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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.15a
Chapter 4, Problem 4.7.15a

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
csc x cot x

Verified step by step guidance
1
Recall that an antiderivative (or indefinite integral) of a function is a function whose derivative is the original function. Here, we want to find a function F(x) such that F'(x) = csc x cot x.
Recognize the derivative of a common trigonometric function: the derivative of csc x is -csc x cot x. This is a key insight because it relates directly to the integrand.
Since the derivative of csc x is -csc x cot x, the antiderivative of csc x cot x will be the negative of csc x plus a constant of integration C.
Write the antiderivative as an integral: \(\int csc\ x\ cot\ x\ d x = -csc\ x + C\).
To verify, differentiate your result \(-csc\ x + C\) and check that it equals the original function \(csc\ x\ cot\ x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Antiderivatives (Indefinite Integrals)

An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often represented as indefinite integrals with a constant of integration. For example, if F'(x) = f(x), then F(x) is an antiderivative of f(x).
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Introduction to Indefinite Integrals

Trigonometric Functions and Identities

Understanding the properties and derivatives of trigonometric functions like csc(x) and cot(x) is essential. Knowing identities such as cot(x) = cos(x)/sin(x) helps simplify expressions and find antiderivatives. Familiarity with these functions' behavior aids in recognizing patterns during integration.
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Introduction to Trigonometric Functions

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original function. This step confirms the correctness of the solution. It reinforces the connection between differentiation and integration, ensuring the antiderivative found is accurate.
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Related Practice
Textbook Question

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 < ∞

Textbook Question

51. Frictionless cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is s = 10 cos πt.

a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then?

Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = sec²x − 2tan x, −π/2 < x < π/2

Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = (x − 2)²ᐟ³.


a. Does f′(2) exist?

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

(3/2)√x

Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = √3cos x + sin x, 0 ≤ x ≤ 2π