Skip to main content

Ch. 3 - Derivatives

Hass - Thomas' Calculus 15th Edition

Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Hass 15th Edition

Ch. 3 - Derivatives

문제 3.9.55b

3장, 문제 3.9.55b

Quadratic approximations

b. Find the quadratic approximation to f(x) = 1/(1 − x) at x = 0.

검증된 단계별 안내

1

Identify the function f(x) = \( \frac{1}{1-x} \) and note that we want to find its quadratic approximation at x = 0.

Recall that the quadratic approximation of a function f(x) at a point a is given by the formula: \( f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 \).

Calculate the first derivative f'(x) of the function f(x) = \( \frac{1}{1-x} \). Use the derivative rule for \( \frac{1}{u} \), which is \( -\frac{u'}{u^2} \).

Calculate the second derivative f''(x) by differentiating f'(x). This involves applying the quotient rule or chain rule again.

Evaluate f(x), f'(x), and f''(x) at x = 0, and substitute these values into the quadratic approximation formula to find the quadratic approximation of f(x) at x = 0.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

7m

도움이 되었나요?

주요 개념

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

Taylor Series

The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For a function f(x) centered at x = a, the series is f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ..., which provides a polynomial approximation of the function near a.

추천 영상:

가이드 코스

Intro to Transformations

Quadratic Approximation

Quadratic approximation is a specific case of the Taylor series where the function is approximated by a polynomial of degree two. It involves using the first three terms of the Taylor series: f(a), f'(a)(x-a), and f''(a)(x-a)^2/2. This approximation is useful for estimating the function's behavior near the point a.

추천 영상:

가이드 코스

Derivatives Applied To Acceleration Example 2

Derivatives

Derivatives measure how a function changes as its input changes. The first derivative, f'(x), represents the rate of change or slope of the function, while the second derivative, f''(x), indicates the curvature or concavity. Calculating these derivatives at a specific point is essential for constructing the quadratic approximation.

추천 영상:

가이드 코스

Derivatives

관련 실천

교과서 질문

Motion Along a Coordinate Line

Exercises 1–6 give the positions s = f(t) of a body moving on a coordinate line, with s in meters and t in seconds.

b. Find the body’s speed and acceleration at the endpoints of the interval.

s = 25/t² − 5/t, 1 ≤ t ≤ 5

교과서 질문

A sliding ladder

A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.

b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?

교과서 질문

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

" style="" width="250">

Find the derivatives with respect to x of the following combinations at the given value of x.

b. f(x)g³(x), x = 0

교과서 질문

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.

b. At what rate is the angle θ changing at this instant (see the figure)?

교과서 질문

By computing the first few derivatives and looking for a pattern, find the following derivatives.

b. d¹¹⁰/dx¹¹⁰ (sin x − 3 cos x)

교과서 질문

Temperature The given graph shows the outside temperature T in °F, between 6 a.m. and 6 p.m.

b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times?