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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.4.13g
Capitolo 4, Problema 4.4.13g

Let ƒ(x) = (x - 3) (x + 3)²


g. Use your work in parts (a) through (f) to sketch a graph of ƒ.

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Step 1: Begin by expanding the expression (x + 3)². This will help simplify the function ƒ(x). The expansion of (x + 3)² is (x + 3)(x + 3) = x² + 6x + 9.
Step 2: Substitute the expanded form back into the function ƒ(x). The function now becomes ƒ(x) = (x - 3)(x² + 6x + 9)g.
Step 3: Distribute (x - 3) across the expanded polynomial (x² + 6x + 9). This involves multiplying each term in the polynomial by (x - 3), resulting in x³ + 6x² + 9x - 3x² - 18x - 27.
Step 4: Combine like terms from the distribution to simplify the expression further. The simplified form of the function is ƒ(x) = x³ + 3x² - 9x - 27.
Step 5: Analyze the behavior of the function ƒ(x) = x³ + 3x² - 9x - 27. Consider the critical points, inflection points, and intercepts to sketch the graph. Determine where the function is increasing or decreasing, and identify any local maxima or minima.

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Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler components, or factors. In the given function ƒ(x) = (x - 3)(x + 3)², recognizing the factors helps identify the roots of the polynomial, which are the x-values where the function equals zero. This is crucial for sketching the graph, as the roots indicate where the graph intersects the x-axis.
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Introduction to Polynomial Functions

Multiplicity of Roots

The multiplicity of a root refers to the number of times a particular root appears in the factored form of a polynomial. In ƒ(x), the root x = -3 has a multiplicity of 2, meaning the graph will touch the x-axis at this point but not cross it. Understanding multiplicity is essential for accurately sketching the behavior of the graph near its roots.
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Derivatives of Inverse Sine & Inverse Cosine Example 1

End Behavior of Polynomials

The end behavior of a polynomial describes how the graph behaves as x approaches positive or negative infinity. For the polynomial ƒ(x) = (x - 3)(x + 3)², the leading term determines this behavior. Since the highest degree term is x^3, the graph will rise to positive infinity as x approaches both positive and negative infinity, which is important for sketching the overall shape of the graph.
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Introduction to Polynomial Functions
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