Suppose that f is differentiable at x0. Modify the argumentof Exercise 53 to prove that

Chapter 2, Problem 54

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Focus on Concepts

Suppose that \(f\) is differentiable at \(x_{0}\). Modify the argument of Exercise 53 to prove that the tangent line to the graph of f at the point \(P\left(x_{0}, f\left(x_{0}\right)\right)\) provides the best linear approximation to \(f\) at \(P\).

[Hint: Suppose that \(y=f\left(x_{0}\right)+m\left(x-x_{0}\right)\) is any line through \(P\left(x_{0}, f\left(x_{0}\right)\right)\) with slope \(m \neq f^{\prime}\left(x_{0}\right)\). Apply Definition 1.4.1 to (5) with \(x=x_{0}+h\) and \(\epsilon=\frac{1}{2}\left|f^{\prime}\left(x_{0}\right)-m\right|\).]

Equation Transcription:

Text Transcription:

f

x_0

f

P(x_0, f(x_0))

f

P

y=f(x_0)+m(x-x_0)

P(x_0, f(x_0))

m neq f'(x_0)

x=x_0+h

in=1/2|f'(x0)-m|