Use the guidelines of this section to make a complete graph of...

Question

Use the guidelines of this section to make a complete graph of f.

f(x) = tan⁻¹ (x²/√3)

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to graph the function F of X, which is equal to the inverse tangent of the quantity of X cube divided by the square of 2 in quantity. So the problem we're given an empty graph on which to plot our function. Now, in order to graph our function F of X, we need to determine several properties of this function. So the first thing we're going to look at is the domain of our function F of X. And so here we have F X equal to the inverse tangent of the quantity of X cube divided by the square root of 2 in quantity. So recall that the domain of the inverse tangent function is all real numbers. And the same is true for X cubed. Its domain is all real numbers. So therefore, our function F of X is defined with all real numbers, so its domain is the open interval from minus infinity to infinity. Next, we'll check for symmetries. So here we're gonna look for even or odd symmetries, so we'll calculate the quantity of F of minus X. Which is going to be equal to the inverse tangent. Of the quantity of minus X in quantity cubed, divided by the square root of 2 in quantity. Which is equal to the inverse tangent. Of the quantity of minus X cubed divided by the square root of 2. Which is equal to minus the inverse tangent of the quantity of X cubed divided by the square root of 2. Which is equal to minus FX. So since F minus X is equal to minus F of X, we have an odd symmetry. So that means our function here is symmetric about the origin. Now, the next quantity that I want to look at are the critical points. For this, we'll need to calculate our first derivative. We'll calculate F of X, which is going to be equal to the derivative with respect to X. Of the inverse tangent of the quantity of X cubed divided by the square root of 2 in quantity. So this is going to be equal to, and here we call that when you take the derivative of the inverse tangent function, you get one. Divided by the quantity of one. Plus, our argument squared, so we'll have X cubed, divided by the square root of 2, and that whole quantity is squared. And here we have to apply our chain rules, so we need to multiply this by the derivative of our argument with respect to X. We'll have the derivative with respect to X of the quantity of X cube divided by the square root 2 in quantity. So you want to take that derivative. I get 3 multiplied by X2, divided by the square root of 2. Now I just need to simplify this expression. So doing so, this is going to be equal to. 3 X2, multiplied by the square root of 2. Divided by the quantity of 2 + X 6. So now to find the critical points, I need to find the X values which our first derivative here is either undefined or equal to 0. So, if I look at the denominator of our fraction here, we see that our denominator here is always a positive quantity. That means it can never be equal to 0. So therefore, we don't have any critical points due to our first derivative not being defined. So, the other condition is to find the points. Where our first derivative is equal to 0, so that means the numerator must be equal to 0. So we'll set. 3 X squad, multiplied by the square of 2 equal to 0, and solved for X. In doing so, we'll see that X is equal to 0. So this is our only critical point. Now we can look to see over what intervals our function is increasing or decreasing. So, if we look at our first derivative here, we see that our denominator is always a positive quantity, and our numerator is either 0 or a positive quantity. So that means that our first derivative here is always going to be greater than or equal to 0. So, since we have a positive first derivative here, that means that our function F of X is always increasing. That will help us when we begin to graph our function. Now, the next quantities that I need to look for are the inflection points, as well as our intervals of concavity. For this, we'll need our second derivatives, so we'll calculate F of X. And that's going to be equal to the derivative with respect to X. Of the quantity of 3 X 2 multiplied by the square root of 2. Divided by the quantity of 2 + X 6. In quantity. So, in order to take this derivative, we need to apply our quotient rule. So this is going to be equal to, and here we call your quotient rule that we'll have our denominator, which is the quantity of 2 plus X 6. Multiplied by the derivative of the numerator, so we'll have the derivative with respect to X of the quantity of 3x2 multiplied by the square root 2. So I want to take that derivative, I will get the 6 X multiplied by the square root of 2. Then we'll have minus our numerator, which is 3x2 multiplied by the square root of 2. Multiplied by the derivative of our denominator. So when we take the derivative with respect to X of the quantity of 2 + X to the 6th, we get 6 multiplied by X to the 5th. And all of this is divided by our denominator squares, so we'll have the quantity of 2 + X 6. In quantity squared. So multiplying out our numerator, this is going to be equal to. 12 X multiplied by the square root of 2. Plus, 6 X to the 7th multiplied by the square root of 2, minus 18, multiplied by X to the 7th, multiplied by the square root of 2, and that whole quantity is divided by. The quantity of 2 + X 6 in quantity squared. So now we just need to collect like terms, so this is going to be equal to. 12 X multiplied by the square root of 2, minus. 12 X the 7th multiplied by the square root of 2. And all of that is divided by The quantity Of 2, plus X 6. In quantity squared. So, in the numerator we can factor out a 12 x multiplied by square 2, this is going to be equal to. 12 X multiplied by the square root of 2. Multiplying the quantity of 1 minus X 6 in quantity, that is divided by the quantity of 2 plus x 6 in quantity squared. So, in order to determine possible inflection points, we need to set our second derivative here equal to 0. Again, the denominator in our fraction here can never be equal to 0, since it's always a positive quantity. So, what we're going to do is just set our numerator equal to 0. So we will have 12 X multiplied by the square root of 2. Multiplying the quantity of 1 minus X 6 in quantity is equal to 0. So, what we're going to do is set each one of these factors here equal to 0 and solve for X. So for the first factor, we'll have 12 X, multiplied by the square 2 is equal to 0, that means that X is equal to 0. And for the second factor, we'll have 1 minus X to the 6th is equal to 0, and here, so for that X is going to be equal to plus or minus 1. So we have 3 possible inflection points. So what we're going to do is check. To see if our concavity changes at these three points. So we're going to make a small little table here. With two rows, in the first row, we'll have the sign of our second derivative F of X, and in the second row, we will have the behavior of F of X, whether it's concave up or concave down. So, the first interval that I'm going to look at is going to be the interval going from minus infinity to X equal to -1. And if I look at our expression for F of X, When X is less than 1, that means Oh sorry, when X is less than -1, that means X is a negative value, but the quantity of 1 minus X to the 6 is going to also be a negative quantity. So, I'll have a negative multiplied by a negative, which gives me an overall positive quantity. And our denominator here is always going to be positive. And so since our second derivative is positive over this interval, that means that our function F of X is concave up. Now the next interval that we want to look at is between X equal to -1 and X equal to 0. And so again, if we look at the numerator in our expression for F of X. We see that X here in this interval is going to be negative, but the quantity of 1 minus X to the 6th is going to be a positive quantity. So we'll have a negative value multiplied by a positive value, which will give us an overall negative value. And so that means that our function F of X is going to be concave down on this enable, since our second derivative is negative. The next interval that we'll look at is between 0 and 1. And here, if we look at our second derivative, X is going to be a positive value, but N1 minus X 6 is also going to be a positive value. So, our second derivative is going to be positive over this interval, so that means our function has to be concave up on this interval. And then the final interval that we'll look at is for X greater than 1, so between 1 and infinity. And so here in this case, X is going to be a positive quantity, but 1 minus X 6 is going to be a negative quantity. So I'll have a positive multiplied by a negative value, and so that'll give us an overall negative sign. And that means that our function F of X is concave down. Now, the final thing that we need to look for are asymptotes. And so recall that the limit as you approaches infinity of the inverse tangent of you. is equal to pi divided by 2, and that the limit. As you approaches minus infinity of the inverse tangent of U is equal to minus pi divided by 2. And so here, in both cases, when X approaches infinity, our argument of the inverse tangent function also approaches infinity. So that means F of X is going to approach pi divided by 2 as X approaches infinity, and it's going to approach minus pi divided by 2 as X approaches minus infinity. So that means we'll have two horizontal asptotes at Y equal to pi divided by 2 and Y equal to minus divide and Y is equal to minus pi divided by 2. Sorry. And so now we'll have enough information to begin plotting our function. So the first thing we're going to do is place our Horizontal asymptotes. So pi divided by 2 is approximately 1.57. So we'll draw a line, a horizontal dash line, indicate that this is a horizontal asptote at that value. And we'll do one also at Y equal to minus 1.5. 7 And so, drawing those two lines will give us our two horizontal asymptotes. So now we need to plots and points in order to graph our function. So the first one is going to be our critical point, which also happens to be an inflection point. Which is X equal to 0. So F of zero turns out to be equal to 0, so I'll plot to 0 00. The next point that we're going to look at is X equal to -1, since I have an inflection point at that location. And so F-1 is equal to -0.62. We'll plot that point on the graph, and by symmetry, that means X equals 1 is equal when X is equal 1 F 1 is equal to 0.62, so we'll plot that point as well as our other inflection point. So now we just need to use The properties for the intervals of increasing and decreasing as well as our concavity. So, we call that our function is always increasing. However, going from minus infinity to X equal to -1, our function is going to be concave up. And then we have an inflection point at X equal to -1. Then over the next interval, our functions is still increasing, but it's concave down. We then have another inflection point at X equal to 0, and then between 0 and X equal to 1, we have our function being concave up and still increasing. And then our function is increasing still approaching our other asymptotic, uh horizontal asymptote. However, our function now is concave down. And so the graph that we have drawn on the screen here is the answer to this question. So I hope that this explanation has been useful, and I'll see you in the next video.

Use the guidelines of this section to make a complete graph of f.
f(x) = tan⁻¹ (x²/√3)

Key Concepts

Inverse Functions

Graphing Techniques

Transformations of Functions

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