{Use of Tech} Equations of tangent lines
b. Use a graphi...

Question

{Use of Tech} Equations of tangent lines

b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = e^x; a = ln 3

Accepted Answer

Hello. In this video, we are going to be plotting the given curve and the tangent line. The problem says that for the given equation, we want to plot the curve and its tangent line on the same rectangular coordinate system. The curve given to us is Y is equal to 2 races to the power of X, and we want to plot its tangent line passing through the value A is equal to logbase 2 of 4. So, how do we start this problem? Well, since we have the equation to the curve, what we first need to do is find the equation to the tangent line. Now, in order to obtain the equation of the tangent line, there are 2 pieces of information we need. First, we need a point of interest that the tangent line will pass through, and we need the slope to the tangent line. Let's first start by finding the point of interest to the tangent line. Now, the value that the tangent line is going to be located at, is given to us as A is equal to logba 2 of 4. This is going to act as the X value to our coordinate points. So we can rewrite this A value as X is equal to logbase 2 of 4. Now what we can do is we can simplify this value. We can rewrite logbase 2 of 4 as logbase 2 of 2 squared. And by using the properties of the logarithm, we can further simplify the logarithm to be 2 multiplied by log-base 2 of 2. This will simplify to be 2 multiplied by 1, which will give us 2. So this tells us that the X value to our coordinate point is going to be positive too. But now we need to obtain the respective Y value. In order to obtain this value, we can take our X coordinate and plug it in to the given curve equation. That will give us Y is equal to 2 raise to the power of 2, which will simplify to 4. What this means is that our tangent line is going to pass through the point 2.4. We have the coordinate that the tangent line will pass through, but now we need to solve for the slope of the tangent line. In order to obtain the slope of the tangent line, what we can do is we can take the derivative of the given curve. So we will take the derivative of the curve with respect to X. Now let's go ahead and take the derivative of both sides of the equation. On the left hand side, the derivative of Y will just be 1, but since we are taking the derivative of Y with respect to X, we will generate a D Y D X term through implicit differentiation. So the derivative of Y on the left hand side will be D Y D X. Now, on the right-hand side, we are taking the derivative of an exponential value. In order to take this derivative, we will first take the original argument, which is to raise the power of X, then multiply this by the natural logarithm of the base, which is going to be the natural logarithm of 2. Then we will multiply this value by the derivative of the exponential function, which will be X. Let's go ahead and simplify. The derivative of X with respect to X will just be 1, and that means that our derivative DYDX will simplify to be 2 X multiplied by the natural logarithm of 2. Now that we have the derivative, what we want to do is we want to plug in the value of X in order to obtain the slope. Now, when we do this, we are going to plug in the value of 2 to the derivative, that is going to give us the slope D Y D X equal to 2 races of the power of 2 multiplied by the natural logarithm of 2. This will further simplify to give us 4 multiplied by the natural logarithm of 2, and this is going to be the slope to the tangent line that we are interested in. So now that we have the coordinate point to the tangent line 2.4, and we have its slope, we can go ahead and use the point slope form to solve for the equation of the tangent line. The point slope form is Y minus Y1 equal to M multiplied by X minus X1, and this is where the X1 and Y1 values will come from the coordinate point. Plugging in our values will give us Y minus 4 equal to 4 multiplied by the natural logarithm of 2, multiplied by the quantity X minus 2. Now let's go ahead and simplify the tangent line. We will distribute the value 4 multiplied by the natural log of 2, into the quantity X minus 2. This will leave us with Y minus 4 equal to 4 X multiplied by the natural logarithm of 2, minus 8 multiplied by the natural logarithm of 2. Finally, we will add 4 to both sides of the equation, and this will leave us with Y is equal to 4X multiplied by the natural logarithm of 2, minus 8 multiplied by the natural logarithm of 2 plus 4. This is going to be our equation to the tangent line that we are going to plot. Now what we want to do is we want to use this equation and the equation of the original curve, and plot them on the same coordinate axis. So, before we continue, let's go ahead and create an XY plot for both of our graphs. Let's start by finding some points for a given curve. Y is equal to 2 races to the power of X. We will go ahead and pick some XY values. For X, let's go ahead and plug in the values -2, -1, 01, and 2. And now what we want to do is we want to plug in these individual values into our given curve. Y is equal to 2 races of the power of X. When we plug in the value of -2, we will get the value 1/4. Plugging in the value of -1 will give us 1/2, plugging in 0 will give us 1, plugging in 1 will give us 2, and plugging in 2 will give us 4. Notice that no matter what value we plug in to two rates to the power of X, we will always get a positive value for Y. What this means is that the curve Y is equal to 2 power of X will always be positive and not intercept the X axis. Now, what about 2 points for the equation of the tangent line? Well, if we pick the value, X is equal to 0, that will leave us with -8, multiplied by the natural logarithm of 2 + 4. When we plug in this, when we plug this into a calculator, this will give us an approximate value of negative 1.545. Now let's go ahead and pick an X value that lies on the curve of Y is equal to 2 power of X. Let's go ahead and pick the value positive 2. If we plug in the value of 2 into our tangent line equation, that will give us 8 multiplied by the natural logarithm of 2, minus 8 multiplied by the natural logarithm of 2 + 4, and this will simplify to give us 4. What this means is that there is a point 2.4 on the tangent line that shares a common point between the tangent line and the given curve. Now let's go ahead and roughly plot these points. Now, again, I am not the best artist, but let's go ahead and plot the points for Y is equal to the power of X. Now, for Y is equal to the power of X, we have the point -2.14, which will be roughly above the X axis. Then we have -1.2, we have a Y intercept at 0.1. We have the 0.1.2. And we have the point 2.4. So what we have is a positive exponential curve that passes through the point 0.1. This is going to be the curve for Y is equal to 2 races the power of X. Now what about the tangent line? Now, we know that the tangent line and the given curve have the common point at 2-4. We also know that the tangent line passes to the point 0.1.5 approximately. So we know that we have a Y intercept roughly about here, between -1 and -2. And since this is the equation to a tangent line, this is just going to be the equation to the shape of a line. And this will be both of the curves, our tangent line and our giving curve plotted on the same coordinate axis. So I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

{Use of Tech} Equations of tangent lines
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = e^x; a = ln 3

Key Concepts

Tangent Line

Graphing Utilities

Exponential Functions

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