62–65. {Use of Tech} Graphing f and f'
a. Graph f...

Question

62–65. {Use of Tech} Graphing f and f'

a. Graph f with a graphing utility.

f(x)=e^−x tan^−1 x on [0,∞)

Accepted Answer

Hello, in this video we are going to be sketching the graph of the given function. The function given to us is F of X is equal to 2 multiplied by Ease the power of negative X multiplied by tangent inverse of 2 X, and we want to plot the graph on the interval from 0 to infinity. So, how can we start this type of process? Well, first, we need to understand the behaviors of the graph. In order to find the behaviors of the graph, let's go ahead and make a number table, X and F of X. We are going to plug in values of X into our function with the help of a calculator, and we are going to calculate the outputs of the function. Let's go ahead and take a few testing points of X. Now, since the interval is stating that we only want positive values from 0 to infinity, some testing points that we can plug into the function will be 01234, and 5. We will go ahead and plug in these values into our given function and view the behaviors of the outputs. So if we plug in zero into the function, we get the output 0. If we plug in 1, we get 0.81. If we plug in 2, we get 0.36. If we plug in 3, we get 0.14. Plugging in 4 will give us 0.05. And plugging in 5 will give us 0.02. So, what does this behavior tell us? Well, first, the 0.00 tells us that the starting point of the graph will be at the origin, 0.0. Now notice that from 0 to 1, the value increases from 0 to 0.81. But after the value of 0.81, the value of the graph graph slowly decreases over time. What this means is that the interval from 0 to 1 indicates that there is a critical point. For the derivative of the graph, this is where the slope is equal to zero, and this means that this implies a local maximum to the graph. Furthermore, notice that the graph slowly decreases on the interval from 1 to 5. This implies that as the value of X approaches positive infinity, the value of the graph is going to approach 0 over time. So, what we need to do now is we need to go ahead and approximate the value of the critical point. In order to do this, we will go ahead and create a 2nd number line. Now, what we want to do is you want to go ahead and approximate values of the critical point. We will go ahead and take values of X, such as 0.4, 0.5, 0.6, and 0.7. We will go ahead and plug this into the function and determine the behaviors of the functions and where the critical point might be located at. So, if we take 0.4 and plug this into the function, we get the value 0.9046. If we plug in 0.5, we get 0.9527. Plugging in 0.6 will give us 0.9616, and plugging in 7 will give us the value of 0.9441. Notice that as we increase towards the value of 0.6, the outputs of the graph are slowly increasing over time. And notice that once we plug in the value 0.7, the value of the graph drops from 0.9616 to 0.9441. This means that the approximate location of the critical point is going to be at 0.6, 0.9616. So now that we've estimated the location of the critical point and the behaviors of the graph, let's go ahead and plot the values of the graph. So, we will go ahead and create a number line. Now, this number line is going to start at the value of 0 and go off to positive infinity. Now looking back at the original table, we know we determined that the graph is going to start at the origin. So let's go ahead and label that point. Next, we approximated the location of the critical point. The critical point will be approximately located at the value of X is equal to 0.6, and based off the first table, we will be increasing to this local maximum. So, we can estimate that value to be roughly around here at 0.6, 0.9616. So the graph will be increasing over time. Now, based off the values that we observed from 1 to 5, this graph is going to slowly decrease to 0 as it approaches positive infinity. So the rest of the behaviors are slowly going to look like this. And this is going to be a rough estimation of the graph of F of X. So I hope this video helps you in understanding on how to graph this type of function, and I will go ahead and see you all in the next video.

62–65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x)=e^−x tan^−1 x on [0,∞)

Key Concepts

Exponential Functions

Inverse Tangent Function

Graphing Utilities

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