Book cover for Calculus with Applications

Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

ISBN #9781292108971

11th Edition

3,612 Questions

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224,424 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section on linear functions introduces fundamental concepts like slope, intercepts, and various forms of linear equations, including slope-intercept and point-slope forms. The material emphasizes the process of constructing linear models to approximate real-world situations, such as predicting tuition increases or analyzing trends in data. Key steps include calculating the slope as the ratio of the changes in y and x, identifying intercepts, and choosing the appropriate equation form based on the information given. Additionally, the section discusses the significance of horizontal and vertical lines, their properties, and the proper use of function notation.

Learning Objectives

1

Explain the concept of a mathematical model and how linear functions can be used to approximate real-world situations.

2

Calculate the slope of a line using the differences in y-values and x-values (?y/?x), and understand its significance.

3

Write equations of lines in various forms including slope-intercept form and point-slope form.

4

Distinguish between horizontal and vertical lines by interpreting slopes (zero or undefined).

5

Apply linear models to practical problems such as predicting tuition increases, interest calculations, and other real-world trends.

Key Concepts

CONCEPT

DEFINITION

Linear Model

A mathematical description of a situation where the relationship between two quantities can be approximated by a straight line.

Slope (m)

A measure of the steepness of a line calculated as the change in y (rise) divided by the change in x (run). Represented as m = (y2 - y1)/(x2 - x1).

Ordered Pair

A pair (x, y) representing a point in the Cartesian coordinate system, where x is the horizontal coordinate and y is the vertical coordinate.

Slope-Intercept Form

An equation of a line expressed as y = mx + b, where m is the slope and b is the y-intercept.

Point-Slope Form

An equation of a line written as y - y1 = m(x - x1), which is useful when the slope and one point on the line are known.

Horizontal Line

A line with a slope of 0, expressed as y = k, indicating the same y-value for all x-values.

Vertical Line

A line with an undefined slope, expressed as x = k, indicating the same x-value for all points on the line.

Function Notation

A notation used to denote functions, such as f(x) = 5 - 3x, where f gives the output corresponding to an input x.

Example Problems

Example 1

Find the slope of each line. Through $(6,-10)$ and $(0,11)$

Example 2

Find the slope of each line. Through $(5,-4)$ and $(1,3)$

Example 3

Find the slope of each line. Through $(3,10)$ and $(3,3)$

Example 4

Find the slope of each line. Through $(1,5)$ and $(-2,5)$

Example 5

Find the slope of each line. $y=2.7 x$
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Step-by-Step Explanations

QUESTION

Find the slope of the line through the points (17, 62) and (-4, 52).

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates: let (x1, y1) = (17, 62) and (x2, y2) = (-4, 52).
Step 2: Compute the change in x: ∆x = x2 - x1 = -4 - 17 = -21.
Step 3: Compute the change in y: ∆y = y2 - y1 = 52 - 62 = -10.
Step 4: Calculate the slope: m = ∆y / ∆x = (-10) / (-21) = 10/21.
Final Answer: The slope of the line is 10/21.

Slope Calculation Example

QUESTION

Find the equation of the line with a slope of 3/4 that passes through the point (0, -3).

STEP-BY-STEP ANSWER:

Step 1: Recognize that (0, -3) is the y-intercept, so b = -3.
Step 2: Use the given slope m = 3/4.
Step 3: Substitute m and b into the slope-intercept form y = mx + b to get y = (3/4)x - 3.
Final Answer: The equation of the line is y = (3/4)x - 3.

Equation in Slope-Intercept Form

QUESTION

Find the equation of the line passing through (13, -7) with a slope of 5/4.

STEP-BY-STEP ANSWER:

Step 1: Use the point-slope form: y - y1 = m(x - x1).
Step 2: Substitute the given point (13, -7) and m = 5/4: y - (-7) = (5/4)(x - 13).
Step 3: Simplify the equation: y + 7 = (5/4)(x - 13).
Step 4: Optionally solve for y to convert to slope-intercept form if needed.
Final Answer: The point-slope form of the line is y + 7 = (5/4)(x - 13).

Using Point-Slope Form

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Common Mistakes

  • Misordering the subtraction when computing ?x and ?y, leading to an incorrect slope.
  • Confusing the x-intercept with the y-intercept or not correctly identifying which coordinate they represent.
  • Incorrectly referring to vertical lines as having 'no slope' instead of specifying that their slope is undefined.
  • Forgetting to solve for y when converting an equation into slope-intercept form.
  • Misunderstanding that the same linear model can sometimes be only an approximation for non-linear real-world phenomena.