Textbook Question
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]
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.R.107
104–107. Functions from derivatives Find the function f with the following properties.
h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3)
Verified step by step guidance1
Step 1: Recognize that the problem involves finding the original function h(x) given its derivative h'(x) = (x⁴ - 2) / (1 + x²) and an initial condition h(1) = -2/3. This is a problem of integration and using the constant of integration.
Step 2: Set up the integral to find h(x). To reverse the derivative, integrate h'(x): ∫((x⁴ - 2) / (1 + x²)) dx. This will give the general form of h(x) plus a constant of integration, C.
Step 3: Break the integral into manageable parts. Notice that the numerator (x⁴ - 2) can be split into two terms: x⁴ and -2. Rewrite the integral as ∫(x⁴ / (1 + x²)) dx - ∫(2 / (1 + x²)) dx.
Step 4: Solve each integral separately. For ∫(x⁴ / (1 + x²)) dx, use polynomial division or substitution techniques to simplify. For ∫(2 / (1 + x²)) dx, recognize it as a standard integral that results in 2 * arctan(x).
Step 5: Combine the results of the integrals and add the constant of integration, C. Use the initial condition h(1) = -2/3 to solve for C by substituting x = 1 into the expression for h(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to understand how a function behaves locally. The notation h'(x) indicates the derivative of the function h with respect to x, which can be used to find slopes of tangent lines and optimize functions.
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Derivatives
Integration
Integration is the process of finding the antiderivative of a function, essentially reversing differentiation. It allows us to recover the original function from its derivative. In this context, to find the function h from its derivative h'(x), we will need to perform integration, which may involve applying techniques such as substitution or integration by parts.
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Initial Conditions
Initial conditions are specific values that help determine a unique solution to a differential equation. In this problem, the condition h(1) = -2/3 provides a point through which the function h must pass. This information is crucial for solving the integral obtained from the derivative, as it allows us to find the constant of integration and fully define the function.
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Related Practice
Textbook Question
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ
Textbook Question
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.
Textbook Question
104–107. Functions from derivatives Find the function f with the following properties.
ƒ'(t) = sin t + 2t; ƒ(0) = 5
Textbook Question
Optimization A right triangle has legs of length h and r and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = (1/3) πr²h.) <IMAGE>
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Textbook Question
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
eˣ and 3ˣ