Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.6.12

Chapter 4, Problem 4.6.12

Suppose f is differentiable on (-∞,∞), f(5.99) = 7, and f(6) = 7.002. Use linear approximation to estimate the value of f'(6).

Verified step by step guidance

First, recall the formula for linear approximation: f(x) ≈ f(a) + f'(a)(x - a). This formula is used to approximate the value of a function near a point a using the tangent line at that point.

To estimate f'(6), we can use the concept of the derivative as the slope of the tangent line. The derivative f'(a) is approximately the change in f(x) divided by the change in x, which is (f(x) - f(a)) / (x - a).

In this problem, we are given f(5.99) = 7 and f(6) = 7.002. We can use these values to estimate f'(6) by considering the interval from x = 5.99 to x = 6.

Calculate the change in f(x) over the interval: Δf = f(6) - f(5.99) = 7.002 - 7.

Calculate the change in x over the interval: Δx = 6 - 5.99. Then, use the formula for the derivative: f'(6) ≈ Δf / Δx.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Differentiability

A function is said to be differentiable at a point if it has a defined derivative at that point, meaning it has a tangent line that approximates its behavior near that point. Differentiability implies continuity, but not vice versa. In this context, since f is differentiable on (-∞,∞), we can apply calculus techniques to analyze its behavior around the point of interest.

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. The formula for linear approximation is f(x) ≈ f(a) + f'(a)(x - a), where a is the point of tangency. This technique is particularly useful when the function is complex, allowing us to use simpler linear functions to estimate values.

Derivative as a Rate of Change

The derivative of a function at a point represents the instantaneous rate of change of the function with respect to its variable at that point. In practical terms, it indicates how much the function's output changes for a small change in input. In this question, estimating f'(6) involves understanding how f changes around x = 6, which is crucial for applying linear approximation effectively.

Recommended video:

04:16

Intro To Related Rates

Related Practice

Textbook Question

Use linear approximation to estimate f (3.85) given that f(4) = 3 and f'(4) = 2.

Textbook Question

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x³ - 6x² - 135x

views

Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)

Textbook Question

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = ln (x² + 1)

Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (1/2y)dy

Textbook Question

Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a cylindrical vertical tree trunk (see figure). The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is Θ. Prove that for a given tree and fixed angle Θ, the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.