Textbook Question
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
h(t) = |t³|
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.1.65
Theory and Examples
The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.
Verified step by step guidance1
Understand the concept of inverse proportionality: If two variables r and s are inversely proportional, it means that as one increases, the other decreases in such a way that their product remains constant. Mathematically, this relationship can be expressed as r * s = k, where k is a constant.
Use the given values to find the constant k: We know that r = 6 when s = 4. Substitute these values into the equation r * s = k to find k. This gives us 6 * 4 = k, so k = 24.
Set up the equation for the new condition: We need to find s when r = 10. Using the inverse proportionality relationship, substitute r = 10 into the equation r * s = k, which becomes 10 * s = 24.
Solve for s: To find the value of s, divide both sides of the equation 10 * s = 24 by 10. This gives s = 24 / 10.
Simplify the expression: Simplify the fraction 24 / 10 to find the value of s. This can be done by dividing the numerator and the denominator by their greatest common divisor.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Proportionality
Inverse proportionality means that as one variable increases, the other decreases in such a way that their product remains constant. In mathematical terms, if r and s are inversely proportional, then r * s = k, where k is a constant. This relationship allows us to find the value of one variable when the other is known.
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Finding the Constant of Proportionality
To solve problems involving inverse proportionality, we first need to determine the constant of proportionality (k). This is done by substituting the known values of r and s into the equation r * s = k. For example, with r = 6 and s = 4, we calculate k = 6 * 4 = 24, which will be used to find other values.
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Solving for Unknown Variables
Once we have the constant of proportionality, we can solve for unknown variables by rearranging the equation. For instance, if we want to find s when r = 10, we set up the equation 10 * s = k. By substituting k = 24, we can solve for s, leading to s = 24 / 10 = 2.4.
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Related Practice
Textbook Question
Evaluate cos (11π/12) as cos (π/4 + 2π/3).
Textbook Question
Increasing and Decreasing Functions
Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.
y = 1/|x|
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Textbook Question
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
cos² 5π/12
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = 1 - cos x
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Textbook Question
A point P in the first quadrant lies on the parabola 𝔂 = 𝔁². Express the coordinates of P as functions of the angle of inclination of the line joining P to the origin.