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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 4.4.49
Chapter 4, Problem 4.4.49

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.


ƒ(x) = 1/3 x³ - 2x² - 5x + 2

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Start by identifying the function you need to graph: \( f(x) = \frac{1}{3}x^3 - 2x^2 - 5x + 2 \). This is a cubic polynomial function.
Use a graphing utility or software to input the function. This will help you visualize the graph and identify key features such as intercepts, local extrema, and inflection points.
Locate the x-intercepts by setting \( f(x) = 0 \) and solving the equation \( \frac{1}{3}x^3 - 2x^2 - 5x + 2 = 0 \). Use the graphing tool to approximate these values.
Identify the local extrema by finding the derivative \( f'(x) \) and setting it to zero to find critical points. The derivative is \( f'(x) = x^2 - 4x - 5 \). Solve \( x^2 - 4x - 5 = 0 \) to find critical points, and use the second derivative test or the graph to determine if they are maxima or minima.
Determine the inflection points by finding the second derivative \( f''(x) \) and setting it to zero. The second derivative is \( f''(x) = 2x - 4 \). Solve \( 2x - 4 = 0 \) to find potential inflection points, and verify by checking the change in concavity on the graph.

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Key Concepts

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

Graphing Functions

Graphing functions involves plotting the curve of a function on a coordinate plane to visualize its behavior. This includes identifying key features such as intercepts, where the graph crosses the axes, and understanding the overall shape and direction of the graph. Graphing utilities, like graphing calculators or software, can assist in accurately plotting complex functions.
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Intercepts

Intercepts are points where the graph of a function crosses the axes. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is found by evaluating the function at x = 0. These points are crucial for understanding the function's behavior and are often used as starting points for sketching graphs.
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Local Extreme Values and Inflection Points

Local extreme values are points where a function reaches a local maximum or minimum. These can be found using the first derivative test. Inflection points occur where the function changes concavity, identified using the second derivative. Both concepts are essential for understanding the function's shape and behavior, especially when graphing.
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Related Practice
Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ ((2 + 3 cos y)/sin² y)dy

Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (3/x⁴ + 2 - 3/x²)

Textbook Question

Cylinder and cones (Putnam Exam 1938) Right circular cones of height h and radius r are attached to each end of a right circular cylinder of height h and radius r, forming a double-pointed object. For a given surface area A, what are the dimensions r and h that maximize the volume of the object?

Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = x³ - (3/2)x² - 36x

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Textbook Question

Verify that the following functions satisfy the conditions of Theorem 4.9 on their domains. Then find the location and value of the absolute extrema guaranteed by the theorem.


f(x) = x√(3-x)

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Textbook Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = eˣ + e⁻ˣ