yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

2015 AP Calculus AB 6a | AP Calculus AB solved exams | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

Consider the curve given by the equation (y^3 - xy = 2). It can be shown that the derivative of (y) with respect to (x) is equal to (\frac{y}{3y^2 - x}).

All right, write an equation for the line tangent to the curve at the point ((-1, 1)).

So, we could figure out the equation for the line if we know the slope of the line and we know a point that it goes through. So that should be enough to figure out the equation of the line.

The line's going to have a form (y = mx + b). (m) is the slope and is going to be equal to (\frac{dy}{dx}) at that point. We know that that's going to be equal to, let's see, (y) is 1 when (x) is -1.

So, (y = 1), so (\frac{1}{3y^2}) - (x), when (y = 1), since (x = -1), we can substitute this in. So this is (\frac{1}{3 \cdot 1^2}) which is (3 - (-1)).

So, this is the same thing as (3 + 1) and so this is equal to (\frac{1}{4}).

And so, the equation of our line is going to be (y = \frac{1}{4}x + b).

Now we need to solve for (b) and we know that the point ((-1, 1)) is on the line. So we can use that information to solve for (b).

This line is tangent to the curve, so it includes this point and only that point. That's what has in common with the curve.

So, when (y = 1) when (x = -1 + b), and so we have (1 = -\frac{1}{4} + b).

You add (\frac{1}{4}) to both sides and you get (b) is equal to, we could either write it as (1) and (\frac{1}{4}) which is equal to (\frac{5}{4}) which is equal to (1.25).

We could write it any of those ways.

So the equation for the line tangent to the curve at this point is (y = \frac{1}{4}x + \frac{5}{4}) and we're done, at least with that part of the problem.

More Articles

View All
Harnessing the Power of the Sun | Origins: The Journey of Humankind
Fusion as an energy source is very attractive. It would be a carbon-free energy source that could power mankind forever. The challenge is making fusion work at the National Ignition Facility. What we’re trying to do is overcome a natural barrier that natu…
Calculating confidence interval for difference of means | AP Statistics | Khan Academy
Kylie suspected that when people exercise longer, their body temperatures change. She randomly assigned people to exercise for 30 or 60 minutes, then measured their temperatures. The 18 people who exercised for 30 minutes had a mean temperature, so this i…
Dilating points example
We’re asked to plot the image of point A under a dilation about point P with a scale factor of three. So, what they’re saying when they say under a dilation is they’re saying stretching it or scaling it up or down around the point P. That’s what we’re go…
Biased and unbiased estimators from sampling distributions examples
Alejandro was curious if sample median was an unbiased estimator of population median. He placed ping-pong balls numbered from zero to 32 in a drum and mixed them well. Note that the median of the population is 16. He then took a random sample of five bal…
Rick and Morty Writer: Ryan Ridley
All right, man. We should probably jump into Rick and Morty at some point. Um, before we do that, uh, how about you just like give your background of up until Rick and Morty? So, I didn’t really know what I was doing with my life. I was a terrible studen…
The Story Behind Europe's Tallest Statue: The Motherland Calls | National Geographic
[Music] Mr. O’Reilly, 300ccs. Don’t name our canoes. No visible earth, it has the scale of America’s National Mall and the seriousness of Pearl Harbor. Combine them, and that’s what it feels like to visit Mammoth Gorgon, the memorial complex for the Batt…