Book cover for Biocalculus Calculus for the Life Sciences

Biocalculus Calculus for the Life Sciences

James Stewart

ISBN #9781133109631

1st Edition

2,565 Questions

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211,110 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces functions of several variables and illustrates their representation from multiple perspectives—verbally, numerically, algebraically, and visually. Key topics include the definitions of domain, range, graphs (surfaces), and level curves, along with procedures for evaluating limits and determining continuity. Real-world examples such as the wind-chill index and BMI emphasize the practical applications of these mathematical concepts. The multivariable approach requires careful attention to different paths when evaluating limits, highlighting a major difference from single-variable calculus.

Learning Objectives

1

Describe and represent functions of several variables in verbal, numerical, algebraic, and visual forms.

2

Determine the domain and range for functions of two and three variables and sketch their graphs.

3

Explain and use level curves (contour maps) to interpret surfaces and real-world data (e.g., wind-chill index, BMI).

4

Evaluate limits and discuss continuity for functions of several variables, emphasizing the importance of path independence.

Key Concepts

CONCEPT

DEFINITION

Function of Several Variables

A rule that assigns a unique real number to each ordered pair (or n-tuple) of numbers in a given domain; e.g., f(x, y) for functions of two variables.

Domain

The set of all input values (ordered pairs or triples) for which the function expression is defined and produces a real number.

Range

The set of all output values that a function can take as the input varies over the domain.

Graph (Surface)

The collection of points (x, y, z) in R³ that satisfy z = f(x, y); it visualizes the behavior of a function of two variables.

Level Curves

Curves in the xy-plane along which the function has a constant value, given by the equation f(x, y) = k.

Limit (of a Multivariable Function)

The value L that f(x, y) approaches as (x, y) approaches a point (a, b) from every possible direction, if such a value exists.

Continuity

A function f(x, y) is continuous at a point (a, b) if the limit as (x, y) approaches (a, b) equals f(a, b), ensuring no breaks or jumps in its graph.

Example Problems

Example 1

Wind chill In Example 1 we considered the function $W=f(T, v),$ where $W$ is the wind-chill index, $T$ is the actual temperature, and $v$ is t (a) What is the value of $f(-15,40) ?$ What is its meaning?he wind speed. A numerical representation is given in Table 1 . (b) Describe in words the meaning of the question "For what value of $v$ is $f(-20, v)=-30 ?$ Then answer the question. (c) Describe in words the meaning of the question "For what value of $T$ is $f(T, 20)=-49 ?^{\prime \prime}$ Then answer the question. (d) What is the meaning of the function $W=f(-5, v) ?$ Describe the behavior of this function. (e) What is the meaning of the function $W=f(T, 50)^{\prime}$ Describe the behavior of this function.

Example 2

The temperature-humidity index $I$ (or humidex, for short) is the perceived air temperature when the actual temperature is $T$ and the relative humidity is $h,$ so we can write $I=f(T, h) .$ The following table of values of $I$ is an excerpt from a table compiled by the National Oceanic \& Atmospheric Administration. (a) What is the value of $f(95,70) ?$ What is its meaning? (b) For what value of $h$ is $f(90, h)=100 ?$ (c) For what value of $T$ is $f(T, 50)=88 ?$ (d) What are the meanings of the functions $I=f(80, h)$ and $I=f(100, h) ?$ Compare the behavior of these two functions of $h .$

Example 3

Body surface area A model for the surface are$S=f(w, h)=0.1091 w^{0.425} h^{0.725}$a of a human body is given by the function where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet (a) Find $f(160,70)$ and interpret it.. (b) What is your own surface area?

Example 4

The wind-chill index $W$ discussed in Example 1 has been modeled by the following function: $W(T, v)=13.12+0.6215 T-11.37 v^{0.16}+0.3965 T v^{0.16}$ Check to see how closely this model agrees with the values in Table 1 for a few values of $T$ and $v .$

Example 5

A manufacturer has modeled its yearly production function $P$ (the monetary value of its entire production) as a so-called Cobb-Douglas function $P(L, K)=1.47 L^{0.65} K^{0.35}$ where $L$ is the number of labor hours (in thousands) and $K$ is the invested capital (in millions of dollars). (a) Find $P(120,20)$ and interpret it. (b) If both the amount of labor and the amount of capital are doubled, verify that the production is also doubled.
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Step-by-Step Explanations

QUESTION

Given the wind-chill function W = f(T, v) (as in Table 1), evaluate f(25, 50) and interpret its meaning.

STEP-BY-STEP ANSWER:

Step 1: Identify the variables: T (actual temperature) = 25°C and v (wind speed) = 50 km/h.
Step 2: Locate the corresponding wind-chill value using the numerical table provided by the National Weather Service.
Step 3: From the table, note that for T = 25 and v = 50 the subjective temperature (wind-chill) is approximately 21°F (or a similar value as provided in the text).
Step 4: Interpret the result: f(25, 50) represents the temperature the human body ‘feels’ given an actual temperature of 25°C and a wind speed of 50 km/h, indicating more severe perceived cold.
Final Answer: f(25, 50) ≈ 21°F (or the indicated value from the table), meaning the apparent temperature under these conditions is significantly lower than the actual temperature.

Evaluating a Function at a Point (Wind-Chill Index Example)

QUESTION

Determine whether the limit of f(x, y) = (xy)/(x² + y²) exists as (x, y) approaches (0, 0).

STEP-BY-STEP ANSWER:

Step 1: Approach (0, 0) along the x-axis by letting y = 0. Then f(x, 0) = (x*0)/(x² + 0) = 0.
Step 2: Approach (0, 0) along the y-axis by letting x = 0. Then f(0, y) = (0*y)/(0 + y²) = 0.
Step 3: Approach (0, 0) along the line y = x. Substitute y = x to get f(x, x) = (x*x)/(x² + x²) = x²/(2x²) = 1/2.
Step 4: Compare the limits: Along the x- and y-axes the limit is 0, but along the line y = x the limit is 1/2.
Final Answer: Since the value of f(x, y) as (x, y) approaches (0, 0) depends on the path taken, the limit does not exist.

Testing Limit Existence for a Function of Two Variables

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Common Mistakes

  • Assuming that evaluating limits along only one or two paths is sufficient to determine the overall limit.
  • Failing to correctly identify the domain of a function, especially when square roots or denominators are involved.
  • Mixing up the representation of the graph (surface in 3D space) with its level curves (2D projection).
  • Overlooking the potential discontinuities that result from functions being defined piecewise or having undefined expressions (like division by zero).
  • Assuming that continuity in single-variable calculus directly extends to multivariable functions without considering path dependence.