Tangent Lines

In Exercises 3...

Question

Tangent Lines

In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.

y = sin x, −3π/2 ≤ x ≤ 2π

x = −π, 0, 3π/2

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to sketch the graph of the curve Y is equal to 2 multiplied by sin of x, only the close interval from -2 to 2, and its tangent lines at X is equal to -3 pi divided by 20, and pi. Label the curve and its tangent lines with the equations. Below the problem we're given into graph in which to plot are curves. So, the first thing we need to do is create a graph of our curve, Y is equal to 2 multiplied by sin of X. And for this, we'll just make a table with some of our key points. So we'll have X in our first column, and we'll have the value of Y in the second column. And so the first point that we're going to want to look at is our end point, which is gonna be -2 pi. And to calculate why, we'll have 2 multiplied by the sign of minus 2 pi. And when we evaluate that expression, that's going to be equal to 0. Next point we'll look at is minus 3 pi divided by 2, and so we'll have for Y 2 multiplied by the sign of minus 3 pi divided by 2, and evaluating in this expression, that's going to be equal to 2. Next, we'll look at minus pi for X, and then for Y, we'll have 2 multiplied by the sign of minus pi. Which is equal to 0. Next value of X we'll look at is minus pi divided by 2. And so for Y we'll have 2 multiplied by sine of minus I divided by 2. And this is going to be equal to -2. The next X value is going to be 0, and for Y we'll have 2 multiplied by the sign of 0, which is equal to 0. The next X value is going to be pi divided by 2, and so we'll have 2 multiplied by the sign of pi divided by 2, which is going to be equal to 2. The next X value is going to be pi, and so, for Y we'll have 2 multiplied by the sign of pi, which is equal to 0. The next point is going to be when X is equal to 3 pi divided by 2, and so for Y, we'll have 2 multiplied by sine. Of 3 pi divided by 2. And evaluating the expression, that's gonna be equal to minus 2. And for the last point, we're gonna have X equal to 2, and so for Y, we will have 2 multiplied by the sign of 2 pi. Which is equal to 0. So now we'll just place each of these points on our graph. So we'll have one at X is equal to -2 pi. We'll have another one when X is equal to -3 pi divided by 2, so we'll have a Y value of 2. The next point is gonna be at X equal to minus pi and Y is going to be equal to 0. The next point is going to be when X is equal to minus pi divided by 2, and a y value is going to be -2. The next point is going to be at the origin. The next point is going to be at I divided by 2.2. The next point is going to be. At 0. Next point is going to be at, at 3 pi divided by 2. Come up -2, and then finally we'll have a point when X is equal to 2 pi and Y is going to be 0. So now all we need to do is just connect these points with a sine curve. So basically we're going to have our sine function but with an amplitude of 2. And then, after we have drawn this function, we're actually going to label our function. With its equation. We'll need to label this. With Y is equal to 2 multiplied by the sine of X. So now we need to determine the equations for our tangent lines at our 3 different points. So to do that, we first need to take a derivative of our function Y. So we'll need to calculate Y because we're going to need this to find the slope of our tangent lines. So Y is going to be equal to the derivative with respect to X. Of the quantity of 2 multiplied by sine of X. And when we take this derivative, this is going to be equal to 2 multiplied by cosine of X. We call that the derivative of sine of Xs with respect to X is cosine of X. So now we need to um evaluate this derivative at each of our points. So we'll look at the first point when X is equal to -3 pi divided by 2. We'll need to calculate Y of minus 3 pi divided by 2, and this is going to be equal to 2 multiplied by the cosine of minus 3 pi divided by 2. And when we evaluate this expression, this is going to be equal to 0. That means that the slope of our tangent line is equal to 0, so we actually have a horizontal line, and that horizontal line is going to have a y value that is equal to 2, hence Y of minus 3 pi divided by 2 was equal to 2. So we're going to draw that on our graph, and so we'll draw that horizontal dash line at Y equal to 2. And that we will label that. With the function, Y is equal to 2. Now, the next point that we need to look at is when X is equal to 0. We'll need to calculate Y. At 0. And that's going to be equal to 2 multiplied by cosine of 0, which is equal to 2. So we actually have a slope of 2 for a tangent line. And so we have our slope here as well as our point. We're going through the origin at 0.0. So that means that we can use our point slope form to determine the equation for this line. So recall for the point slope form, we'll have Y minus our Y coordinate, which in this case is 0, and that's could be equal to our slope. 2 multiplied with the quantity of X minus our X coordinate. Which is 0 in quantity. That means that the equation for a line is going Y is equal to 2 X. So we'll go ahead and draw that on our graph. And so we're gonna need to have a line that has a slope of 2 that passes through the origin. So we'll draw this with a dash line. Index in that tangent line out towards the edge of the graph. So that equation that we need to label in that graph is gonna be Y is equal to 2 X. Now for the final point, Which occurs when X is equal to I. We're gonna need to calculate Y of pi, and this is going to be equal to 2 multiplied by the cosine of pi. Which is equal to -2. So, here we have a slope for a tangent line of minus 2. Again, we'll use our point slope form. So we'll have Y minus. Our Y coordinate, which in this case is 0, and that's gonna be equal to -2, multiplied by the quantity of X minus pi. In quantity. So simplifying this expression, we'll get Y is equal to minus 2 X. Plus 2 pi. So we need to draw this equation. On our graph So we're gonna have a line. That has a slope. Of -2, and it's going to pass through our point. At P.0. And we'll just extend that line out towards the edge of the graph. And so we'll label that line as Y is equal to -2 X plus 2 pi. And so the graph that we have drawn on the screen is going to be the answer for this problem. So I hope this explanation has been useful, and I'll see you in the next video.

Tangent Lines

In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.

y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2

Key Concepts

Tangent Line

Derivative of Trigonometric Functions

Graphing Trigonometric Functions

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