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Verify the given linear approximation at $ a = 0. $ Then determine the values of $ x $ for which the linear approximation is accurate to within 0.1.$ e^x \cos x \approx 1 + x $

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Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 10

Linear Approximation and Differentials

Derivatives

Differentiation

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

01:54

Verify the given linear ap…

04:03

00:31

04:30

Find the linear approximat…

01:41

01:45

we know that f of X is e to the X coastline exploding in 0 to 0 co sign zero anything to zero powers once. This is one times coastlines here, which is one. Therefore, the derivative is either the ex co sign X minus you the axe sign ex plug n zero and we end up with one. Therefore, we have each of the AKs. Co sign of acts is equivalent to approximately one plus acts. Now we know that we have negative 0.1 is less than either the axe co sign axe minus X minus one, which is less than 0.1. And we know 0.9 is less than either the axe co sign X minus fax, which is less than 1.1. So now, if we are to Graff, thus negative 0.763 is less than acts, which is less than 0.6 year of seven juice. It's between these two values

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