Book cover for Calculus

Calculus

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

ISBN #9781119379331

7th Edition

5,101 Questions

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33,688 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the concept of functions as relationships between independent and dependent variables, with a focus on continuous and linear functions. Key themes include determining domains and ranges in context, calculating the slope and difference quotient, and understanding the meaning behind intercepts and slopes in real-world settings. Real-life models such as the cricket chirp rate and Olympic records demonstrate how linear functions can represent practical situations, while discussions on proportionality reveal the constant relationship between variables.

Learning Objectives

1

Explain the concept of a function as a relationship between independent and dependent variables, including continuous and linear functions.

2

Determine the domain and range of functions in real-world contexts, such as the cricket chirp rate problem.

3

Identify and compute key aspects of linear functions, including slope, y-intercept, and difference quotients.

4

Apply the ideas of proportionality and linear models to analyze real-world scenarios like sports records, depreciation, and resource consumption.

5

Differentiate between increasing and decreasing functions and understand the implications of their slopes in various applications.

Key Concepts

CONCEPT

DEFINITION

Function

A rule or relationship that assigns each input exactly one output, often written as y = f(x).

Domain

The set of all possible input values (typically x-values) for which the function is defined.

Range

The set of all possible output values (typically y-values) produced by a function.

Continuous Function

A function whose graph can be drawn without lifting the pen, meaning there are no breaks, jumps, or holes.

Linear Function

A function of the form y = b + mx, where m is the slope (or rate of change) and b is the y-intercept.

Slope

The rate of change of a function, computed as the ratio of the change in the output (rise) to the change in the input (run).

Difference Quotient

A formula (f(x2) - f(x1))/(x2 - x1) used to calculate the slope between two points on the graph of a function.

Proportionality

A relationship where one quantity is a constant multiple of another, expressed as y = kx (direct) or inversely proportional as y = k(1/x).

Increasing/Decreasing Function

A function is increasing if its output values rise as the input increases, and decreasing if its output values fall as the input increases.

Example Problems

Example 1

The population of a city, $P$, in millions, is a function of $t,$ the number of years since $2010,$ so $P=f(t) .$ Explain the meaning of the statement $f(5)=7$ in terms of the population of this city.

Example 2

The pollutant PCB (polychlorinated biphenyl) can affect the thickness of pelican eggshells. Thinking of the thickness, $T,$ of the eggshells, in $\mathrm{mm},$ as a function of the concentration, $P$, of $\mathrm{PCBs}$ in ppm (parts per million), we have $T=f(P) .$ Explain the meaning of $f(200)$ in terms of thickness of pelican eggs and concentration of PCBs.

Example 3

Describe what Figure 1.8 tells you about an assembly line whose productivity is represented as a function of the number of workers on the line. (FIGURE CAN'T COPY)

Example 4

Find an equation for the line that passes through the given points. (0,0) and (1,1)

Example 5

Find an equation for the line that passes through the given points. (0,2) and (2,3)
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Step-by-Step Explanations

QUESTION

Given the cricket chirp function C = 4T - 160, where C must be positive and temperature T is limited to a maximum of 134°F, what is the domain of T?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the chirp rate must be nonnegative, so set C ≥ 0.
Step 2: Solve 4T - 160 ≥ 0 → 4T ≥ 160 → T ≥ 40°F.
Step 3: The context provides an upper limit: T must be at most 134°F.
Step 4: Thus, the domain of T is all values from 40°F to 134°F, inclusive.
Final Answer: The domain is [40, 134].

Determining the Domain of a Contextual Function

QUESTION

For the linear function representing Olympic pole vault heights given by y = 130 + 2t, using the points corresponding to t = 0 (y = 130) and t = 4 (y = 138), calculate the slope.

STEP-BY-STEP ANSWER:

Step 1: Identify the two points: (0, 130) and (4, 138).
Step 2: Compute the rise: 138 - 130 = 8 inches.
Step 3: Compute the run: 4 - 0 = 4 years.
Step 4: Calculate the slope as rise/run: 8/4 = 2 inches per year.
Final Answer: The slope of the function is 2 inches/year.

Calculating the Slope Using the Difference Quotient

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Common Mistakes

  • Assuming the domain of a function is always all real numbers without considering practical context (e.g., temperature must be positive for cricket chirps).
  • Incorrectly computing the slope by mixing up the rise and run or not using consistent units.
  • Confusing dependent and independent variables, especially when roles are reversed in interpretation.
  • Over-extrapolating linear models beyond the range of given data, leading to unrealistic predictions.
  • Misidentifying intercepts: For instance, not recognizing the significance of the vertical intercept as the initial value when the input is zero.