Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
d. Determine all extrema of f.
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.109e
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫[f(x) + g(x)] dx
Verified step by step guidance1
First, identify the functions f(x) and g(x) as derivatives of given expressions. Specifically, f(x) = \(\frac{d}{dx}\) (1 - \(\sqrt{x}\)) and g(x) = \(\frac{d}{dx}\) (x + 2).
Next, compute f(x) by differentiating the function inside: recall that \(\sqrt{x}\) = x^{1/2}, so use the power rule for derivatives to find \(\frac{d}{dx}\) (1 - x^{1/2}).
Similarly, compute g(x) by differentiating the function inside: \(\frac{d}{dx}\) (x + 2), which involves differentiating a linear function.
After finding explicit expressions for f(x) and g(x), write the integral as \(\int\) [f(x) + g(x)] \, dx = \(\int\) f(x) \, dx + \(\int\) g(x) \, dx.
Finally, integrate each term separately. Since f(x) and g(x) are derivatives of known functions, integrating them will return the original functions (up to a constant). Combine the results and include the constant of integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative and Differentiation
The derivative of a function represents its instantaneous rate of change with respect to the variable. Differentiation rules, such as the power rule, allow us to find derivatives of functions like √x or polynomials. Understanding how to compute derivatives is essential to identify f(x) and g(x) in the problem.
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Integration as the Inverse of Differentiation
Integration is the reverse process of differentiation, used to find the original function given its derivative. The integral of a sum of functions equals the sum of their integrals. Recognizing that ∫[f(x) + g(x)] dx can be simplified by integrating each term separately is key to solving the problem.
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Properties of Definite and Indefinite Integrals
Indefinite integrals represent families of functions differing by a constant. When integrating derivatives, the result returns the original function plus a constant of integration. This concept helps in understanding that integrating f(x) + g(x), where f and g are derivatives, recovers the sum of the original functions.
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Related Practice
Textbook Question
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
y = (x² - 49) / (x² + 5x - 14)
Textbook Question
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
y=1-(x+1)^3
Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁻⁴ + 2x + 3
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Textbook Question
The Mean Value Theorem
a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .
[Technology Exercises] b.Find the solution to as many decimal places as you can.
Textbook Question
106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (d) When is the acceleration positive? Negative?