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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 4.4.80
Chapter 4, Problem 4.4.80

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
80. y' = 1 - cot²θ, for 0 < θ < π

Verified step by step guidance
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First, find the second derivative y'' by differentiating y' = 1 - cot²θ with respect to θ. Use the chain rule and the derivative of cotangent: d(cotθ)/dθ = -csc²θ.
Differentiate y' = 1 - cot²θ: y'' = d/dθ [1 - cot²θ] = -2cotθ * (-csc²θ) = 2cotθ * csc²θ.
Simplify the expression for y'': y'' = 2cotθ * csc²θ. This is the second derivative of the function.
To sketch the graph of f, analyze the critical points and concavity. Critical points occur where y' = 0 or is undefined. Since y' = 1 - cot²θ, set 1 - cot²θ = 0 to find critical points.
Determine the intervals of concavity using y''. If y'' > 0, the graph is concave up; if y'' < 0, the graph is concave down. Analyze the sign of y'' = 2cotθ * csc²θ over the interval 0 < θ < π.

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Key Concepts

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

First Derivative

The first derivative of a function, denoted as f'(x), represents the rate of change of the function with respect to its variable. It provides information about the slope of the tangent line to the graph of the function at any given point. Understanding the first derivative is crucial for determining critical points, where the function may have local maxima or minima.
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The First Derivative Test: Finding Local Extrema

Second Derivative

The second derivative, denoted as f''(x), is the derivative of the first derivative. It indicates the rate of change of the slope of the function, providing insights into the concavity of the graph. A positive second derivative suggests the graph is concave up, while a negative second derivative indicates concave down, which is essential for sketching the function's general shape.
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The Second Derivative Test: Finding Local Extrema

Graphing Procedure

The graphing procedure involves several steps to analyze the behavior of a function based on its derivatives. Steps typically include identifying critical points, determining intervals of increase or decrease, analyzing concavity using the second derivative, and sketching the graph based on this information. This systematic approach helps in visualizing the function's overall shape and key features.
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Graphing The Derivative
Related Practice
Textbook Question

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dv/dt = (1/2)sec t tan t, v(0) = 1

Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x³ / (3x² + 1)

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Textbook Question

Theory and Examples


Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

Textbook Question

30. Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

Textbook Question

Theory and Examples

67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then

[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd ≥ 16