Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Properties of Functions

Calculus

0. Functions

Properties of Functions

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1:09 minutes

Problem 5.4.7

Textbook Question

Is x¹² an even or odd function? Is sin x² an even or odd function?

Verified step by step guidance

Step 1: Recall the definitions of even and odd functions. An even function satisfies f(-x) = f(x), meaning it is symmetric about the y-axis. An odd function satisfies f(-x) = -f(x), meaning it is symmetric about the origin.

Step 2: Analyze the function x¹². Substitute -x into the function: f(-x) = (-x)¹². Since raising -x to an even power results in x¹² (because the negative sign disappears), f(-x) = f(x). Therefore, x¹² is an even function.

Step 3: Analyze the function sin(x²). Substitute -x into the function: f(-x) = sin((-x)²). Since squaring -x results in x² (because the negative sign disappears), f(-x) = sin(x²). Therefore, sin(x²) satisfies f(-x) = f(x), making it an even function.

Step 4: Summarize the results. Both x¹² and sin(x²) are even functions because they satisfy the condition f(-x) = f(x).

Step 5: Reflect on the symmetry of these functions. x¹² is symmetric about the y-axis due to the even power, and sin(x²) is symmetric about the y-axis because the argument x² is unaffected by the sign of x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Even and Odd Functions

A function is classified as even if it satisfies the condition f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, a function is odd if it meets the condition f(-x) = -f(x), indicating that its graph is symmetric about the origin. Understanding these definitions is crucial for determining the nature of specific functions.

Polynomial Functions

Polynomial functions, such as x¹², are expressions that involve variables raised to whole number powers. The degree of the polynomial and the coefficients determine its behavior. In the case of even-degree polynomials, they are always even functions, as substituting -x yields the same result as substituting x.

Trigonometric Functions

Trigonometric functions, like sin(x), have specific properties regarding symmetry. The sine function is an odd function, meaning sin(-x) = -sin(x). When considering sin(x²), we must analyze the argument x², which is always non-negative, affecting the overall symmetry and classification of the function.

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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 = x³/8

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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 = (−x)²/³

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Textbook Question

Find the largest interval on which the given function is increasing.

b. ƒ(x) = (x + 1)⁴

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Textbook Question

Find the largest interval on which the given function is increasing.

c. g(x) = (3x - 1)¹/³

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Textbook Question

State whether the functions represented by graphs A , B , C and in the figure are even, odd, or neither. <IMAGE>

1009

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Textbook Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$ƒ (x) = x^{4} + 5 x^{2} - 12$

130

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Textbook Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$ƒ (x) = x^{5} - x^{3} - 2$

125

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Textbook Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$

134

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