Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the foundational concept of limits as a tool to formally define the instantaneous rate of change and the slope of tangent lines to curves. By exploring both average and instantaneous rates of change through practical examples like free fall and parabola slopes, we understand how continuous change can be analyzed. The careful transition from secant slopes to the tangent slope via limits underpins much of differential calculus.

Learning Objectives

1

Explain the concept of a limit and its role in defining instantaneous rates of change.

2

Differentiate between average rate of change and instantaneous rate of change in various contexts.

3

Apply limits to determine the slope of a tangent line to a curve at a given point.

4

Analyze the behavior of functions near points where they may not be defined (holes) through limit concepts.

5

Use geometric interpretations (secant and tangent lines) to understand motion and change in physical phenomena.

Key Concepts

CONCEPT

DEFINITION

Limit

The value that a function approaches as the input approaches a certain point, even if the function is not defined at that point.

Average Rate of Change

The change in the function's value divided by the change in the independent variable over a specified interval; graphically, this is the slope of the secant line connecting two points.

Instantaneous Rate of Change

The limit of the average rate of change as the interval shrinks to zero, representing the slope of the tangent line at a point.

Secant Line

A line that passes through two points on a curve, used to compute the average rate of change between those points.

Tangent Line

A line that touches a curve at a single point and has a slope equal to the instantaneous rate of change (limit of secant slopes) at that point.

Free Fall Equation

An equation such as y = 16t² (with y in feet and t in seconds) that models the distance fallen by an object under the influence of gravity, ignoring air resistance.

Example Problems

Example 1

Find the average rate of change of the function over the given interval or intervals. $f(x)=x^{3}+1$ a. $[2,3] \quad$ b. $[-1,1]$

Example 2

Find the average rate of change of the function over the given interval or intervals. $g(x)=x^{2}-2 x$ a. $[1,3] \quad$ b. $[-2,4]$

Example 3

Find the average rate of change of the function over the given interval or intervals. $h(t)=\cot t$ a. $[\pi / 4,3 \pi / 4] \quad$ b. $[\pi / 6, \pi / 2]$

Example 4

Find the average rate of change of the function over the given interval or intervals. $g(t)=2+\cos t$ a. $[0, \pi] \quad$ b. $[-\pi, \pi]$

Example 5

Find the average rate of change of the function over the given interval or intervals. $R(\theta)=\sqrt{4 \theta+1} ; \quad[0,2]$
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Step-by-Step Explanations

QUESTION

How do we determine the instantaneous speed of a falling rock at a specific time using limits?

STEP-BY-STEP ANSWER:

Step 1: Write the expression for the distance fallen: y = 16t².
Step 2: Set up the average speed over the interval from t = 1 to t = 1 + h: Average speed = [16(1 + h)² - 16(1)²] / h.
Step 3: Expand the numerator: 16[(1 + 2h + h²) - 1] = 16(2h + h²).
Step 4: Simplify the expression: Average speed = 16(2h + h²) / h = 32 + 16h, for h ≠ 0.
Step 5: Take the limit as h approaches 0 to obtain the instantaneous speed: Instantaneous speed = 32 + 16(0) = 32 ft/sec.
Final Answer: The instantaneous speed at t = 1 second is 32 ft/sec.

Instantaneous Speed of a Falling Rock at t = 1 sec

QUESTION

How do we find the slope of the tangent line to the parabola y = x² at the point (2, 4)?

STEP-BY-STEP ANSWER:

Step 1: Write the expression for the slope of the secant line between P(2, 4) and Q(2 + h, (2 + h)²): Slope = [(2 + h)² - 2²] / h.
Step 2: Expand (2 + h)² to get 4 + 4h + h²; then subtract 4 to have 4h + h² in the numerator.
Step 3: Factor out h: h(4 + h) / h = 4 + h, for h ≠ 0.
Step 4: Take the limit as h approaches 0: Slope = 4 + 0 = 4.
Step 5: Write the equation of the tangent line using the point-slope form: y - 4 = 4(x - 2).
Final Answer: The slope of the tangent line is 4 and its equation is y = 4x - 4.

Slope of the Parabola y=x² at Point (2, 4)

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Common Mistakes

  • Failing to recognize the distinction between average rate of change (secant slope) and instantaneous rate of change (tangent slope).
  • Attempting to directly substitute h = 0 into the difference quotient without taking the limit, leading to division by zero.
  • Misinterpreting the geometric meaning of limits, which leads to errors in understanding the behavior of functions near points of discontinuity (holes).
  • Forgetting to simplify expressions properly before taking the limit, which can result in algebraic mistakes.