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$7-10$ Verify the given linear approximation at $a=0 .$ Thendetermine the values of $x$ for which the linear approximation isaccurate to within $0.1 .$$e^{x} \approx 1+x$

$L(x)=1+(x-a)=1+x$for accuracy $y=0.1$ then $x$ must be $-0.4831< x <0.4163$

Calculus 1 / AB

Chapter 3

Derivatives

Section 8

Linear Approximations and Taylor Polynomials

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

01:07

$7-10$ Verify the given li…

00:31

Verify the given linear ap…

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01:17

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01:45

01:08

01:03

02:21

01:22

03:31

Verify the linear approxim…

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01:32

Linear approximation a. Fi…

05:19

Find the linear approximat…

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02:27

04:10

03:51

Linear approximationa.…

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In this problem, we are asked to prove that function. A to the x, the inertion of that function at point x, equals to 0 is 1 plus x and then further we are asked to determine the values for which the lineation is accurate to within .4. So, let's remember the equation for lineation of a function at point x, not the neston at the point x. Not is the function at the at that point, plus the derivative of that functionary at that point, multiplied by x, minus x, 1 point so, let's first remember extent in this case is 0. So, let's eat a function. 1 x, equals to 00 is e to the 0. This is 1 point. How about the derivative, while f prime of x, is e to the x right, since our initial function is e to the x now from this, the derivative of this function as 0 is also 1. So if you combine these 2, we find a liner to be 1 plus 1 x 0, and from this we find it to be x, plus 1 point, so we just proved it next. We need to find this rain. We need to find the xv for which this lineation is cured to within .1, so we can write this inequality as each of the x minus .1 of the x plus .1 point so to find the values of x. We'Re going to say that from this we find the actual .416, and from this we find it to be negative. .483 point. So if you combine these 2, we consider f x, is between .416 and negative orange, for the bandeliero is accurate to within.

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