Textbook Question
The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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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 3.9.46a
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
a. surface area?
Verified step by step guidance1
First, understand that the surface area of a cube with edge length x is given by the formula: \( A = 6x^2 \).
Next, we need to find the relationship between the error in x and the error in A. Use differentials to approximate the change in A: \( dA = \frac{dA}{dx} \cdot dx \).
Calculate the derivative of the surface area with respect to x: \( \frac{dA}{dx} = 12x \).
Substitute \( dx = 0.005x \) (since the error in x is 0.5% of x) into the differential equation: \( dA = 12x \cdot 0.005x \).
Finally, express the percentage error in A as \( \frac{dA}{A} \times 100\% \) and simplify using the expressions for dA and A.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differential Approximation
Differential approximation is a method used to estimate the change in a function's value based on the change in its input. For a function f(x), the differential df is approximately f'(x)dx, where f'(x) is the derivative and dx is the small change in x. This concept helps in estimating errors in measurements.
Recommended video:
05:53
Finding Differentials
Surface Area of a Cube
The surface area of a cube with edge length x is given by the formula 6x^2. Understanding this formula is crucial because it allows us to relate changes in the edge length to changes in the surface area, which is necessary for calculating the percentage error in the surface area.
Recommended video:
09:07Example 1: Minimizing Surface Area
Percentage Error
Percentage error quantifies the accuracy of a measurement by comparing the error to the actual value, expressed as a percentage. It is calculated as (error/actual value) * 100%. In this context, it helps in determining how a small error in measuring the edge length of a cube affects the calculated surface area.
Recommended video:
04:57Determining Error and Relative Error
Related Practice
Textbook Question
Interpreting Derivative Values
Growth of yeast cells In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number P of cells present at hourly intervals. The number after t hours is shown in the accompanying figure.
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a. Explain what is meant by the derivative P'(5). What are its units?
Textbook Question
If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s
a. surface area change?
Textbook Question
Assume that f'(3) = −1, g'(2) = 5, g(2) = 3, and y = f(g(x)). What is y' at x = 2?
Textbook Question
Find y⁽⁴⁾ = d⁴y/dx⁴ if:
a. y = −2 sin x
Textbook Question
Computer Explorations
Use a CAS to perform the following steps in Exercises 55–62.
a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.
xy³ + tan(x + y) = 1, P(π/4, 0)