Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.R.45
Chapter 4, Problem 4.R.45

45–46. Linear approximation


a. Find the linear approximation to f at the given point a.
b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?


ƒ(x) = x²⸍³ ; a =27; ƒ(29)

Verified step by step guidance
1
To find the linear approximation of a function f(x) at a point a, we use the formula L(x) = f(a) + f'(a)(x - a). This is derived from the tangent line to the curve at x = a.
First, calculate f(a) for the given function f(x) = x^(2/3) at a = 27. This involves evaluating 27^(2/3).
Next, find the derivative f'(x) of the function f(x) = x^(2/3). Use the power rule for derivatives: if f(x) = x^n, then f'(x) = n*x^(n-1). Here, n = 2/3.
Evaluate the derivative at the point a = 27 to find f'(27). Substitute x = 27 into the derivative expression you found in the previous step.
Now, substitute f(a), f'(a), and a into the linear approximation formula L(x) = f(a) + f'(a)(x - a) to find the linear approximation L(x) at x = 29. Compare this approximation to the actual value of f(29) to determine if it is an underestimate or overestimate.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

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

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. It is based on the idea that a function can be closely approximated by a linear function (a straight line) when the input values are near a specific point. The formula for linear approximation is f(x) ≈ f(a) + f'(a)(x - a), where f'(a) is the derivative of the function at point a.
Recommended video:
07:17
Linearization

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the function at that point. In the context of linear approximation, the derivative is crucial because it determines the slope of the tangent line, which is used to create the linear approximation.
Recommended video:
05:44
Derivatives

Estimation and Error Analysis

Estimation in calculus involves using methods like linear approximation to predict the value of a function without calculating it exactly. After estimating, it is important to analyze whether the approximation underestimates or overestimates the actual function value. This can be determined by examining the concavity of the function or the behavior of the derivative, which helps in understanding the accuracy of the approximation.
Recommended video:
04:57
Determining Error and Relative Error
Related Practice
Textbook Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→1 ( x- 1)^sinπx

1
views
Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = x³ - 6x² on [-1, 5]

1
views
Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

g(x) = x sin⁻¹ x on [-1, 1]

1
views
Textbook Question

90–103. Indefinite integrals Determine the following indefinite integrals.


∫ (x² / (x⁴ + x²)) dx

Textbook Question

104–107. Functions from derivatives Find the function f with the following properties.


ƒ'(t) = sin t + 2t; ƒ(0) = 5

Textbook Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.


eˣ and 3ˣ