Question

Explain how to apply the First Derivative Test.

Short answer

Question: Explain the steps of the First Derivative Test to determine local extrema of a function. Answer: The First Derivative Test is used to determine local extrema of a function. The steps are as follows: 1. Find the function's first derivative, which represents its slope at different points. 2. Identify critical points by setting the first derivative equal to zero or finding where it is undefined. 3. Choose test intervals by dividing the function's domain based on the critical points. 4. Test the sign of the first derivative in each interval using a test point to determine if the function is increasing or decreasing in that interval. 5. Apply the First Derivative Test to classify critical points as local maximum, local minimum, or neither, based on the sign of the first derivative.

Step-by-step solution

Find the function's first derivative

First, find the derivative of the given function with respect to its variable. The first derivative represents the function's slope at different points on its graph. We can use it to find out when the slope is increasing or decreasing, which corresponds to the function's local extrema.

Identify critical points

To find the critical points of the function, we need to set the first derivative equal to zero or find where the derivative is undefined. These points are where the function may have a local maximum, local minimum, or a point of inflection.

Choose test intervals

After finding the critical points, divide the domain of the function into various intervals based on the critical points. Each interval contains one critical point. We will examine each interval separately to determine the local behavior of the function around its critical points.

Test the sign of the first derivative in each interval

Within each interval, choose a test point, and compute the value of the function's first derivative at that point. If the first derivative is positive at the test point, the function is increasing on that interval. If the first derivative is negative at the test point, the function is decreasing on that interval. By finding the signs of the first derivative in each interval, we can determine the local behavior of the function around the critical points.

Apply the First Derivative Test

Now, apply the First Derivative Test to classify the critical points as local maximum, local minimum, or neither. If the first derivative changes from positive to negative, the function has a local maximum at the critical point. If the first derivative changes from negative to positive, the function has a local minimum at the critical point. If the first derivative does not change signs, the critical point is neither a local maximum nor a local minimum.

Key concepts

Critical Points

In exploring the landscape of mathematics, critical points act as signposts to intriguing features of a function's graph. Discovering these points is akin to uncovering hidden treasures in the realm of calculus. Mathematically speaking, critical points occur where the first derivative of a function is either zero or undefined. Visualize a mountain trail; these points are like flat or steep areas where you stop to ponder the path ahead.

To unearth these points, set the function's first derivative equal to zero and solve for the variable. Remember, not all solutions may be physical points on the graph, especially if the derivative is undefined there. It's like checking a map for possible rest stops, but some may not exist in reality. These critical points are the candidates for the highest highs and the lowest lows of a function – the peaks and valleys of our mathematical adventure.

Function's Local Extrema

Summit or Trough: Unveiling a Function's Peaks and Valleys

Local extrema represent the highs and lows in the terrain of a function's graph. Think of these as the summits and troughs of a rolling landscape, points where the function's value doesn't merely stop increasing or decreasing, but does a complete turnaround. We label these as local maximums and minimums – the respective peaks and dips on our journey through calculus.

To spot these extrema, we investigate the critical points. A local maximum is where the function takes on its highest value in a neighborhood, often where the derivative swings from positive to negative. Conversely, a local minimum rests where the function is at its lowest, frequently where the derivative transitions from negative to positive. This delicate dance of dips and peaks forms the crux of understanding many natural phenomena described by mathematical functions.

Test Intervals

Dividing the Journey: Setting the Stage for Analysis

Imagine segmenting a lengthy road trip into manageable stretches – this is the practical idea of test intervals in calculus. Once critical points are pinpointed, they serve as boundaries that carve the domain of the function into test intervals. Within these intervals, the function's behavior is scrutinized, much like observing the changing landscapes between cities on a road map.

We endeavor to explore each interval independently. By considering sample points – not unlike rest stops within these stretches – we can assess the function's behavior in these sections. Doing so equips us with a clearer vision of the function's overall journey, preparing us for understanding its local extrema and other intriguing features that lie between the critical points.

Sign of the First Derivative

The Telltale Slope: Deciphering the Function's Ascent and Descent

In the realm of calculus, the sign of the first derivative is the guiding compass that indicates whether a function is climbing upward or descending downward. A positive sign suggests that the function is ascending within an interval, similar to an uphill trek where the altitude increases. Conversely, a negative sign signals a decline in the function's graph, much like a downhill path where elevation drops.

Detecting the sign of the first derivative at strategically chosen test points within intervals prevents us from getting lost in the calculations. This vital signpost enables us to anticipate whether we're approaching a peak (local maximum) or a valley (local minimum), adding a crucial layer of insight to our mathematical journey through the graph's terrain.