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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 3 - Derivatives문제 3.9.45b
3장, 문제 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?

검증된 단계별 안내
1
Understand that the area A of a circle is given by the formula: A=πr2, 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, dr, is 2% of r.
Use the concept of differentials to relate the change in area to the change in radius. The differential of the area is: dA=2πrdr.
Calculate the relative error in the area by dividing the differential of the area by the area itself: dAA=2rdr/πr2.
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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
가이드 코스
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.
추천 영상:
05:06
Finding 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.
추천 영상:
가이드 코스
04:57
Determining Error and Relative Error
관련 실천
교과서 질문

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)

191
views
교과서 질문

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


c. ƒ(x) , x = 1

g(x) + 1

208
views
교과서 질문

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)

187
views
교과서 질문

The folium of Descartes (See Figure 3.27)


c. Find the coordinates of the point A in Figure 3.29 where the folium has a vertical tangent line.


384
views
교과서 질문

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?

198
views
교과서 질문

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?

241
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