Textbook Question
The accompanying graph shows the total distance s traveled by a bicyclist after t hours.
b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 2, Problem 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?
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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Related Practice
Textbook Question
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim x →π cos² (x― tan x)
Textbook Question
The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after it has been driven for t days.
c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.
Textbook Question
Finding Deltas Algebraically
Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.
f(x) = mx, m > 0, L = 2m, c = 2, ε = 0.03
Textbook Question
The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.
a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.
b. About how fast was the object going when it hit the surface?
Textbook Question
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim x →π sin (x/2 + sin x)