Thread: Lines Tangent to Curves

Hey, these are some multiple choice questions I can't figure out. Please feel free to answer all or selectively answer some of them. Thank you!

If m1 is the slope of the curve xy=2 and m2 is the slope of the curve x^2+y^2=3, then the point of intersection of the two curves is:

a) m1= - m2 b) m1*m2= -1 c) m1= m2 d) m1*m2 = 1

e) m1*m2 = -2

---------------------------------------------------------------------

The tangent to the curve y^2 -xy +9 = 0 is vertical when:

a) y=0 b) y= + or - the square root of 3 c) y= .5 d) y= + or - 3

e) none of these

---------------------------------------------------------------------

The equation of the tangent to the curve y= xsinx at the point (pi/2, pi/2) is

a) y= x- pi b) y= .5 c) y= pi- x d) y=x+ .5 e) y= x

For #2, we could solve for y and differentiate, but let's do it implicitly.

..............[1]



Divide by y' and get:



Now, take the limit:



Sub into [1]:





Solve for x and we get

That leaves

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~

Let's do it using 'regular' differentiation and see if we get the same thing.

Solve for y and get:



Differentiate:

Set the denominator to 0 and solve for x:



We can see that

Plugging this into y, we see

I showed you this so you may know how to find a vertical tangent in the future. Study both methods. They may come in handy.