Textbook Question
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = ln (1 - x)
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 4.9.61
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (eˣ⁺²) dx
Verified step by step guidance1
Step 1: Recognize that the integral ∫(e^(x+2)) dx involves an exponential function. The general rule for integrating e^(u) is ∫e^(u) du = e^(u) + C, where u is a function of x.
Step 2: Identify the exponent in the given integral. Here, u = x + 2. To apply the rule, we need to check if the derivative of u with respect to x is present in the integral.
Step 3: Compute the derivative of u = x + 2 with respect to x. The derivative is du/dx = 1. Since the derivative is 1, we can directly integrate without additional adjustments.
Step 4: Apply the integration rule for e^(u). The integral becomes e^(x+2) + C, where C is the constant of integration.
Step 5: Verify your result by differentiating e^(x+2) + C. The derivative of e^(x+2) is e^(x+2), and the derivative of the constant C is 0. This confirms that the original integrand is recovered, ensuring the solution is correct.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integrals
Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is known as integration, which is the reverse operation of differentiation.
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Introduction to Indefinite Integrals
Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of natural logarithms, approximately equal to 2.71828. In the context of integration, the integral of e^(kx) is (1/k)e^(kx) + C, which is crucial for solving integrals involving exponential terms.
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Differentiation Check
Checking work by differentiation involves taking the derivative of the obtained integral to verify that it matches the original integrand. This process ensures that the integration was performed correctly and helps identify any potential errors in the integration process.
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05:02Determining Differentiability Graphically
Related Practice
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (ln(3x + 5eˣ)) / (ln(7x + 3e²ˣ)
Textbook Question
Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.
v(t) = 2t + 4; s(0) = 0
Textbook Question
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = tan x/2 on (-π,π)
Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x² - 2 ln x
Textbook Question
{Use of Tech} Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations.
f(x) = x³ - 3x² + x + 1