2.14 Linear Approximations
By the end of your studying, you should know:
On-screen applet instructions:
The initial configuration is an implementation of the figure at the beginning of Section 2.14 of the text, except that the slope of the line is not fixed. The applet is to be used to investigate the nature of the error function error(x) = f(x) − L(x) as x → a. First, for a fixed x from the pull-down menu use the slider to change the slope m of the line: in fact, let the slope of the line approach f′(a). For each m, the normalized error function error(t) is shown in the inset: watch its shape. Can you explain the change in the error function as the slope of the line approaches f′(a)? What can you conclude about the error as x → a? Click here for more details and an explanation.
ExamplesWhen solving problems in geometric optics, engineers and physicists often use the simplifying assumption that, for small angles θ, sin(θ) is approximately equal to θ. Find a linear approximation for sin(x) that shows why this is a reasonable assumption.
A pizza restaurant sells an average of 80 pizzas per day at its usual price of $12.95. It experiments with sales and coupons for dollars off the usual price, and finds that the number of pizzas sold when the price decreases by 2 dollars is 135. It estimates that the number of pizzas sold when the price does down by x dollars is modeled by the function 50 ln(x + 1) + 80. Use linear approximation to find the change in the number of pizzas sold when the price drops from $10.95 to $9.95.
AppletsBest Linear Approximation
VideosSee short videos of worked problems for this section.
ExercisesSee Exercises for 2.14 Linear Approximations (PDF).
Work online to solve the exercises for this section, or for any other section of the textbook.
Resources on the WebInformation on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"
Nothing yet has been found. Any ideas?
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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel