{Use of Tech} Tangent line Find the equation of the line tange...

Question

{Use of Tech} Tangent line Find the equation of the line tangent to y=2^sin x at x=π/2. Graph the function and the tangent line.

Accepted Answer

Hello. In this video, we will go ahead and approach the given tangent line problem. So we are asked to calculate the slope of the tangent line to the graph of Y is equal to 3, to raise the power of cosine X at X is equal to 0. We also want to graph the function and the tangent line on the same coordinate system. So, how do we approach this problem? Well, first we are asked to calculate the slope of the tangent line to the graph of the given function. So, since we are asked to find the slope of a tangent line, let's go ahead and take the derivative of the given function. The function given to us is Y is equal to 3, raise the power of cosine of X. In order to take the derivative, we will have to use an exponential rule, since we have a function of X in the exponential value. The derivative will be calculated as the following. First, we will take the original function, which is 3, raise the power of cosine of X, then we will multiply this quantity by the natural logarithm of the base of the exponential value, which in this case is going to be 3, then multiply this by the derivative with respect to X of the exponential value, which is going to be cosine of X. So How do we take the derivative of cosine of X? Well, recall that the derivative of cosine is negative sine. So that is going to simplify to give us Y equal to 3 race the power of cosine of X multiplied by the natural logarithm of 3, multiplied by the derivative of cosine of X, which be negative sine of X. And that will simplify to give us a derivative Y equal to -3, raise the power of cosine of X, multiplied by the natural logarithm of 3, multiplied by sin of X. This is going to be the derivative to the original function given to us. Now that we have the derivative, we want to calculate the slope of the tangent line at the value X is equal to 0. So let's go ahead and take this X value and plug it into our derivative in order to get the value of the slope at this point. That is going to be -3, raise to the power of cosine of zero, multiplied by the natural logarithm of 3, multiplied by sine of zero. Now let's go ahead and simplify. Cosine of zero will simplify to be positive 1. But sign of zero will simplified to be 0. And anything multiplied by 0 is going to be 0. That means that the derivative, the value of the derivative at X is equal to 0, is just going to be zero, and that tells us that the value of the slope of the tangent line at the value X is equal to 0, is going to be zero. So that's one of the solutions we need for this problem. Now what we want to do is you want to graph the original function and the tangent line on the same coordinate system. So since we want to graph the tangent line, we are going to need to calculate the equation of the tangent line. We can do this by using the point slope form. We have the slope at the value we're interested in, but now what we need to do is we need to find a corresponding point. In order to do this, we can take our X value that we're interested in, which again, is X is equal to 0. And plug it into the original function. That will give us the corresponding Y value at X is equal to 0. So, that will give us Y is equal to 3, race to the power of cosine of zero. And in just like the previous step, cosine of zero will simplify to be 1, and 3 raise the power of 1 is just going to be 3. So that means that the corresponding point at X is equal to 0 is going to be 0.3. Now let's go ahead and use the slope we calculated, and the point to find the equation to the tangent line. We will use the point slope form, which is Y minus Y1 equal to the slope M multiplied by the quantity X minus X1. And this is where the point is going to be our corresponding X1 and Y1 values. Plugging in the information, this will give us Y minus 3 is equal to 0, multiplied by the quantity X minus 0. And notice that the right-hand side of this equation is going to simplify to be zero. This will leave us with Y minus 3 is equal to 0, and adding 3 to both sides of the equation will give us the equation of the tangent line Y is equal to 3. So, this is the equation of the tangent line, and now we want to take this equation and graph it with the original equation. 3, raise the power of cosine of X. So, let's go ahead and calculate some points to plug in for 3 cosine of X. Let's go ahead and create an XY table. And let's go ahead and plug in some values that are within the first rotation of the unit circle. These values are going to include 0, i divided by 2. Pie 3 pi divided by 2. And to pie. Now, if we plug in the values of 0 and 2 pi to cosine, cosine will simplify to be 1. That will give us the corresponding Y values 33 and 3. Next, if we plug in pi divided by 2 in a cosine, cosine will simplified to be 0, and 3 ranks the power of 0 will be 1. The same will apply to the value 3 pi divided by 2. And if we plug in the value of pi to the value of cosine, cosine at i will be -1, and 3 rates of the power of -1 will be 1/3. So let's go ahead and graph one rotation of this of this equation. That is going to be at the value 0. i i divided by 2. I'm sorry, punt. I got these values wrong. It goes from 0 to pi divided by 2. Pie, 3 pie divided by 2, and finally finishing at 2 pie. We will plug in the first point, which is going to be at 0.3. Pi divided by 21. At Pi, we have a height of 1/3. At 3 pi divided by 2, we have a height of 1. And finally at 2 pai, we have a height of 3. And if we connect the circles. This is going to be the rough graph of one rotation of 3 races of the power of cosine of X. Now, let's go ahead and graph the tangent line we calculated earlier, which is Y is equal to 3. That is just going to be the horizontal line at the Y value of 3. And that is going to look roughly like this. And this is going to be the graph of the original function with the tangent line. So I hope this video helps you in understanding how to approach this type of problem, and I will go ahead and see you all in the next video.

{Use of Tech} Tangent line Find the equation of the line tangent to y=2^sin x at x=π/2. Graph the function and the tangent line.

Key Concepts

Tangent Line

Derivative

Exponential Functions

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