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Determine the values of $x$ for which the quadratic approximation $f(x) \approx T_{2}(x)$ in Example 7 is accurate to within 0.1 $\left[$ Hint: Graph $y=T_{2}(x), y=\cos x-0.1,$ and \right. $y=\cos x+0.1$ on a common screen. $]$

$-1.261< x <1.261$

Calculus 1 / AB

Chapter 3

Derivatives

Section 8

Linear Approximations and Taylor Polynomials

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Lectures

04:40

In mathematics, a derivati…

30:01

In mathematics, the deriva…

04:37

Use Taylor's formula …

01:16

Verify the given linear ap…

02:58

In each part, find the loc…

04:17

02:21

02:32

Verify the linear approxim…

02:48

01:07

$7-10$ Verify the given li…

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Compute the linear approxi…

01:22

03:18

Find the linear approximat…

02:20

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Use Newton’s Method to app…

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Find the lincar approximat…

02:27

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01:43

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(a) Find the linear approx…

In this problem, we are asked to determine the x for which the approximation, in example, 7, is accurate to wooden into so mathematically. We can write this form as the sine x minus 1 minus 1. Half x square is accurate within .1, where this part is our either polynomial order 2. So this equation, we say this equal is less say. This is 1, is same as cosine x, minus .11 minus 1, half x square cosine, x, plus point to the initial. In the first negro of these same as the second 1, now, if we solve for the values of x, that satisfies basicalty, we find x, 1.2 6 point now, if we sold for the values of x that satisfies now. This part that we find x to be negative 1.26, so we're going to say when x is when x is between negative 1.26 and positive 1.26. Then the approximation is accurate to with .1.

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