Textbook Question
Computer Explorations
Use a CAS to perform the following steps in Exercises 55–62.
b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.
2y² + (xy)¹/³ = x² + 2, P(1,1)
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 3.9.45b
The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s
b. area?
Verified step by step guidance1
Understand that the area A of a circle is given by the formula: , where r is the radius.
Recognize that the error in the radius is given as a percentage, specifically 2%. This means the change in radius, , is 2% of .
Use the concept of differentials to relate the change in area to the change in radius. The differential of the area is: .
Calculate the relative error in the area by dividing the differential of the area by the area itself: .
Simplify the expression for the relative error in the area to find that it is twice the relative error in the radius. Therefore, the maximum percentage error in the area is 2 times 2%, which is 4%.
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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 given by f'(x)dx, where f'(x) is the derivative of f with respect to x, and dx is the change in x. This concept helps in estimating small changes in the output, such as the area of a circle, due to small changes in the input, like the radius.
Recommended video:
05:53
Finding Differentials
Derivative of Area with Respect to Radius
The area A of a circle is given by the formula A = πr². To find how the area changes with respect to the radius, we take the derivative of A with respect to r, which is dA/dr = 2πr. This derivative indicates how a small change in the radius affects the area, which is crucial for determining the percentage error in the area based on the percentage error in the radius.
Recommended video:
05:06Finding Area When Bounds Are Not Given
Percentage Error Calculation
Percentage error is a way to express the error in a measurement as a percentage of the true value. It is calculated as (error/true value) × 100%. In this context, if the radius has a percentage error of 2%, the percentage error in the area can be found using the relationship between the differential of the area and the radius, considering the derivative and the given percentage error in the radius.
Recommended video:
04:57Determining Error and Relative Error
Related Practice
Textbook Question
Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.
c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?
Textbook Question
Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is
y = 37 sin[(2π/365)(x − 101)] + 25
and is graphed in the accompanying figure.
b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?
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Textbook Question
Computer Explorations
Use a CAS to perform the following steps in Exercises 55–62.
b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.
x√(1 + 2y) + y = x², P(1,0)
Textbook Question
Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).
b. How is ds/dt related to dy/dt and dz/dt if x is constant?
Textbook Question
Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is
A = (1/2) ab sinθ.
b. How is dA/dt related to dθ/dt and da/dt if only b is constant?