Textbook Question
Finding Derivative Functions and Values
Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.
f(x) = 4 – x²; f′(−3), f′(0), f′(1)
246
views
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 2.9.46b
The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s
b. volume?
Verified step by step guidance1
First, understand that the volume V of a cube with edge length x is given by the formula: .
To find the maximum percentage error in the volume, we need to use the concept of differentials. The differential of the volume, dV, can be expressed in terms of the differential of x, dx, as: .
Calculate the derivative of the volume with respect to x: .
Substitute the derivative into the differential equation: . The relative error in x is given as 0.5%, which means .
The relative error in volume is then: . Multiply this by 100 to convert to a percentage error: percent.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differential Error Analysis
Differential error analysis involves using calculus to approximate the change in a function's output based on small changes in its input. For a function f(x), the differential df represents the approximate change in f due to a small change dx in x. This concept is crucial for understanding how measurement errors propagate in calculations, such as finding the volume of a cube.
Recommended video:
04:57
Determining Error and Relative Error
Volume of a Cube
The volume of a cube is calculated using the formula V = x^3, where x is the length of an edge. Understanding this formula is essential because it shows how the volume depends on the cube's edge length. Any error in measuring x will affect the calculated volume, and this relationship is key to determining the percentage error in the volume.
Recommended video:
04:48Finding Volume Using Disks
Percentage Error Calculation
Percentage error quantifies the relative error in a measurement as a percentage of the true value. It is calculated by dividing the absolute error by the true value and multiplying by 100. In the context of the cube's volume, understanding how to compute percentage error helps in assessing the impact of measurement inaccuracies on the calculated volume.
Recommended video:
04:57Determining Error and Relative Error
Related Practice
Textbook Question
In Exercises 19–22, find the slope of the curve at the point indicated.
y = x³ − 2x + 7, x = −2
245
views
Textbook Question
Slopes, Tangent Lines, and Normal Lines
In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.
y = 2 sin(πx – y), (1,0)
240
views
Textbook Question
Derivatives
In Exercises 1–18, find dy/dx.
y = (sec x + tan x)(sec x − tan x)
420
views