Domanda del libro di testo
Solving Trigonometric Equations
For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.
sin² θ = cos² θ
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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
Capitolo 1, Problema 1.1.37
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
Guida verificata passo dopo passo1
First, identify the basic shape of the graph of the function y = -x³. The function y = x³ is a cubic function, which typically has an S-shaped curve. The negative sign in front of x³ indicates that the graph will be reflected over the x-axis, resulting in a flipped S-shape.
Next, determine the symmetry of the function. A function is symmetric about the origin if replacing x with -x results in the negative of the original function. For y = -x³, replacing x with -x gives y = -(-x)³ = -x³, which is the same as the original function. Therefore, the graph is symmetric about the origin.
To find the intervals where the function is increasing or decreasing, calculate the derivative of the function. The derivative of y = -x³ is y' = -3x². This derivative will help us determine the slope of the tangent line at any point on the graph.
Analyze the sign of the derivative y' = -3x². Since x² is always non-negative, -3x² is always non-positive. This means the derivative is always less than or equal to zero, indicating that the function is always decreasing.
Finally, conclude that the function y = -x³ is decreasing on the entire real line, from negative infinity to positive infinity. There are no intervals where the function is increasing.
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Increasing and Decreasing Functions
A function is considered increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, a function is decreasing on an interval if the function's value at the second point is less than at the first. Understanding these concepts is crucial for analyzing the behavior of functions and determining where they rise or fall.
Video consigliato:
07:32
Determining Where a Function is Increasing & Decreasing
Critical Points
Critical points are values of the independent variable where the derivative of a function is either zero or undefined. These points are essential for identifying intervals of increase and decrease, as they often indicate where the function changes from increasing to decreasing or vice versa. Analyzing critical points helps in sketching the graph and understanding the function's overall behavior.
Video consigliato:
04:50Critical Points
Symmetry in Functions
Symmetry in functions refers to the property where a function exhibits a specific reflective behavior about a line or point. For example, a function is even if it is symmetric about the y-axis, and odd if it is symmetric about the origin. Recognizing these symmetries can simplify the graphing process and provide insights into the function's characteristics.
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Percorso guidato
06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
sin x²
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Domanda del libro di testo
Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.
Graph the function f (x) = sin 2x + cos 3x.
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Domanda del libro di testo
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
sin (A − B) = sin A cos B − cos A sin B
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Domanda del libro di testo
Algebraic Combinations
In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.
f(x) = √(x + 1), g(x) = √(x − 1)
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Domanda del libro di testo
Graph the functions in Exercises 37–56.
y = (x + 1)²/³
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