Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.3.2

Chapter 4, Problem 4.3.2

Explain how to apply the First Derivative Test.

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Identify the critical points of the function. These are the points where the derivative is zero or undefined. To find them, take the derivative of the function and solve for when the derivative equals zero or does not exist.

Determine the intervals around each critical point. This involves dividing the number line into intervals where each critical point is a boundary.

Choose a test point from each interval. A test point is any value within the interval that is not a critical point.

Evaluate the derivative at each test point. This will tell you whether the function is increasing or decreasing in that interval. If the derivative is positive, the function is increasing; if negative, the function is decreasing.

Apply the First Derivative Test: If the function changes from increasing to decreasing at a critical point, that point is a local maximum. If it changes from decreasing to increasing, that point is a local minimum. If there is no change, the critical point is neither a maximum nor a minimum.

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Key Concepts

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

First Derivative Test

The First Derivative Test is a method used to determine the local maxima and minima of a function. It involves analyzing the sign of the first derivative of the function at critical points, which are points where the derivative is zero or undefined. If the derivative changes from positive to negative at a critical point, it indicates a local maximum, while a change from negative to positive indicates a local minimum.

Critical Points

Critical points are values of the independent variable where the derivative of a function is either zero or undefined. These points are essential for applying the First Derivative Test, as they are potential locations for local extrema. To find critical points, one must first compute the derivative of the function and then solve for where this derivative equals zero or is undefined.

Sign Chart

A sign chart is a visual tool used to analyze the behavior of a function's derivative across its domain. By plotting the critical points and testing intervals between them, one can determine where the derivative is positive or negative. This information helps in identifying the intervals of increase and decrease of the function, which is crucial for applying the First Derivative Test effectively.

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