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Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 4.7.35
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(−2cost) dt
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int (-2 \cos t) \, dt\).
Recall the basic integral formula for cosine: \(\int \cos t \, dt = \sin t + C\).
Apply the constant multiple rule for integrals: \(\int k f(t) \, dt = k \int f(t) \, dt\), where \(k\) is a constant.
Rewrite the integral using the constant multiple: \(\int (-2 \cos t) \, dt = -2 \int \cos t \, dt\).
Integrate \(\cos t\) and multiply by \(-2\): \(-2 \sin t + C\), where \(C\) is the constant of integration.
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Indefinite Integral
An indefinite integral represents the family of all antiderivatives of a function and is expressed with a constant of integration, C. It reverses differentiation, meaning if F'(x) = f(x), then ∫f(x) dx = F(x) + C.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Integration of Trigonometric Functions
Integrating trigonometric functions like cosine involves knowing their antiderivatives. For example, the integral of cos(t) dt is sin(t) + C, and constants can be factored out to simplify the process.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Verification by Differentiation
After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and helps identify any errors in the integration process.
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Percorso guidato
05:53Finding Differentials
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Domanda del libro di testo
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
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Applications
Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft/sec) to 0 in 45 ft. What constant deceleration does it take to do that?
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Domanda del libro di testo
Finding Extrema from Graphs
In Exercises 11–14, match the table with a graph.
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Domanda del libro di testo
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dy/dx = 3x⁻²ᐟ³, y(−1) = −5
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Domanda del libro di testo
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
4. y=9/14x^(1/3)(x^2-7)
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