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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 40
Chapter 4, Problem 40

Use the guidelines of this section to make a complete graph of f.
f(x) = 2 - 2x2/3 + x4/3

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1
Identify the domain of the function f(x) = 2 - 2x^{2/3} + x^{4/3}. Since the function involves fractional exponents, check for any restrictions. In this case, the domain is all real numbers because the exponents are positive and the base x can be any real number.
Find the first derivative f'(x) to determine the critical points and analyze the increasing or decreasing behavior of the function. Use the power rule for derivatives: f'(x) = -\(\frac{4}{3}\)x^{-1/3} + \(\frac{4}{3}\)x^{1/3}.
Set the first derivative f'(x) equal to zero to find critical points: -\(\frac{4}{3}\)x^{-1/3} + \(\frac{4}{3}\)x^{1/3} = 0. Solve this equation to find the values of x where the slope of the tangent is zero.
Find the second derivative f''(x) to determine the concavity and points of inflection. Differentiate f'(x) to get f''(x) = \(\frac{4}{9}\)x^{-4/3} - \(\frac{4}{9}\)x^{-2/3}.
Analyze the behavior of f(x) as x approaches positive and negative infinity to understand the end behavior of the graph. Also, use the second derivative test to confirm the nature of the critical points found in step 3, and identify any points of inflection from the second derivative.

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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 points on a coordinate plane to visualize the relationship between the input (x) and output (f(x)). Understanding how to identify key features such as intercepts, symmetry, and asymptotes is essential for creating an accurate representation of the function.
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Graph of Sine and Cosine Function

Polynomial Functions

The function f(x) = 2 - 2x^(2/3) + x^(4/3) is a polynomial function, which is a sum of terms consisting of variables raised to non-negative integer powers. Recognizing the degree and leading coefficient of the polynomial helps in determining the end behavior and shape of the graph.
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Introduction to Polynomial Functions

Critical Points and Derivatives

Finding critical points involves calculating the derivative of the function and setting it to zero to identify where the function's slope changes. Analyzing these points helps in determining local maxima, minima, and points of inflection, which are crucial for sketching the complete graph accurately.
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Critical Points
Related Practice
Textbook Question

Evaluate and simplify y'.

y = (3t²−1 / 3t²+1)^−3

Textbook Question

First Derivative Test


a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = x² + 3 on [-3,2]

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Textbook Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

ln 1.05

Textbook Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

√146

Textbook Question

Use the guidelines of this section to make a complete graph of f.

f(x) = x + 2 cos x on [-2π,2π)

Textbook Question

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

e⁰·⁰⁶