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) = sin⁻¹ x
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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 4.9.43
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (sec² x - 1) dx
Verified step by step guidance1
Step 1: Recognize the integral ∫(sec² x - 1) dx as a sum of two separate terms. You can split the integral into ∫sec² x dx - ∫1 dx.
Step 2: Recall the standard integral formula for sec² x. The integral of sec² x is tan(x) + C, where C is the constant of integration.
Step 3: Recall the standard integral formula for 1. The integral of 1 with respect to x is x + C.
Step 4: Combine the results from Step 2 and Step 3. The indefinite integral becomes tan(x) - x + C, where C is the constant of integration.
Step 5: To check your work, differentiate the result tan(x) - x + C. The derivative of tan(x) is sec² x, and the derivative of -x is -1. Adding these together gives sec² x - 1, which matches the original integrand.
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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 often referred to as antiderivation, where we seek a function F(x) such that F'(x) equals the integrand.
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Basic Integration Rules
Basic integration rules are fundamental techniques used to compute integrals. For example, the integral of sec²(x) is a standard result, yielding tan(x) as its antiderivative. Understanding these rules is essential for solving integrals efficiently and accurately, as they provide a foundation for more complex integration techniques.
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Verification by Differentiation
Verification by differentiation involves checking the correctness of an indefinite integral by differentiating the result. If the derivative of the antiderivative matches the original integrand, the integration is confirmed to be correct. This step is crucial in calculus to ensure that the integration process has been performed accurately.
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Related Practice
Textbook Question
Snell’s Law Suppose a light source at A is in a medium in which light travels at a speed v₁ and that point B is in a medium in which light travels at a speed v₂ (see figure). Using Fermat’s Principle, which states that light travels along the path that requires the minimum travel time (Exercise 55), show that the path taken between points A and B satisfies (sinΘ₁/v₁ = (sin Θ₂) /v₂ . <IMAGE>
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Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 1 (x² + 2x) / (x +3)
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Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)
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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) = tan x
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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) = eˣ(x - 2)²
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