Graph each function. Then determine any critical values, inflection points, intervals over which

Graph each function. Then determine any critical values,...

Key Concepts

Graph Analysis Using Derivatives

Graph analysis utilizes the first and second derivatives to provide comprehensive insights into the behavior of a function. The first derivative reveals where the function is increasing or decreasing, while the second derivative informs about the concavity and potential inflection points. Together, these analyses enable a detailed sketch of the function, highlighting critical features such as local extrema, inflection points, and intervals of monotonicity and concavity.

Exponential Functions

Exponential functions are mathematical functions in which the variable appears in the exponent. They generally have the form f(x) = a^x, where the base a is a constant, and they exhibit rapid growth or decay depending on whether a is greater than or less than one. These functions have unique properties, such as a constant relative rate of change, and are fundamental in modeling processes with exponential growth or decay.

Function Transformations

Function transformations involve applying operations such as scaling, translating, or reflecting to a basic function in order to obtain a new function. In the context of exponential functions, transformations can change the base graph's shape, orientation, or position. Common transformations include vertical and horizontal shifts, reflections across an axis, and stretching or compressing, all of which affect how the function is graphed.

Critical Points and the First Derivative Test

Critical points of a function are values in the domain where the first derivative is zero or undefined. These points are candidates for local extrema (maximum or minimum points). The first derivative test is a tool used to determine whether a critical point is a maximum, minimum, or neither by examining the sign of the derivative on either side of the point, thereby giving insight into the function's increasing or decreasing behavior.

Inflection Points and Concavity (Second Derivative Test)

Inflection points occur where a function changes its concavity, that is, from concave up to concave down or vice versa. The concavity of a function is determined by its second derivative: if the second derivative is positive, the function is concave up; if it is negative, the function is concave down. The second derivative test helps identify these points, providing additional understanding of the function's curvature and behavior over its domain.

Transcript

00:01 Hey guys, in this problem we're going to be graphing the function g of x equals negative e to the one half x power and also finding the critical data points.

00:09 So first off let's go ahead and get some data points at x equals negative 1 the function will be negative e to the negative one half power.

00:17 Let's put that in our calculator it's going to be negative 0 .607 at x equals 0 this whole term will become 1 negative so negative 1 at x equals 1 we have one half times 1 that's negative e to the negative e to the 1 half power and that should be negative 1 .65 at x equals 2 this should become 1 negative e to the first power that should just be negative 2 .7 2 so let's go ahead and graph at x equals negative 1.

01:21 We had negative 6 .607.

01:26 Then we had negative 1, negative 1 .65, and let's see 1, 2, closer to 3.

01:42 So the graph should look something like this.

01:51 So next the question asks, determine the critical points or the critical values.

01:57 So to do this, we have to take the first derivative.

02:06 So to differentiate this, we need to differentiate the power.

02:10 The derivative of 1 .5x is 1⁄2 times negative.

02:14 So negative 1 .5 .e to the 1 .5x power.

02:21 Next, we can set this equal to 0.

02:24 So let's ask yourselves, let's set the constant for a second and look at this part here.

02:30 When is this equal to 0? well, let's go ahead and look at this.

02:38 Now i know that this is a negative, so this is the negative counterpart to this, but let's just look at this for a second.