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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 3.9.15a
3장, 문제 3.9.15a

Faster than a calculator Use the approximation (1 + x)ᵏ ≈ 1 + kx to estimate the following.


a. (1.0002)⁵⁰

검증된 단계별 안내
1
Identify the expression to approximate: (1.0002)⁵⁰. Here, we can see that x = 0.0002 and k = 50.
Recognize that the approximation (1 + x)ᵏ ≈ 1 + kx is useful when x is small, which is the case here.
Substitute the values of x and k into the approximation formula: 1 + kx = 1 + 50 * 0.0002.
Calculate the product kx: 50 * 0.0002 = 0.01.
Add the result to 1 to complete the approximation: 1 + 0.01.

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주요 개념

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Binomial Approximation

The binomial approximation (1 + x)ᵏ ≈ 1 + kx is a simplification used when x is small and k is a constant. It allows for quick estimates of expressions raised to a power without complex calculations. This approximation is derived from the binomial theorem and is particularly useful for small perturbations around 1.
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가이드 코스
04:57
Determining Error and Relative Error

Small x Assumption

The small x assumption is crucial for the validity of the binomial approximation. It implies that x is close to zero, making higher-order terms in the binomial expansion negligible. This assumption simplifies calculations and is often used in physics and engineering to approximate values efficiently.
추천 영상:
03:39
Integrals of Natural Exponential Functions (e^x)

Exponentiation

Exponentiation is the mathematical operation involving numbers raised to a power, denoted as (1 + x)ᵏ. Understanding how to manipulate and approximate powers is essential in calculus, especially when dealing with series expansions and approximations. It forms the basis for many calculus concepts, including derivatives and integrals.
추천 영상:
6:13
Exponential Functions
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교과서 질문

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.


a. On what day is the temperature increasing the fastest?


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교과서 질문

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


a. d⁹⁹⁹/dx⁹⁹⁹ (cos x)

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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.


a. How fast is the boat approaching the dock when 10 ft of rope are out?


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교과서 질문

Consider the function f graphed here. The domain of f is the interval [−4, 6] and its graph is made of line segments joined end to end.


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b. Graph the derivative of f. The graph should show a step function.

227
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교과서 질문

a. Let f(x) be a function satisfying |f(x)| ≤ x² for −1 ≤ x ≤ 1. Show that f is differentiable at x = 0 and find f′(0).

247
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교과서 질문

The accompanying figure shows the velocity v = ds/dt = f(t) (m/sec) of a body moving along a coordinate line.


a. When does the body reverse direction?

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