Textbook Question
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) = 3x⁴ + 4x³ - 12x²
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.8.45
{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.
Where are the inflection points of f(x) = (9/5)x⁵ - (15/2)x⁴ + (7/3)x³ + 30x² + 1 located?
Verified step by step guidance1
To find the inflection points of the function \( f(x) = \frac{9}{5}x^5 - \frac{15}{2}x^4 + \frac{7}{3}x^3 + 30x^2 + 1 \), we first need to find the second derivative, \( f''(x) \).
Calculate the first derivative \( f'(x) \) by differentiating \( f(x) \): \( f'(x) = \frac{d}{dx}\left(\frac{9}{5}x^5 - \frac{15}{2}x^4 + \frac{7}{3}x^3 + 30x^2 + 1\right) \).
Differentiate \( f'(x) \) to find the second derivative \( f''(x) \). This involves applying the power rule to each term of \( f'(x) \).
Set \( f''(x) = 0 \) to find potential inflection points. Solve this equation to find the critical points where the concavity might change.
Use Newton's method to approximate the roots of \( f''(x) = 0 \). Start with an initial guess \( x_0 \) and iterate using the formula \( x_{n+1} = x_n - \frac{f''(x_n)}{f'''(x_n)} \) until the values converge to a satisfactory level of precision.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Newton's Method
Newton's Method is an iterative numerical technique used to find approximate solutions to equations, particularly for finding roots of functions. It utilizes the function's derivative to refine guesses, starting from an initial estimate. The method is effective for functions that are continuous and differentiable, allowing for rapid convergence to a solution when the initial guess is close enough.
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Evaluating Composed Functions
Inflection Points
Inflection points are points on a curve where the concavity changes, indicating a shift in the direction of curvature. To find inflection points, one must compute the second derivative of the function and determine where it equals zero or is undefined. These points are significant in curve sketching as they help identify changes in the behavior of the graph, such as transitions from concave up to concave down.
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Second Derivative Test
The Second Derivative Test is a method used to classify critical points of a function by analyzing the sign of the second derivative. If the second derivative is positive at a critical point, the function is concave up, indicating a local minimum; if negative, it indicates a local maximum. If the second derivative is zero, the test is inconclusive, and further analysis is needed to determine the nature of the critical point.
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Related Practice
Textbook Question
{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = ln x - x² + 3x - 1
Textbook Question
Approximating changes
Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).
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Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x ln x - 2x + 3 on (0,∞)
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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) = 2x³ - 3x² + 12
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_u→ π/4 (tan u - cot u) / (u - π/4)