Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

Group icon
63,186 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the derivative, beginning with the slope of a straight line and extending to the instantaneous slope of a curve using limits. It establishes that the derivative is not only a foundational concept in calculus but also a practical tool in various real-world applications. The relationship between continuity and differentiability is explored, and essential differentiation rules are presented to equip learners with techniques to compute derivatives efficiently.

Learning Objectives

1

Explain and compute the slope of a straight line and the slope of a curve at a point.

2

Understand the definition of the derivative through limits and how it connects with the concept of continuity.

3

Apply the fundamental rules of differentiation to various functions.

4

Differentiate between differentiability and continuity and explain their relationship.

5

Interpret the derivative as a rate of change in real-world contexts.

Key Concepts

CONCEPT

DEFINITION

Derivative

A measure of how a function changes as its input changes; formally defined as the limit of the difference quotient.

Slope of a Straight Line

A constant value representing the steepness of a line, calculated as the ratio of the change in y to the change in x.

Slope of a Curve at a Point

The instantaneous rate of change of the function at a point, found using the derivative.

Limit

The value that a function approaches as the input approaches some value.

Differentiability

A property of a function indicating that it has a derivative at a certain point or over an interval.

Continuity

A property of a function which means there are no breaks, jumps, or holes in the graph of the function.

Differentiation Rules

A set of guidelines such as the power rule, product rule, and chain rule that simplify finding derivatives of functions.

Example Problems

Example 1

Find the slopes and $y$-intercepts of the following lines. $$y=3-7 x$$

Example 2

Find the slopes and $y$-intercepts of the following lines. $$y=\frac{3 x+1}{5}$$

Example 3

Find the slopes and $y$-intercepts of the following lines. $$x=2 y-3$$

Example 4

Find the slopes and $y$-intercepts of the following lines. $$y=6$$

Example 5

Find the slopes and $y$-intercepts of the following lines. $$y=\frac{x}{7}-5$$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

How do you determine the slope of a straight line given two points?

STEP-BY-STEP ANSWER:

Step 1: Identify two points on the line, (x1, y1) and (x2, y2).
Step 2: Compute the difference in y-coordinates (y2 - y1).
Step 3: Compute the difference in x-coordinates (x2 - x1).
Step 4: Divide the change in y by the change in x to obtain the slope.
Final Answer: Slope = (y2 - y1) / (x2 - x1).

Slope of a Straight Line

QUESTION

How is the derivative of a function f(x) at a point defined using limits?

STEP-BY-STEP ANSWER:

Step 1: Consider a small increment h added to x, resulting in x + h.
Step 2: Form the difference quotient [f(x + h) - f(x)] / h.
Step 3: Take the limit as h approaches 0.
Step 4: This limit, if it exists, is the derivative f'(x).
Final Answer: f'(x) = lim (h→0) [f(x + h) - f(x)] / h.

Derivative via Limits

QUESTION

What is the relationship between differentiability and continuity of a function?

STEP-BY-STEP ANSWER:

Step 1: Understand that a function f is continuous at a point if there is no interruption in its graph.
Step 2: Recognize that differentiability at a point means the derivative exists at that point.
Step 3: Note that if a function is differentiable at a point, it is also continuous at that point.
Step 4: However, a function can be continuous without being differentiable (e.g., at a sharp corner).
Final Answer: Differentiability implies continuity, but the converse is not necessarily true.

Differentiability and Continuity

QUESTION

How do you apply basic differentiation rules to compute the derivative of a polynomial function?

STEP-BY-STEP ANSWER:

Step 1: Identify each term of the polynomial and its exponent.
Step 2: Apply the power rule: for any term ax^n, the derivative is a*n*x^(n-1).
Step 3: Sum the derivatives of each term.
Step 4: Simplify the expression to obtain the final derivative.
Final Answer: The derivative is computed term-by-term following the power rule.

Rules for Differentiation

Scroll left
Scroll right

Common Mistakes

  • Assuming that a continuous function must always be differentiable.
  • Confusing the average rate of change with the instantaneous rate of change.
  • Applying differentiation rules incorrectly, such as misusing the power rule.
  • Errors in limit calculations when defining the derivative.