Find an equation of the line tangent to the following curves a...

Question

Find an equation of the line tangent to the following curves at the given value of x.

y = csc x; x = π/4

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 c X at X equals pi divided by 3. Awesome. So it appears for this particular problem, we're ultimately trying to figure out what this equation is, the equation of this line that is tangent to the graph of Y. Awesome, and we need to note that X equals pi divided by 3. So now that we know that we're ultimately trying to figure out the equation of the line that is tangent to this graph of Y, let's read off our multiple choice answers to see what our final answer might be, noting that they all state that Y is equal to some expression. So A is 2 multiplied by the square root of 3 X minus pi divided by 3 plus 2. B is 2 multiplied by the square root of 3 X minus 2 multiplied by the square root of 3 pi divided by 3, minus 2. C is 2 multiplied by the square root of 3 X minus 2 multiplied by the square root of 3 pi divided by 3 plus 2. And D is X minus 2 multiplied by the square root of 3 pi divided by 3 + 2. Awesome. So first off, in order to solve this problem, you must first figure out what the derivative of our of our function of Y is. So we're trying to figure out what Y is, and Y is a fancy way of saying the first derivative of this function of Y, and let us know that our function of Y states that Y is equal to cant of X. So, as we should recall, when we take the derivative, the first derivative to be exact of C X, we will find that Y is equal to Ccant of X multiplied by tangent of X. Fantastic. And now our next step is we need to find the slopem of the tangent line, and we can do this by recalling that we can figure out this value by taking the derivative of the function at X equals divided by 3. So once again we can find the slopem of the tangent line by finding the value of the derivative of the function at X equals pi divided by 3. Awesome. So with that said, as we should recall, we could say that in the slope is equal to Y subscript X equals pi divided by 3 because we're plugging in a value of pi divided by 3 N for any value of X. So this will be equal to C can of divided by 3, multiplied by Tangent of pi divided by 3, and when we evaluate that expression, we will find that it's equal to 2 multiplied by the square root of 3. So 2 multiplied by the square root of 3. So that will be our answer when we evaluate this expression in our calculator, this expression for the slope. So now, our next step is we need to find the value of the function, the original function of Y, when X is equal to pi divided by 3. So when we do that, we will find that Y of pi divided by 3, when we're plugging in pi divided by 3 in for any value of X, we will find that C can of pi divided by 3 is equal to 2. So now, Once again, once again, to quickly recap, we're plugging in our X equals pi divided by 3, N for X into our original function of Y, and when we get that, we will get from C can of pi divided by 3 will be equal to 2 when we evaluate it in our calculator. So now our next step that we need to take is we need to find the equation of the tangent line to our given function at. I divided by 3, so you take I divided by 32. So when we evaluate using point slope form, we will get Y minus 2 is equal to 2 multiplied by the square root of 3, multiplied by X minus pi divided by 3. And when we rearrange this expression to isolate and solve for Y, we will find, and I'll skip a few steps, but you just, it's simple. You just move everything around and simplify it as much as you can to get Y all by itself, and when we do, we will find that Y is equal to 2 multiplied by the square root of 3 X minus 2 multiplied by the square root of 3 pi divided by 3. + 2, and that is our final answer. Hooray, we did it. And this corresponds with multiple choice answer letter C, which once again states that Y is equal to 2 multiplied by the square root of 3 X minus 2 multiplied by the square root of 3 pi divided by 3 plus 2. 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 following curves at the given value of x.y = csc x; x = π/4

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Find an equation of the line tangent to the following curves at the given value of x.
y = csc x; x = π/4