Textbook Question
Applications
A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 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
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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
Related Practice
Textbook Question
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.
Textbook Question
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) = {sinx / x, −π ≤ x < 0
0, x = 0
Textbook Question
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
Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.
f(x) = x⁴ᐟ³, −1 ≤ x ≤ 8
Textbook Question
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.
∫csc θ/(csc θ − sin θ) dθ