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

Question

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

b. Compute and graph f'.

f(x)=e^−x tan^−1 x on [0,∞)

Accepted Answer

Hello. In this video, we are going to be graphing the derivative of the given function. The function given to us is F of X equal to 2 multiplied by E raises the power of negative X multiplied by tangent inverse of 2 X, and we want to plot the derivative on the interval from 0 to infinity. Now, in order to plot the derivative, the first thing we need to do is we need to calculate the derivative of the given function. In order to calculate the derivative, we are going to need to use the product rule. So In order to calculate derivative, we will first call the derivative F of X. Then we will multiply the coefficient 2 to the product rule between E to the power of negative X and tangent inverse of 2 X. So, the product rule states the following. We will go ahead and take the original first function, which is E, raise the power of negative X, and multiply this by the derivative of tangent inverse of 2 X. Now, we will go ahead and take this derivative by using chain rule. By the definition of the derivative of tangent inverse, that derivative will be 1 divided by 1 + 2 x squared. Then by the chain rule we will multiply this by the innermost function, which will be the derivative of 2 and 2 X, and that is going to be 2. Then we will proceed with the rest of the product rule. So we took the original first function, multiplied the derivative of the second function, then we will add the original second function, which is tangent inverse of 2 X, and multiply this by the derivative of the first function. Now, we will go ahead and use the exponential rule for the derivative of E to the power of negative X. The exponential rule tells us to first take the original first function, which is E to the negative X, multiply this by the natural logarithm of the base, which is going to be the natural logarithm of E, then multiply this by the derivative of the exponential value, and the derivative of negative X will be -1. And this is going to be the entire product rule. Now what we need to do is we need to simplify. So if we multiply the terms together, 1 divided by 1 + 2 X2 multiplied by 2 will simplify to be 2 divided by 1 + 44 X2. So we have E to the power of negative X multiplied by 2 divided by 1 + 4 X squad. Then we will add tangent inverse. Of 2 X multiplied by E raise the power of negative X. Then multiplied by the natural logarithm of E, and that is going to simplify to be 11 multiplied by 1 will give us -1, and if we multiply that to the tangent inverse, that will give us negative tangent inverse of 2 X. Now notice that each term has a multiple of e to the power of negative X. We can go ahead and simplify by factoring out E to the power of negative X, and that is going to give us 2 multiplied by E to the power of negative X multiplied by the quantity, 2 divided by 1 + 4 X squared minus tangent inverse of 2 X. This is going to be the function for the derivative of FXX. So, how do we start to graph this function? Well, in order to graph this function, the derivative, let's go ahead and just plot a few points of X. We will take some values of X and plug them into to F of X. So, let's try the values, 01234, and 5. We will go ahead and use the help of a calculator to evaluate the value of F of X at these independent values. By plugging in 0 into the derivative, we will have an output of 4. Plugging in 1 to the derivative will give us negative, 0.52. Plugging in 2 will give us -0.33. Plugging in 3 will give us negative 0.13, plugging in 4 will give us -0.05, and plugging in 5 will give us the value of negative 0.02. Now, what do these values mean? Well, first, notice that between the values of 0 and 1, we have a positive output that switches to a negative output. Then when we out input the values from 1 to 5, our negative value slowly decreases or increases over time to 0. So this implies two things about the graph of FX. First, this implies that there is a critical point. There is a critical point, approximately around the value of 1. And since we go from positive value to a negative value, and this negative value slowly approaches 0 over time, as the value of X approaches infinity, F of X will approach the value of 0. What we need to do now is we need to approximate the value of this critical point. So, in order to approximate the value, let's just go ahead and plug in some more values of X around the value of 1 into the function. So we will create a 2nd number table. And again, with the help of a calculator, we will go ahead and plug in the values of 1, 1.1, and 1.2. If we plug in one into the given function, we will get the approximate output of negative 0.5203. If we plug in 1.1, we get negative 0.5337. And if we plug in 1.2, we get negative 0.5302. Now, what do these three values slowly tell us? Well, what we want to do is you want to take the median value of these three, and that median value is going to exist at X is equal to 1.1. So this tells us that the critical point will approximately be located at 1.1.0.5337. So, now that we've approximated the critical point and the behaviors of the graph, let's go ahead and graph the derivative. So we will go ahead and create an axis. And this is going to be the graph of F of X. Now, going back to the first number table, when we plugged in the value of 0, we got the output of 4. What this means is that we are going to start on the Y axis at the value Y is equal to 4. So this is going to be the starting point of our graph. Next, we will go ahead and plot the approximate value of the critical point. That critical point will roughly be located here at X is equal to 1.1. And finally, we know that the behavior of the graph will go below the x axis and slowly increase over time to zero. So this graph, and again I apologize, I'm not the best artist, but this graph will start at 0.4, pass through the X value 1.1, and slowly increase to 0 over time as X approaches positive infinity. And this is going to be the graph of the derivative of the function. So I go, I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

62–65. {Use of Tech} Graphing f and f'
b. Compute and graph f'.
f(x)=e^−x tan^−1 x on [0,∞)

Key Concepts

Derivative

Graphing Functions

Behavior at Infinity

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