{Use of Tech} Tangent lines Determine equations of the lines t...

Question

{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the tangent lines.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the equation of the line tangent to the graph of Y equals X multiplied by the square root of 17 minus X2 at the point 1.4. Awesome. So it's important to note that for this particular problem, the final answer that we're ultimately trying to solve for is we're trying to create or figure out an equation for this specific line that is tangent to our graph of this function of Y at a point of 1.4, which is our coordinate plane, which is X. Y. So that's our coordinate at 1.4. Awesome, so 1 is X and 4 is Y. So now that we know that we're trying to solve for the equation for this line that is tangent to this graph of this function of Y at this specific point, in order to achieve this goal, we need to recall and note that in order to find the equation of the line of the tangent line to the graph of Y. For this function of Y, which is Y equals X multiplied by the square root of 17 minus X2d at the point 1.4, we need to follow three simple steps. So our first step is we need to find the derivative of Y with respect to. So with respect to X, so we need to find the derivative of Y with respect to X, which will give us the slope. Of the tangent line and we'll use the product rule and chain rule for differentiation. So we need to recall at this point when you start remembering in the back of our mind, we need to we need to remember how to use the product rule and the chain rule to help us perform this differentiation. Our second step is we need to evaluate the derivative at. So, let's use a capital E to denote evaluate, so we need to evaluate at. X equals 1, and that is to help us find the slope of the tangent line at our 0.1.4. Fantastic. And our third and final step is we need to use point. Slope form. So we need to use point slope form of a line to write our equation of the tangent line using the slope that we found from step 2 and the point 1.4. Awesome, so we need to use point slope form. In order to find our equation. Fantastic. So now, let's start with our first step. So let us find the derivative of Y. So we're given that once again that Y equals X multiplied by the square root of 17 minus X2. So when we differentiate with respect to X, we'll get that Y. is equal to D by DX of X multiplied by the square root of 17 minus X2. So using the product rule, so we need to recall, so let's make a note of this up above here in red. So the product rule states that you multiplied by V is equal to U multiplied by V plus U multiplied by V. Where you will be equal to X in this case, and V will be equal to the square root of 17 minus X to the power of 2. So, with that in mind, that will mean we will get that to the power, so U is equal to 1. So U is equal to 1, and we will find that V is equal to D by DX. Of the square root of 17 minus X2. Which is equal to 1 divided by 2 multiplied by the square root of 17 minus X2. And then this will be multiplied by -2 X, and this will be equal to Which I think maybe we can fit it down below here. So this will be equal to negative, negative. X divided by the square root of 17 minus X2. Fantastic. So, moving right along here, so now our next order of business is we need to note that Y will be equal to 1 multiplied by the square root of 17 minus X2. Plus x multiplied by negative x divided by the square root of 17 minus X2. Fantastic. So that will mean that Y prime. So once again we found Y and and we're still trying to get it down. We're still trying to solve for Y, so Y is equal to the square root of 17 minus X2 minus X 2 divided by the square root of 17 minus X2. So now we can start using, so that was, let's make a note of this. So this was all part of our first step. So now we can move on to our second step. So now we're evaluating the derivative at X equals 1. So that's what we are doing now. So, with that said, we need to take Y. Of 1, and make sure it's 1 is equal to the square root of 17 minus 12 minus 12 divided by the square root of 17 minus 12. These are plugging in 1 for the value of X. And when we do that, we will find that this is equal to, when we write it in a fraction, when we plug this into our calculator and we write it as a fraction, it will be equal to 15/4, so 15 divided by 4. So, therefore, the slope of the tangent line at 1.4 is 15/4, so let's make a note of that. So, our slope, so M denotes the slope is equal to 15/4. Awesome. So let's put a box around that because that's an important answer that we need to write our final answer with. Awesome. So now our next step is for our 3rd and final step is we need to use point slope form of a line. So as we should recall, point slope form of a line states that Y minus Y1 is equal to M multiplied by X minus X1. So where once again M is the slope, and once again we need to note that X1. Y1 is the point on the line, so this equals our point. So, we need to use our slope and our 1.4 coordinate point. So when we do that, when we plug in our known variables, we'll get Y minus 4 is equal to 15/4 multiplied by X minus 1. Fantastic. So, when we rearrange this equation, To isolate and solve for Y, we will find that Y is equal to 15/4 X minus 15/4 + 4. Awesome. So we could actually simplify this. Once again, to get Y equals 1544 X plus 1/4, and that will be our final answer. Hooray, we did it. So this is our final answer that we were ultimately trying to solve for. We finally officially figured out the derivative of our equation, that is the tangent to the line of graph of this function of Y. So that's our final derivative. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the tangent lines.

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