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Sketch the graphs of the rational functions in Exercises 53–60.
𝓍²
y = ------------------
𝓍² ― 4
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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 4.7.60
Checking Antiderivative Formulas
Verify the formulas in Exercises 57–62 by differentiation.
∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C
Guida verificata passo dopo passo1
Identify the given antiderivative formula: \(\int \csc^{2}\left(\frac{x - 1}{3}\right) \, dx = -3 \cot\left(\frac{x - 1}{3}\right) + C\).
Recall that to verify an antiderivative, you differentiate the right-hand side and check if you get the original integrand.
Differentiate the function \(-3 \cot\left(\frac{x - 1}{3}\right)\) using the chain rule. Start by differentiating \(\cot(u)\) where \(u = \frac{x - 1}{3}\).
Use the derivative formula \(\frac{d}{du} \cot(u) = -\csc^{2}(u)\), so \(\frac{d}{dx} \cot\left(\frac{x - 1}{3}\right) = -\csc^{2}\left(\frac{x - 1}{3}\right) \cdot \frac{d}{dx} \left(\frac{x - 1}{3}\right)\).
Calculate \(\frac{d}{dx} \left(\frac{x - 1}{3}\right) = \frac{1}{3}\), then multiply all parts together and simplify to confirm the derivative equals \(\csc^{2}\left(\frac{x - 1}{3}\right)\), which verifies the formula.
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Antiderivatives and Indefinite Integrals
An antiderivative of a function is another function whose derivative equals the original function. Indefinite integrals represent the family of all antiderivatives and include a constant of integration, C, since differentiation of a constant is zero.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Differentiation of Composite Functions (Chain Rule)
The chain rule is used to differentiate composite functions, where one function is inside another. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). This is essential when verifying antiderivatives involving expressions like (x - 1)/3.
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03:58The Chain Rule for 3+ Functions
Derivatives of Trigonometric Functions
Knowing the derivatives of basic trig functions like cotangent and cosecant squared is crucial. For example, the derivative of cot(u) is -csc²(u) times the derivative of u. This knowledge helps verify if the given antiderivative formula is correct by differentiation.
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06:35Derivatives of Other Inverse Trigonometric Functions
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Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {x² − x, −2 ≤ x ≤−1
2x² − 3x − 3, −1 < x ≤ 0
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56. Airplane landing path An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the accompanying figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function y = ax^3+bx^2+cx+d, where y(-L)= H and y(0)=0.
a. What is dy/dx at x = 0?
b. What is dy/dx at x = -L?
c. Use the values for dy/dx at x = 0 and x =- L together with y(0) = 0 and y(-L) = H to show that y(x)=H[2(x/L)^3+3(x/L)^2]
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Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
g(x) = √(4 − x²), −2 ≤ x ≤ 1
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The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.
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Domanda del libro di testo
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.
∫2x(1 − x⁻³) dx
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