교과서 질문
Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.
180
views
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
3장, 문제 3.9.38
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the lateral surface area S = πr√(r² + h²) of a right circular cone when the radius changes from r₀ to r₀ + dr and the height does not change
검증된 단계별 안내1
Identify the formula for the lateral surface area of a right circular cone: \( S = \pi r \sqrt{r^2 + h^2} \).
Recognize that the problem asks for the change in the lateral surface area when the radius changes from \( r_0 \) to \( r_0 + dr \), while the height \( h \) remains constant.
To estimate the change in \( S \), use the concept of differentials. The differential \( dS \) is given by the derivative of \( S \) with respect to \( r \), multiplied by \( dr \).
Calculate the derivative \( \frac{dS}{dr} \) using the product rule and chain rule: \( \frac{dS}{dr} = \pi \left( \sqrt{r^2 + h^2} + \frac{r^2}{\sqrt{r^2 + h^2}} \right) \).
The differential formula that estimates the change in the lateral surface area is \( dS = \pi \left( \sqrt{r^2 + h^2} + \frac{r^2}{\sqrt{r^2 + h^2}} \right) dr \).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
1m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Differential Calculus
Differential calculus focuses on the concept of the derivative, which represents the rate of change of a function. In this context, it is used to estimate how small changes in the radius of a cone affect its lateral surface area, assuming the height remains constant.
추천 영상:
가이드 코스
06:11
Fundamental Theorem of Calculus Part 1
Lateral Surface Area of a Cone
The lateral surface area of a right circular cone is given by the formula S = πr√(r² + h²), where r is the radius and h is the height. Understanding this formula is crucial for determining how changes in the radius impact the surface area when the height is fixed.
추천 영상:
09:07Example 1: Minimizing Surface Area
Differential Formula
A differential formula is used to approximate the change in a function's value due to small changes in its variables. For the cone's surface area, the differential formula involves calculating the derivative of the surface area with respect to the radius, providing an estimate of the change when the radius increases by a small amount dr.
추천 영상:
가이드 코스
05:53Finding Differentials
관련 실천
교과서 질문
Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.
248
views
교과서 질문
Using the Alternative Formula for Derivatives
Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.
g(x) = 1 + √x
239
views
교과서 질문
The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).
203
views
교과서 질문
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x² sin² (2x²)
364
views
교과서 질문
Find the derivatives of all orders of the functions in Exercises 29–32.
y = x⁵ / 120
183
views