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

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the concept of the derivative as the limit of the difference quotient and its central role in finding the slope of the tangent line to a curve. It emphasizes that differentiability implies continuity, even though continuity alone does not guarantee differentiability. In addition, several basic differentiation rules such as the Constant Rule and Power Rule are introduced, equipping students with tools to compute derivatives efficiently and apply them to problems in calculus ranging from geometry to real-world rate changes.

Learning Objectives

1

Explain the concept of the derivative as the limit of the difference quotient and its geometric interpretation as the slope of the tangent line.

2

Use the limit definition to compute the derivative of a function at a point.

3

Describe the relationship between differentiability and continuity, including cases of non-differentiability such as sharp turns and vertical tangents.

4

Apply basic differentiation rules (such as the Constant Rule, Power Rule, and Sum/Difference Rules) to calculate derivatives efficiently.

5

Utilize derivatives to determine instantaneous rates of change and equations of tangent lines.

Key Concepts

CONCEPT

DEFINITION

Derivative

The derivative of a function at a point is the limit of the difference quotient as the increment approaches zero. It represents the slope of the tangent line to the function at that point and the instantaneous rate of change.

Tangent Line

A line that touches a curve at a single point and has the same instantaneous slope as the curve at that point. Its slope is given by the derivative of the function.

Difference Quotient

An expression of the form [f(x+c) - f(c)]/Δx used to approximate the slope of the secant line through two points on a curve; its limit as Δx approaches 0 gives the derivative.

Continuity

A function is continuous at a point if there is no break, jump, or hole at that point. Differentiability at a point implies continuity there, though the converse is not always true.

Differentiability

A function is differentiable at a point if the derivative exists at that point, meaning that the limit defining the derivative exists (including equality of the left-hand and right-hand limits).

Constant Rule

A basic differentiation rule stating that the derivative of any constant is zero.

Example Problems

Example 1

Estimating Slope In Exercises 1 and $2,$ estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$

Example 2

Estimating Slope In Exercises 1 and $2,$ estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$

Example 3

Slopes of Secant Lines In Exercises 3 and $4,$ use the graph shown in the figure. To print an enlarged copy of the graph, go to MathGraphs.com. Identify or sketch each of the quantities on the figure. $$ \begin{array}{l}{\text { (a) } f(1) \text { and } f(4) \quad \text { (b) } f(4)-f(1)} \\ {\text { (c) } y=\frac{f(4)-f(1)}{4-1}(x-1)+f(1)}\end{array} $$

Example 4

Slopes of Secant Lines In Exercises 3 and $4,$ use the graph shown in the figure. To print an enlarged copy of the graph, go to MathGraphs.com. Insert the proper inequality symbol $(<\text { or }>)$ between the given quantities. $$ \begin{array}{l}{\text { (a) } \frac{f(4)-f(1)}{4-1} \quad \frac{f(4)-f(3)}{4-3}} \\ {\text { (b) } \frac{f(4)-f(1)}{4-1} \quad f^{\prime}(1)}\end{array} $$

Example 5

Finding the Slope of a Tangent Line In Exercises $5-10$ , find the slope of the tangent line to the graph of the function at the given point. $$ f(x)=3-5 x, \quad(-1,8) $$
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Step-by-Step Explanations

QUESTION

Find the derivative of f(x) = x³ – 2x at an arbitrary point x = c using the limit definition.

STEP-BY-STEP ANSWER:

Step 1: Write the definition of the derivative: f'(c) = lim (Δx → 0) [f(c + Δx) – f(c)] / Δx.
Step 2: Substitute f(x) into the formula: f(c + Δx) = (c + Δx)³ – 2(c + Δx) and f(c) = c³ – 2c.
Step 3: Expand (c + Δx)³ to get c³ + 3c²Δx + 3c(Δx)² + (Δx)³, and expand –2(c + Δx) to get –2c – 2Δx.
Step 4: Compute the difference f(c + Δx) – f(c): [c³ + 3c²Δx + 3c(Δx)² + (Δx)³ – 2c – 2Δx] – [c³ – 2c] = 3c²Δx + 3c(Δx)² + (Δx)³ – 2Δx.
Step 5: Factor Δx from the numerator: Δx(3c² + 3cΔx + (Δx)² – 2).
Step 6: Divide by Δx: [3c² + 3cΔx + (Δx)² – 2].
Step 7: Take the limit as Δx approaches 0: f'(c) = 3c² – 2.
Final Answer: f'(c) = 3c² – 2, and for any x, f'(x) = 3x² – 2.

Using the Limit Definition to Find a Derivative

QUESTION

Explain why a function can be continuous at a point but not differentiable there.

STEP-BY-STEP ANSWER:

Step 1: Recognize that continuity means the function has no breaks or holes at a point.
Step 2: Understand that differentiability requires the limit for the derivative (from both sides) to exist and be equal.
Step 3: Identify examples such as functions with sharp turns (e.g., |x| at x=0) where the left-hand and right-hand derivatives differ.
Step 4: Conclude that even if a function is continuous (no gaps), a sudden change in direction can prevent a well-defined tangent line, meaning the function is not differentiable at that point.
Final Answer: A function may be continuous at a point but not differentiable there if the instantaneous slope (left and right limits) are not equal.

Understanding Differentiability and Continuity

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Common Mistakes

  • Confusing the tangent line with a secant line; the tangent only touches the curve at one point with the same instantaneous slope.
  • Failing to cancel the ?x factor completely when simplifying the difference quotient, leading to erroneous limits.
  • Assuming that continuity implies differentiability; neglecting the fact that functions may be continuous yet have sharp turns or vertical tangents.
  • Not verifying that one-sided limits of the derivative agree, leading to incorrect conclusions about differentiability at endpoints or points of non-smooth behavior.