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

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

Key Concepts

Domain of a Function

The domain is the complete set of input values for which the function is defined. In the context of function analysis, it is important to establish the domain before considering other features, as it constrains where the function and its derivatives exist.

Critical Points

Critical points are values in the domain at which the first derivative of the function is zero or undefined. These points serve as candidates for local extrema (maximums or minimums) and are crucial in determining the overall behavior of the function.

First Derivative and Increasing/Decreasing Intervals

The first derivative of a function indicates its instantaneous rate of change. By examining the sign of the first derivative, one can determine on which intervals the function is increasing (positive derivative) or decreasing (negative derivative), thereby identifying local extrema through the first derivative test.

Second Derivative and Concavity

The second derivative provides information about the curvature of the function, revealing its concavity. A positive second derivative indicates the function is concave up (shaped like a cup), while a negative second derivative indicates it is concave down (shaped like a cap).

Inflection Points

Inflection points are specific points on the graph where the concavity of the function changes from up to down or vice versa. These are found by detecting where the second derivative changes sign, and they are key to understanding the overall shape of the graph.

Graphing Functions Using Derivative Analysis

Graphing functions in calculus involves integrating the insights gained from analyzing the function's domain, critical points, intervals of increase or decrease, and concavity. This multi-step approach helps in constructing an accurate sketch of the function’s behavior and identifying important features like extrema and inflection points.

Transcript

00:01 Hey guys, in this problem, we're going to be graphing and finding the critical points of this function here.

00:05 G of x equals 2 times the quantity 1 minus e to the negative x.

00:09 So first off, let's go ahead and get some points here.

00:12 We can start off with the point negative 1.

00:15 So let's go ahead and plug that in.

00:16 And by the way, this function, we're going to be only plotting it for x values greater than are equal to 0.

00:23 So to make sure that i don't cross this y -axis here and go into the negative x values, i'm going to put this.

00:30 Little asymptote here, although it's not quite an asymptote as we can cross it, it can be equal to zero.

00:38 So we can sit on the line, but we can't go past it.

00:42 So anyway, plugging in the value x equals negative 1, we have 2 times 1 minus e to the 1.

00:50 We can plug that into our calculator.

00:52 Let's see what i get.

00:54 1 minus e to the times 2.

01:01 So that i got about negative 3 .44.

01:07 Okay, we can plug in the value.

01:09 And by the way, we're not going to be plotting this point here.

01:14 Anything in the next axis, not plotting it, just kind of like testing the value and kind of getting a feel for how the function is going to behave, whether it's going to go up like this or down like this.

01:26 So not actually plotting this value, just kind of seeing what the function does.

01:30 So we have negative 3.

01:32 It's good to know.

01:34 Let's plug in the value x equals 0.

01:37 At x equals 0, this term becomes 1 here.

01:40 1 minus 1 is 0.

01:41 So this is 0.

01:43 So right here.

01:48 All right.

01:49 And at x equals 1, 2 times a quantity 1 minus e to the negative 1 1 1.

02:03 I got 1 .26.

02:13 At 1, we have 1 .26 right here.