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Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = −x² − 6x − 9,−4 ≤ x < ∞
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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 4.7.12a
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
πcos πx
Guida verificata passo dopo passo1
Identify the function you need to find the antiderivative of, which is \(\pi \cos \pi x\).
Recall that the antiderivative (indefinite integral) of \(\cos(ax)\) with respect to \(x\) is \(\frac{1}{a} \sin(ax) + C\), where \(a\) is a constant and \(C\) is the constant of integration.
Apply this rule to the function \(\pi \cos \pi x\). Since the function has a constant multiplier \(\pi\), factor it out and integrate \(\cos \pi x\).
Integrate \(\cos \pi x\) to get \(\frac{1}{\pi} \sin \pi x + C\). Then multiply by the constant \(\pi\) outside the integral.
Combine the results to write the antiderivative as \(\pi \times \frac{1}{\pi} \sin \pi x + C\), which simplifies to \(\sin \pi x + C\). Remember to verify your answer by differentiating it to check if you get back the original function.
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Antiderivative (Indefinite Integral)
An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is expressed with an arbitrary constant since differentiation of a constant is zero.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Integration of Trigonometric Functions
Integrating trigonometric functions like cosine involves recognizing standard integral forms. For example, the integral of cos(kx) dx is (1/k) sin(kx) + C, where k is a constant multiplier inside the function's argument.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the antiderivative and helps identify any missing constants or errors in integration.
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Percorso guidato
05:53Finding Differentials
Pratica correlata
Domanda del libro di testo
53. Distance between two ships At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yd) and continued to do so all day. Ship B was sailing east at 8 knots and continued to do so all day.
a. Start counting time with t=0 at noon and express the distance s between the ships as a function of t.
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Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).
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Domanda del libro di testo
Finding displacement from an antiderivative of velocity
a. Suppose that the velocity of a body moving along the s-axis is
ds/dt = v = 9.8t − 3.
i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.
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Domanda del libro di testo
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = sec³θ / 3 + C
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Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
a. What are the critical points of f?
f′(x) = (x − 1)(x + 2)(x − 3)
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