Calculus I · Unit 2B · review

Approximation and Newton's Method Review

Review

Summary

L(x)=f(a)+f'(a)(x-a), \qquad dy=f'(x)dx,

\Delta y\approx dy, \qquad x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.

Linear approximations work near the contact point. Differentials estimate changes and propagated error. Newton's method repeatedly uses tangent-line roots.

Exercise

Linearize $f(x)=1/x$ at $a=2$ and estimate $1/2.03$ .

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Exercise 1 answer

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Exercise

Estimate $\sqrt[4]{15.8}$ using a nearby easy value.

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Exercise 2 answer

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Exercise

Use differentials to estimate the change in $V=\pi r^2h$ when $r$ changes and $h$ is fixed.

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Exercise 3 answer

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Exercise

Estimate percentage error in the area of a square from percentage error in side length.

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Exercise 4 answer

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Exercise

Perform two Newton steps for $x^3-5=0$ starting at $x_0=2$ .

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Exercise 5 answer

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Exercise

Give an example where a tangent-line estimate is an overestimate and explain the role of concavity.

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Exercise 6 answer

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After the explanation

Use the section idea

Reading lens

A differentiable curve behaves like its tangent line over a small enough neighborhood, with concavity explaining the direction of error.

Mental model

Linearization, differentials, and Newton's method reuse one local line for estimation, uncertainty, or a better root guess.

Decision

Choose a nearby easy input, record the local slope, and state why the requested change is small enough for the model.

Common trap

Presenting a tangent estimate as exact or running Newton iterations without checking residuals and failure modes.

Check yourself

Can you give the estimate, error direction, units, and a reason the local line is trustworthy here?

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