Question

Use the tangent line approximation. Given $f(5)=3, f^{\prime}(5)=-2,$ approximate $f(5.03)$.

Name: Problem 52, Chapter 3: Calculus Single Variable
Uploaded: 2021-04-16T16:05:38-08:00
Duration: 1 min 47 s
Channel: Gregory Higby
Description: Use the tangent line approximation. Given f(5)=3, f^'(5)=-2, approximate f(5.03).

   Use the tangent line approximation.
Given $f(5)=3, f^{\prime}(5)=-2,$ approximate $f(5.03)$.

Calculus Single Variable

Deborah… 7th Edition

Chapter 3, Problem 52

Instant Answer

Solved by Expert Gregory Higby

04/16/2021

Step 1

We can use the point-slope form of a line to find the equation of the tangent line at $x=5$. The point-slope form of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. Show more…

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Use the tangent line approximation. Given $f(5)=3, f^{\prime}(5)=-2,$ approximate $f(5.03)$.

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Key Concepts

Tangent Line Approximation

This concept refers to using the tangent line at a point on a function to approximate the function's values near that point. The idea is that if a function is smooth and differentiable, then for inputs close to a known value, the tangent line provides a good estimation because it has the same slope as the function at that point.

Derivative

The derivative of a function at a given point represents the instantaneous rate of change of the function at that point. It is crucial for forming the tangent line, as it provides the slope needed to construct the linear approximation.

Linear Approximation

Linear approximation, also known as linearization, is the process of estimating the value of a function near a known point using the tangent line. By combining the function's value and its derivative at the given point, one can create a simple linear formula to approximate the function for small changes in the input.

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