1MileCrash said:Homework Statement
Find the equation for the tangent line at (2,2)
f(x)=xy+y^3=12
Homework Equations
The Attempt at a Solution
I really feel like I know what I'm doing here, but the key disagrees.
My equation for the tangent slope comes out to be:
(-y)/(x+3y^2)
when solved for my (2,2), I get a slope of -1/7
Yielding an entire equation of y= -1/7x + 12/7
1MileCrash said:If my slope is correct, but my tangent line equation is wrong, I can only assume I did something wrong in y - y = M(x-x)?
1MileCrash said:I used y - 2 = M (x - 2), y = 2 and x = 2.
What else could I have used? I think I'm getting confused..
1MileCrash said:Ahh, I think I see now, a sign error?
y - 2 = -1/7x + 2/7
y = -1/7x + 16/7
Dick said:Yes. Sign error. Looks ok now. If you'd post all of your work to begin with it would be easier to diagnose these things.
1MileCrash said:Sorry about that, but thanks for the help!
The equation for a tangent line is y = mx + b, where m represents the slope of the line and b represents the y-intercept.
The slope of a tangent line can be calculated using the derivative of the function at the point of tangency. Alternatively, it can also be found by using the slope formula (m = (y2 - y1)/(x2 - x1)) using two points on the tangent line.
Yes, a tangent line can have a negative slope. The slope of a tangent line depends on the slope of the original function at the point of tangency.
The y-intercept of a tangent line can be determined by finding the y-value of the point of tangency. This can be done by substituting the x-value of the point into the original function.
Yes, it is possible for a curve to have more than one tangent line at a given point. This can happen when the curve has a sharp point or a cusp at that point, or when the curve is not differentiable at that point.