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

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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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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.

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.

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.

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