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📐 MathLevel 6 — 🔵 IntermediateCalculus👁 17 views⏱ 2 min read

Estimate function value using derivative at a point

📝 Problem

A device is used to measure the value of a quantity that changes over time , described by a function . At time f(3) = 10$ units. Simultaneously, an integrated sensor indicates that the instantaneous rate of change of this quantity at f'(3) = 2$ units/second. Based on this information, estimate the value of the quantity at time $t=3.1$ seconds.

Detailed Solution

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In Calculus, the derivative of a function at a point represents the instantaneous rate of change of the function at that point. When we need to estimate the function's value at a point very close to a known point, we can use the linear approximation (or tangent line approximation).

The linear approximation formula for a function near a point $t_0$ is:

In this problem, we have:

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