Question

Use the tangent line approximation. Given $f(4)=5, f^{\prime}(4)=7,$ approximate $f(3.92)$.

Name: Problem 51, Chapter 3: Calculus Single Variable
Uploaded: 2021-04-16T16:01:08-08:00
Duration: 1 min 49 s
Channel: Gregory Higby
Description: Use the tangent line approximation. Given f(4)=5, f^'(4)=7, approximate f(3.92).

   Use the tangent line approximation.
Given $f(4)=5, f^{\prime}(4)=7,$ approximate $f(3.92)$.

Calculus Single Variable

Deborah… 7th Edition

Chapter 3, Problem 51

Instant Answer

Solved by Expert Gregory Higby

04/16/2021

Step 1

We can use these values to create a linear approximation of the function $f(x)$ around the point $(4,5)$. Show more…

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Use the tangent line approximation. Given $f(4)=5, f^{\prime}(4)=7,$ approximate $f(3.92)$.

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Key Concepts

Tangent Line Approximation

This is a method used in calculus to estimate the value of a function near a given point using the equation of the tangent line at that point. It is based on the idea that, close to the point of tangency, a function behaves similarly to its tangent line.

Derivative

The derivative of a function at a point provides the rate at which the function is changing at that point. In tangent line approximation, the derivative is used as the slope of the tangent line, which guides how the function is linearly approximated near a particular point.

Linearization

Linearization refers to the process of approximating a non-linear function by a linear function at a specific point. This linear approximation uses the function’s value and its derivative at that point, making it an effective tool for estimating nearby function values when the exact function is difficult to compute.

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