교과서 질문
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|
214
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 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.
검증된 단계별 안내1
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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
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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.
추천 영상:
4:49
Inverse Cosine
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.
추천 영상:
01:56Finding Limits by Direct Substitution Example 1
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.
추천 영상:
04:27Substitution With an Extra Variable
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Evaluate cos (11π/12) as cos (π/4 + 2π/3).
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교과서 질문
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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교과서 질문
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
cos² 5π/12
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교과서 질문
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = 1 - cos x
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교과서 질문
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.
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