Find an equation of the line tangent to the given curve at a.

Question

y = 2x² / (3x - 1); a = 1

Accepted Answer

Below there. Today we're gonna 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 tangent line to the curve defined by Y equals X2 + 2 X all divided by X minus 1 at the point where X equals 3. So it appears for this particular problem, we're asked to figure out what the equation of the tangent line to the curve defined as Y equals X2 + 2 X all divided by X minus 1 at the point where X equals 3. So ultimately we're trying to figure out what this equation is for this equation that is a tangent to the curve. So we're trying to figure out this equation of the tangent line that's To the, so it's going to be a tangent line to the curve defined by Y. So we're trying to figure out what this equation of the tangent line is. So with that said, our first step in order to help us solve for this problem is we need to recall and use the quotient rule, and the quotient rule states that D by DX of F of X divided by G of X. is equal to G of X multiplied by F of X minus F of X multiplied by G of X, all divided by G of X2. Fantastic. We also need to recall and note that the slope of the line tangent to the curve at parenthesis X F of X is the derivative of F of X. So to say that again, the slope of the line tangent to the curve at X F of X. Is going to be the derivative of the function F of X. Awesome. So that means we could go ahead and write that Y is equal to x minus 1 multiplied by D by DX of X2 + 2 X. Minus X2 + 2 X. Multiplied by D by DX of X minus 1. All divided by X minus 1 squad, and this is all equal to X minus 1 multiplied by 2 x 2 X plus 2, so X minus 1 multiplied by 2 X + 2. Minus X2 + 2 X. Multiplied by 1. Fantastic. And then this will be all divided by X minus 1 to the power of 2. And this will be all equal to 2 X to the power of 2. Plus 2 X minus 2X minus 2. Minus X to the power of 2, minus 2 X. All divided by X minus 1 to the power of 2, which will mean that this is all equal to X2 minus 2X minus 2, all divided by X minus 12. So now, our next step is we need to evaluate the derivative at X equals 3. So when we evaluate the derivative at X equals 3, we can state that Y of 3, and noting that we're plugging in 3 for X will give us that Y of 3 is equal to 3 to the power of 2 minus 2 multiplied by 3 minus 2. All divided by 3 minus 1 to the power of 2, which when we plug this expression into our calculator, we will find that it's equal to 1/4. So therefore, the slope of 1/4 is the slope of the tangent line or the line tangent to the curve at X equals 3. So now our next step is we need to evaluate the function at X equals 3, so we need to say that Y is equal to 32 + 2 multiplied by 3. Once again, we're plugging in 3 for X divided by 3 minus 1. Which when we plug that expression into our calculator, we will find that it's equal to 15 divided by 2. Now, we need to recall and use point slope form, so that states that Y minus Y1 is equal to M, where M is our slope, multiplied by X minus X1, fantastic. And now, at this point, we need to recall a note that X1 is equal to 3, and Y1 is equal to 15 halves, and our slope M once again is equal to 1/4. So when we plug in our known variables into our equation for the point slope form, you will find that Y minus 15 halves is equal to 1/4 multiplied by X minus 3. So in an effort to save time, I'm gonna skip a few steps, but essentially we're taking this expression and we're rearranging it to isolate and solve for a Y. So when we do that, we will find that Y all by itself. It's going to be equal to 1/4 X plus 27 divided by 4, and that is our final answer. Hooray, we did it. So therefore, without a doubt, our final answer is Y equals 1/4 X plus 27 divided by 4. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find an equation of the line tangent to the given curve at a.
y = 2x² / (3x - 1); a = 1

Key Concepts

Tangent Line

Derivative

Point-Slope Form

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