Calculus I · 2B · review
Approximation and Newton's Method Review
Review approximation and newton's method review with mixed practice and links back to the exact lessons behind each skill.
Section overview
Local linearity, differentials, and Newton's methodWhat this section is building
Review approximation and newton's method review with mixed practice and links back to the exact lessons behind each skill.
Linearization, differentials, and Newton's method reuse one local line for estimation, uncertainty, or a better root guess.
Choose a nearby easy input, record the local slope, and state why the requested change is small enough for the model.
Presenting a tangent estimate as exact or running Newton iterations without checking residuals and failure modes.
Review
Linear approximations work near the contact point. Differentials estimate changes and propagated error. Newton's method repeatedly uses tangent-line roots.
Linearize at and estimate .
Exercise 1 answer
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Estimate using a nearby easy value.
Exercise 2 answer
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Use differentials to estimate the change in when changes and is fixed.
Exercise 3 answer
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Estimate percentage error in the area of a square from percentage error in side length.
Exercise 4 answer
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Perform two Newton steps for starting at .
Exercise 5 answer
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Give an example where a tangent-line estimate is an overestimate and explain the role of concavity.
Exercise 6 answer
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