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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.4.13g
Chapter 4, Problem 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.

Verified step by step guidance
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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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Key Concepts

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

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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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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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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Related Practice
Textbook Question

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51). 


d. Find the number and location of the fixed points of g for a = 2, 3, and 4 on the interval 0 ≤ x ≤ 1. 

Textbook Question

Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>


f. Sketch one possible graph of f.

Textbook Question

For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.


a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .


b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.



ƒ(x) = x⁵/16 ; [-2, 2] <IMAGE>

Textbook Question

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

Plot a possible graph of f.

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Textbook Question

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.


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

Textbook Question

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>


d. Find the time and the displacement when the object reaches its high point for the second time.

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