Book cover for Calculus Early Transcendental Functions

Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

ISBN #9781285774770

6th Edition

8,973 Questions

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64,375 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, which provides the slope of the tangent line to a function's graph. It emphasizes the process of approximating the tangent slope via secant lines and clarifies the relationship between differentiability and continuity. Additionally, it presents several basic differentiation rules, such as the Constant Rule and Power Rule, that simplify finding derivatives without directly computing limits each time. The derivative not only measures the slope but also serves as a tool for calculating rates of change in various applications, from geometry to physics.

Learning Objectives

1

Use the limit definition of the derivative to find the slope of a tangent line to a curve at a given point.

2

Apply the difference quotient as an approximation tool and understand its convergence to the derivative.

3

Understand and explain the relationship between differentiability and continuity, including conditions involving vertical tangents and sharp turns.

4

Utilize basic differentiation rules such as the Constant Rule, Power Rule, Constant Multiple Rule, and Sum/Difference Rules to compute derivatives.

5

Employ derivatives to determine instantaneous rates of change in practical applications.

Key Concepts

CONCEPT

DEFINITION

Derivative

A function that gives the slope of the tangent line to the graph of a function at any point where the derivative exists. It is defined as the limit of the difference quotient as the change in the independent variable approaches zero.

Tangent Line

A line that touches a curve at a single point without crossing it (locally) and has the same instantaneous rate of change as the curve at that point.

Secant Line

A line that intersects a curve at two distinct points. Its slope, given by the difference quotient, approximates the slope of the tangent line as the two points become arbitrarily close.

Difference Quotient

An expression of the form [f(c + Δx) - f(c)] / Δx used to approximate the slope of the tangent line; its limit as Δx → 0 defines the derivative.

Differentiability

A property of a function if its derivative exists at a particular point or over an interval. Differentiability at a point implies that the function is continuous there.

Continuity

A function is continuous at a point if there is no interruption in its graph at that point. However, continuity does not guarantee differentiability.

Constant Rule

A basic differentiation rule stating that the derivative of any constant function is 0, reflecting the fact that a horizontal (constant) line has no slope.

Example Problems

Example 1

Estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$. (GRAPH CAN'T COPY)

Example 2

Estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$. (GRAPH CAN'T COPY)

Example 3

Use the graph shown in the figure. (GRAPH CAN'T COPY) Identify or sketch each of the quantities on the figure. (a) $f(1)$ and $f(4)$ (b) $f(4)-f(1)$ (c) $y=\frac{f(4)-f(1)}{4-1}(x-1)+f(1)$

Example 4

Use the graph shown in the figure. (GRAPH CAN'T COPY) Insert the proper inequality symbol $(<\text { or }>)$ between the given quantities. (a) $\frac{f(4)-f(1)}{4-1} \quad \frac{f(4)-f(3)}{4-3}$ (b) $\frac{f(4)-f(1)}{4-1} \quad f^{\prime}(1)$

Example 5

Use the graph shown in the figure. (GRAPH CAN'T COPY) $$f(x)=3-5 x, \quad(-1,8)$$
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Step-by-Step Explanations

QUESTION

How do you find the derivative of f(x) = x² at a general point x = c using the definition of the derivative?

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: Compute f(c + Δx) for f(x) = x²: f(c + Δx) = (c + Δx)².
Step 3: Expand (c + Δx)² to get: c² + 2cΔx + (Δx)².
Step 4: Subtract f(c) = c² from the expression: (c² + 2cΔx + (Δx)²) - c² = 2cΔx + (Δx)².
Step 5: Form the difference quotient: [2cΔx + (Δx)²] / Δx.
Step 6: Factor out Δx: Δx(2c + Δx) / Δx, then cancel Δx (with Δx ≠ 0) to obtain 2c + Δx.
Step 7: Take the limit as Δx → 0: lim (Δx → 0) (2c + Δx) = 2c.
Final Answer: The derivative of f(x) = x² at x = c is f'(c) = 2c.

Finding the Derivative of f(x) = x² using the Limit Definition

QUESTION

What is the derivative of the constant function f(x) = 7?

STEP-BY-STEP ANSWER:

Step 1: Recognize that f(x) = 7 is a constant function.
Step 2: Apply the Constant Rule, which states that the derivative of a constant is 0.
Final Answer: f'(x) = 0 for all x.

Using the Constant Rule

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Common Mistakes

  • Confusing the slope of a secant line with the slope of the tangent line; it is important to let the interval shrink to zero to get the tangent slope.
  • Failing to properly cancel the ?x factor in the difference quotient, potentially leading to undefined or incorrect limits.
  • Assuming that continuity automatically implies differentiability; while differentiability implies continuity, the converse is not always true.
  • Forgetting that the derivative of a constant function is zero, which can lead to incorrect conclusions when applying differentiation rules.