than variation in the radius. If we measure the diameter of the sphere rather than the radius, then we want .
Practice 7: You want to determine the height of the rocket to within 10 meters when it is 4000 meters high (Fig. 6). How accurate must your angle measurement be? (Do your calculations in radians).
Relative Error and Percentage Error
The "error" we have been examining is called the absolute error to distinguish it from two other commonly used terms, the relative error and the percentage error which compare the absolute error with the magnitude of the number being measured. An "error" of 6 inches in measuring the circumference of the earth would be extremely small, but a 6 inch error in measuring your head for a hat would result in a very bad fit.
Example 6: If the relative error in the calculation of the area of a circle must be less than , then what relative error can we tolerate in the measurement of the radius?
Solution: so and . The Relative Error of is
. We can guarantee that the Relative Error of , is less than if the Relative Error of , is less than .
Practice 8: If you can measure the side of a cube with a percentage error less than 3%, then what will the percentage error for your calculation of the surface area of the cube be?
The Differential of f
In Fig. 7, the change in value of the function near the point is and the change along the tangent line is . If is small, then we have used the approximation that . This leads to the definition of a new quantity, , called the differential of .
The differential of represents the change in , as changes from to , along the tangent line to the graph of at the point . If we take to be the number , then the differential is an approximation of .
Example 7: Determine the differential df of each of , and . Solution: , and .
Practice 9: Determine the differentials of , and .
We will do little with differentials for a while, but are used extensively in integral calculus.
The Linear Approximation "Error" | f(x) – L(x)
An approximation is most valuable if we also have have some measure of the size of the "error", the distance between the approximate value and the value being approximated. Typically, we will not know the exact value of the error (why not?), but it is useful to know that the error must be less than some number. For example, if one scale gives the weight of a gold pendant as grams with an error less than grams grams) and another scale gives the weight of the same pendant as grams with an error less than grams grams), then we can have more faith in the second approximate weight because of the smaller "error" guarantee. Before finding a guarantee on the size of the error of the linear approximation process, we will check how well the linear approximation process approximates some functions we can compute exactly. Then we will prove one bound on the possible error and state a somewhat stronger bound.
Example 8: Let . Evaluate and for
and for a general value of .
Cutting the value of in half makes the error as large. Cutting to as large makes the error as large. In general, .
This function and error also have a nice geometric interpretation (Fig. 8): is the area of a square of side so is the area of a square of side , and that area is the sum of the pieces with areas , and . The linear approximation to the area of the square includes the 3 largest pieces and , but it omits the small square with area so the approximation is in error by the amount .
Practice 10: Let . Evaluate and
for and for a general value of . Use Fig. 9 to give a geometric interpretation of , and .
In both the example and practice problem, the error turned out to be very small, proportional to , when was small. In general, the error approaches as approaches .
Theorem : If is differentiable at a and
Not only does the difference approach , but this difference approaches so fast that we can divide it by , another quantity approaching , and the quotient still approaches .
In the next chapter we can prove that the error of the linear approximation process is proportional to . For now we just state the result.
Practice Problem Answers
Practice 1: so . At the point on the graph of , the slope of the tangent line is . The equation of the tangent line is
Practice 2: so . At , the slope of the tangent line is . The equation of the tangent line is or . Then and .
Practice 5: and so and . Then "error" . When and , "error" .
Finally, "error" so and radians .
Practice 8: so and We are also told that Percentage error is .
is the actual volume of the cube with side length .
is the volume of the cube with side length plus the volume of the 3 "slabs"
is the volume of the "leftover" pieces from : the 3 "rods" and the tiny cube .