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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 2 - Limits and Continuity문제 24
2장, 문제 24

[Technology Exercise] Let f(t) = 1/t for t≠0.
         
a. Find the average rate of change of f with respect to t over the intervals (i) from t=2 to t=3, and (ii) from t=2 to t=T.
         
b. Make a table of values of the average rate of change of f with respect to t over the interval [2,T], for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001, and 2.000001.
         
c. What does your table indicate is the rate of change of f with respect to t at t=2?

검증된 단계별 안내
1
To find the average rate of change of the function \( f(t) = \frac{1}{t} \) over an interval \([a, b]\), use the formula: \( \frac{f(b) - f(a)}{b - a} \).
For part (a)(i), calculate the average rate of change from \( t = 2 \) to \( t = 3 \) by substituting into the formula: \( \frac{f(3) - f(2)}{3 - 2} = \frac{\frac{1}{3} - \frac{1}{2}}{1} \). Simplify the expression to find the result.
For part (a)(ii), calculate the average rate of change from \( t = 2 \) to \( t = T \) using the formula: \( \frac{f(T) - f(2)}{T - 2} = \frac{\frac{1}{T} - \frac{1}{2}}{T - 2} \). Simplify this expression to find a general formula in terms of \( T \).
For part (b), create a table of values for the average rate of change using the expression from part (a)(ii) for \( T = 2.1, 2.01, 2.001, 2.0001, 2.00001, \) and \( 2.000001 \). Calculate each value by substituting \( T \) into the expression.
For part (c), observe the values in the table as \( T \) approaches 2. The pattern of these values will indicate the instantaneous rate of change of \( f \) at \( t = 2 \), which is the derivative of \( f \) at that point.

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주요 개념

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

Average Rate of Change

The average rate of change of a function over an interval is calculated as the change in the function's value divided by the change in the input variable. Mathematically, for a function f(t) over the interval [a, b], it is given by (f(b) - f(a)) / (b - a). This concept is essential for understanding how a function behaves over specific intervals and is foundational for more advanced topics like instantaneous rates of change.
추천 영상:
가이드 코스
06:37
Average Value of a Function

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is crucial for defining derivatives and understanding continuity. In this exercise, as T approaches 2, the limit helps determine the instantaneous rate of change of the function at that point, which is the derivative of f(t) at t=2.
추천 영상:
05:50
One-Sided Limits

Derivative

The derivative of a function at a point quantifies the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change as the interval shrinks to zero. In this context, finding the derivative of f(t) at t=2 will provide the exact rate of change of the function at that point, which is the ultimate goal of the exercise.
추천 영상:
가이드 코스
05:44
Derivatives
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