Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Calculus extends the static methods of precalculus by introducing the dynamic concept of limits. It transforms familiar topics such as slopes and areas into powerful tools for understanding change in real-life situations. The tangent line and area problems exemplify how the limit process works to provide more precise and general formulations, laying the groundwork for differentiation and integration.

Learning Objectives

1

Describe how calculus extends precalculus by using the limit process to analyze change.

2

Explain the fundamental concepts of the tangent line problem and the area problem in calculus.

3

Compare the static nature of precalculus mathematics with the dynamic approach of calculus.

4

Estimate limits both numerically and graphically and understand their role in finding slopes and areas.

Key Concepts

CONCEPT

DEFINITION

Calculus

The branch of mathematics that studies change through concepts such as derivatives and integrals, relying on the limit process to move from discrete approximations to continuous formulations.

Limit

A value that a function approaches as the input approaches a particular point. It is the foundation used to formulate both derivatives and integrals.

Precalculus Mathematics

The study of static mathematical concepts such as line slopes and areas of simple shapes, which serve as building blocks for calculus when extended via limits.

Tangent Line

A line that touches a curve at a single point and has the same slope as the curve at that point; its slope is found by taking the limit of secant slopes.

Secant Line

A line that intersects a curve at two or more points; as the second point approaches the first, its slope approximates the tangent line’s slope.

Riemann Sum

An approximate method for determining the area under a curve by summing the areas of multiple rectangles, which becomes exact in the limit.

Example Problems

Example 1

Precalculus or Calculus In Exercises $1-5,$ decide whether the problem can be solved using precalculus or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. Find the distance traveled in 15 seconds by an object traveling at a constant velocity of 20 feet per second.

Example 2

Precalculus or Calculus In Exercises $1-5,$ decide whether the problem can be solved using precalculus or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. Find the distance traveled in 15 seconds by an object moving with a velocity of $v(t)=20+7 \cos t$ feet per second.

Example 3

Precalculus or Calculus In Exercises $1-5,$ decide whether the problem can be solved using precalculus or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. A bicyclist is riding on a path modeled by the function $f(x)=0.04\left(8 x-x^{2}\right),$ where $x$ and $f(x)$ are measured in miles (see figure). Find the rate of change of elevation at $x=2$

Example 4

Precalculus or Calculus In Exercises $1-5,$ decide whether the problem can be solved using precalculus or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. A bicyclist is riding on a path modeled by the function $f(x)=0.08 x,$ where $x$ and $f(x)$ are measured in miles (see figure). Find the rate of change of elevation at $x=2$ .

Example 5

Precalculus or Calculus In Exercises $1-5,$ decide whether the problem can be solved using precalculus or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. Find the area of the shaded region.
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Step-by-Step Explanations

QUESTION

How do you find the slope of the tangent line to f(x) = x² at x = c using secant lines?

STEP-BY-STEP ANSWER:

Step 1: Identify the point of tangency P = (c, c²) on the graph of f(x) = x².
Step 2: Select a nearby point Q = (c + Δx, (c + Δx)²) on the curve.
Step 3: Compute the slope of the secant line using the formula: [f(c + Δx) - f(c)] / Δx = [((c + Δx)² - c²)] / Δx.
Step 4: Expand the numerator: (c² + 2cΔx + (Δx)² - c²) simplifies to 2cΔx + (Δx)².
Step 5: Factor Δx from the numerator to get Δx(2c + Δx) and then cancel Δx with the denominator, leaving 2c + Δx.
Step 6: Take the limit as Δx approaches 0, so the expression becomes 2c.
Final Answer: The slope of the tangent line at x = c is 2c.

Tangent Line Slope via Secant Approximation

QUESTION

Estimate the limit limₓ→2 (x² - 3x + 2)/(x - 2) using a numerical approach.

STEP-BY-STEP ANSWER:

Step 1: Recognize that direct substitution leads to an indeterminate form (0/0).
Step 2: Factor the numerator: x² - 3x + 2 factors as (x - 1)(x - 2).
Step 3: Cancel the common term (x - 2), leaving f(x) = x - 1 (for x ≠ 2).
Step 4: Substitute x = 2 into the simplified expression: 2 - 1.
Final Answer: The limit is 1.

Numerical Limit Evaluation

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Common Mistakes

  • Treating calculus as a mere collection of formulas rather than understanding the underlying limit processes.
  • Failing to differentiate between static (precalculus) techniques and dynamic (calculus) methods.
  • Confusing secant lines (approximation) with tangent lines (instantaneous rates of change).
  • Neglecting to cancel common factors before evaluating limits, leading to indeterminate forms.