Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 1 - Functions

Problema 1.2.43

Capitolo 1, Problema 1.2.43

Graph the functions in Exercises 37–56.

y = (x + 1)²/³

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Identify the type of function: The given function is y = (x + 1)^(2/3). This is a power function with a fractional exponent, which indicates a root function.

Determine the domain: Since the base of the power (x + 1) is raised to a fractional power with an even numerator, the expression is defined for all real numbers x. Therefore, the domain is all real numbers.

Find the critical points: To find critical points, take the derivative of the function with respect to x. Use the chain rule to differentiate y = (x + 1)^(2/3).

Analyze the behavior at critical points: Evaluate the derivative to find where it is zero or undefined. These points will help determine local maxima, minima, or points of inflection.

Sketch the graph: Use the information from the derivative and critical points to sketch the graph. Consider the symmetry, intercepts, and end behavior of the function to complete the graph.

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Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the behavior of a function. It requires understanding the function's domain, range, and any transformations applied to the basic function. For y = (x + 1)²/³, identifying key features like intercepts, asymptotes, and symmetry helps in sketching an accurate graph.

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Graph of Sine and Cosine Function

Fractional Exponents

Fractional exponents, such as ²/³, indicate roots and powers. The expression (x + 1)²/³ can be interpreted as the cube root of (x + 1) squared. Understanding fractional exponents is crucial for determining the function's behavior, especially near critical points where the base might be zero or negative.

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Introduction to Exponent Rules

Transformations of Functions

Transformations involve shifting, stretching, or compressing the graph of a function. The expression y = (x + 1)²/³ includes a horizontal shift left by 1 unit due to the (x + 1) term. Recognizing transformations helps in predicting how the graph will move or change shape compared to the parent function y = x²/³.

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Intro to Transformations

Pratica correlata

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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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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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

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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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Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.