1. Given y=x 2 /(x + 3). Find where y has vertical and horizontal tangents. 2. I

Question

1. Given y=x2/(x + 3).  Find where y has vertical and horizontal tangents.

2. If y= sqrt(6 + 4x3).  Find d2y/dx2.   Write the equation of the tangent line at (-1,sqrt(2)).

3.State where y=abs(4+x) is not differentiable and then use the alternative form for the derivative at a point to show that it is not differentiable.

4.Find dy/dx where y = sin(3x + 2y).

5.Air is being pumped into a spherical balloon at a rate of 32 cubic feet per min.  At what rate is the radius increasing when the radius is 3 feet when V=4r3/3.

Explanation / Answer

1)Given y=f(x)=x2/(x + 3)

domain of f(x) is (-infinity,-3)U(-3,infinity)

differentiate with respect to x . (u/v)'=(u'v -uv')/v2

dy/dx=[2x(x + 3) -x2(1 + 0)]/(x + 3)2

dy/dx=[2x2 + 6x -x2]/(x + 3)2

dy/dx=[x2 + 6x]/(x + 3)2

horizontal tangent occurs when dy/dx =0

=>[x2 + 6x]/(x + 3)2=0

=>x2+6x =0

=>x(x+6)=0

=>x =0,x =-6

at x =0,y=0

x=-6,y=(-6)2/(-6 + 3) =-12

horizontal tangent occurs at (x,y)=(0,0)and (-6,-12)

vertical tangent occurs when 1/(dy/dx) =0

(x + 3)2/[x2 + 6x]=0

=>x =-3

but x=-3 is not in domain of function, so there is no vertical tangent

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