Answered: 1.9 1.99 2.01 2.1 f(x) T(x) 2.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Find the tangent line approximation T to the graph of f at the given point. Then complete the table. f(x) = sin x, (2, sin 2)

Expert Solution

Step 1

The given function $f (x) = \sin x$ .

We have to find the tangent line approximation at the point $x = 2$ .

Step 2

The tangent line approximation of $f (x)$ at $x = c$ is $T (x) = f (c) + f' (c) (x - c)$

$f (x) = \sin x$

$f (2) = \sin 2$

Differentiate with respect to $x$ ,

$f' (c) = \cos x$

$f' (2) = \cos 2$

Now, the tangent line approximation of $f (x)$ at $x = 2$ ,

$T (x) = f (2) + f' (2) (x - c) T (x) = \sin 2 + (\cos 2) (x - 2)$

The tangent line approximation is $T (x) = \sin 2 + (\cos 2) (x - 2)$ .

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