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

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

Key Concepts

Graphing Functions

Graphing functions involves plotting the curve of a function over its domain based on key characteristics such as intercepts, asymptotic behavior, and curvature. This process often starts by identifying the function’s domain, range, and special points derived from calculus, enabling a visual understanding of the function’s behavior and trends.

Critical Values

Critical values are points in the domain of a function at which the first derivative is either zero or undefined. These points are significant because they are potential locations for local maxima or minima, and their identification is a crucial step in analyzing the overall behavior of the function.

Inflection Points

Inflection points are points on a curve where the concavity changes from upward to downward or vice versa. Determining inflection points involves evaluating the second derivative of the function and identifying where it changes sign, indicating a shift in the curvature of the graph.

Intervals of Increase and Decrease

Intervals of increase or decrease of a function are determined by the sign of its first derivative. When the derivative is positive, the function is increasing, and when it is negative, the function is decreasing. Analyzing these intervals helps in understanding the function's overall trend and the locations of its extrema.

Concavity

Concavity describes the direction in which a function curves. A function is concave upward (convex) if its second derivative is positive, meaning the graph opens upward, and concave downward if the second derivative is negative. This concept is important for identifying the nature of inflection points and the overall shape of the function's graph.

Exponential Functions

Exponential functions, characterized by a constant base raised to a variable exponent or vice versa, exhibit unique behaviors such as rapid growth or decay. Their derivatives and integrals maintain the exponential form, which simplifies analysis and makes them common in modeling real-world processes.

Transcript

00:01 Hey guys, how's it going? in this problem, we're going to be graphing and finding the critical values of the function f of x equals 3 minus e to the negative x for all x values greater than are equal to 0.

00:11 So before i start graphing, let me quickly go ahead and make this asymptote here to kind of remind me not to go past this point.

00:22 I can sit on the line here, but i can't go past it.

00:28 So at x equals negative 1, you can plug that into your calculator.

00:32 I got 0 .28.

00:36 At x equals 0, i got the point y equals 2.

00:43 At x equals 1, i got 2 .63.

00:48 And at x equals 2, i got 2 .86.

00:53 So i'm going to just graph kind of a phantom point here to help me figure out how exactly the function is behaving.

01:01 So at x equals negative 1, we had something kind of like right here.

01:08 At x equals 0, we had y equals 2.

01:12 At 1, we had 2 .63, right about here.

01:19 And at 2, we had 2 .86.

01:24 So the graph kind of is going to go up like this.

01:35 And approach infinity.

01:39 But the slope, as you can see, is kind of slowing down here.

01:42 The change in y was about 0 .6.

01:47 The change of y between these two points was about 0 .2 something.

01:55 So this will tend off to positive infinity.

01:59 So next let's go ahead and take the first derivative find the critical values.

02:04 So this becomes 0 negative e to the negative x times by negative 1, e to the negative x.

02:14 To find the critical values, we set this equal to 0.

02:18 When does this function become zero? well, excuse me, sorry, i was moving something.

02:36 So when does this function become zero? well, we can set up another graph very quickly to test the values.

02:46 If x equals negative 1, we have 2 .71, about something like that.

02:52 Zero, we have 1.

02:54 One, we have e to the negative 1.

02:58 I got about 0 .37.

03:11 So overall, this function will behave something like this.

03:23 So we ask yourselves, when does this function cross the x -axis? well, it never does, but it will approach the x -axis as it approaches infinity.

03:39 So therefore, this function will never be equal to zero.

03:42 So there are no critical points...

Please give Ace some feedback