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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 55e
3장, 문제 55e

Quadratic approximations


[Technology Exercise] e. Find the quadratic approximation to h(x) = √(1 + x) at x = 0. Graph h and its quadratic approximation together. Comment on what you see.

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Identify the function for which you need the quadratic approximation: h(x) = √(1 + x).
Recall the formula for the quadratic approximation of a function f(x) at a point a: f(x) ≈ f(a) + f'(a)(x - a) + (f''(a)/2)(x - a)^2.
Calculate the first derivative of h(x) = √(1 + x) using the chain rule: h'(x) = (1/2)(1 + x)^(-1/2).
Evaluate the first derivative at x = 0: h'(0) = (1/2)(1 + 0)^(-1/2) = 1/2.
Calculate the second derivative of h(x): h''(x) = -(1/4)(1 + x)^(-3/2), and evaluate it at x = 0: h''(0) = -(1/4)(1 + 0)^(-3/2) = -1/4.

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주요 개념

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

Quadratic Approximation

Quadratic approximation is a method used to approximate a function using a second-degree polynomial. It is derived from the Taylor series expansion, where the function is approximated around a point, typically x = 0, using the function's value, first derivative, and second derivative at that point. This provides a more accurate approximation than a linear one, especially near the point of expansion.
추천 영상:
가이드 코스
06:51
Derivatives Applied To Acceleration Example 2

Taylor Series

The Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a single point. For a function f(x), the Taylor series expansion around x = a is given by f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ..., which can be truncated to form polynomial approximations. The quadratic approximation is a specific case where the series is truncated after the second derivative term.
추천 영상:
가이드 코스
5:25
Intro to Transformations

Graphing Functions

Graphing functions involves plotting the function's output values against its input values on a coordinate plane. This visual representation helps in understanding the behavior of the function and its approximations. When graphing h(x) = √(1 + x) and its quadratic approximation, one can observe how closely the approximation follows the actual function, particularly near the point of expansion, and where it starts to diverge.
추천 영상:
가이드 코스
5:53
Graph of Sine and Cosine Function
관련 실천
교과서 질문

Quadratic approximations


[Technology Exercise] c. Graph f(x) = 1/(1 − x) and its quadratic approximation at x = 0. Then zoom in on the two graphs at the point (0,1). Comment on what you see.

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교과서 질문

Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + π.

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교과서 질문

Assume that a particle’s position on the x-axis is given by


x = 3 cos t + 4 sin t,


where x is measured in feet and t is measured in seconds.


a. Find the particle’s position when t = 0, t = π/2, and t = π.

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교과서 질문

Assume that a particle’s position on the x-axis is given by


x = 3 cos t + 4 sin t,


where x is measured in feet and t is measured in seconds.


b. Find the particle’s velocity when t = 0, t = π/2, and t = π.

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교과서 질문

A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of x = 10 cos t, where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

b. Find the spring’s velocity when t = 0, t = π/3, and t = 3π/4.

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교과서 질문

Quadratic approximations


d. Find the quadratic approximation to g(x) = 1/x at x = 1. Graph g and its quadratic approximation together. Comment on what you see

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