Basic Exercises on Local Linear Approximation

1Problem 1

Find the local linear approximation of $$f\left(x\right)=\sqrt{x}$$f(x)=\sqrt{x} near $$x=9$$x=9 and use it to approximate $$\sqrt{9.2}$$\sqrt{9.2}.

1.1Sample Solution

Let $$f\left(x\right)=\sqrt{x}$$f(x)=\sqrt{x}. Rationalizing the difference quotient gives

so $$f^{\prime }\left(9\right)=1/6$$f'(9)=1/6. Since $$f\left(9\right)=3$$f(9)=3, the local linear approximation is

Therefore,

1.2Explanation

The approximation has the same value and slope as the function at the base point $$x=9$$x=9. The actual value is $$\sqrt{9.2}\approx 3.03315$$\sqrt{9.2}\approx3.03315, confirming that the approximation error is small.

1.3Common Error

The approximation need not retain the same accuracy far from the base point. The word local indicates that the approximation is valid in a neighborhood of that point.

2Problem 2

Find the tangent line to $$f\left(x\right)=x^2$$f(x)=x^2 at $$x=1$$x=1.

2.1Sample Solution

For $$f\left(x\right)=x^2$$f(x)=x^2, $$f\left(1\right)=1$$f(1)=1, and the difference quotient is

Thus $$f^{\prime }\left(1\right)=2$$f'(1)=2, and the tangent line is

2.2Explanation

For the increment $$h=x-1$$h=x-1, the linear map is $$L\left(h\right)=2h$$L(h)=2h. The tangent line $$y=1+2h$$y=1+2h is the affine approximation obtained by adding the constant term.

3Problem 3

Let $$f\left(x\right)=x^2$$f(x)=x^2 and fix an arbitrary base point $$a\in \mathbb{R}$$a\in\mathbb{R}. Find the remainder $$r\left(h\right)$$r(h) satisfying

and verify that $$r\left(h\right)=o\left(h\right)$$r(h)=o(h).

3.1Sample Solution

Since $$f^{\prime }\left(a\right)=2a$$f'(a)=2a,

so $$r\left(h\right)=h^2$$r(h)=h^2. Moreover,

Therefore, $$r\left(h\right)=o\left(h\right)$$r(h)=o(h).

3.2Explanation

The first-order term $$2ah$$2ah forms the linear part acting on the increment, while the second-order term $$h^2$$h^2 is negligible relative to $$\left|h\right|$$|h|. Thus the complete affine approximation is $$a^2+2ah$$a^2+2ah. This calculation illustrates the equivalence between differentiability defined by difference quotients and its characterization using an $$o\left(h\right)$$o(h) remainder.

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