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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) = 2x + 1
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.9.27
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (5s + 3)² ds
Guida verificata passo dopo passo1
Step 1: Recognize that the integral ∫ (5s + 3)² ds involves a polynomial raised to a power. To simplify, expand the square (5s + 3)² using the distributive property: (5s + 3)² = (5s)² + 2(5s)(3) + 3².
Step 2: Expand the terms: (5s)² = 25s², 2(5s)(3) = 30s, and 3² = 9. Combine these to rewrite the integrand as 25s² + 30s + 9.
Step 3: Break the integral into separate terms: ∫ (25s² + 30s + 9) ds = ∫ 25s² ds + ∫ 30s ds + ∫ 9 ds.
Step 4: Apply the power rule for integration to each term. For ∫ sⁿ ds, the rule is ∫ sⁿ ds = (sⁿ⁺¹)/(n+1) + C, where n ≠ -1. For constants, ∫ k ds = k * s + C.
Step 5: Combine the results of the integration for each term, ensuring to include the constant of integration (C). Finally, check your work by differentiating the result to verify it matches the original integrand.
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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 whose derivative matches the given function.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Integration Techniques
To solve integrals like ∫ (5s + 3)² ds, one may need to apply specific integration techniques. In this case, expanding the integrand before integrating can simplify the process. Techniques such as substitution or integration by parts may also be useful for more complex integrals, but recognizing when to apply these methods is key.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
Verification by Differentiation
After finding an indefinite integral, it is essential to verify the result by differentiation. This involves taking the derivative of the antiderivative obtained and checking if it equals the original integrand. This step ensures that the integration was performed correctly and reinforces the fundamental relationship between differentiation and integration.
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Percorso guidato
05:53Finding Differentials
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Domanda del libro di testo
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ ((e²ʷ - 5eʷ + 4)/(eʷ - 1))dw
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Domanda del libro di testo
105–106. {Use of Tech} Races The velocity function and initial position of Runners A and B are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other.
A : v(t) = sin t; s(0) = 0 B. V(t) = cos t; S(0) = 0
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Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = √(9 - x²) + sin⁻¹ (x/3)
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Domanda del libro di testo
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) = 2x² ln x - 11x²
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Domanda del libro di testo
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = tan⁻¹ (x/(x²+2))
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