Math111 Make-Up Exam
16-12-2015
1. Compute the following limits
(a) limx!0+
R x4
x2
p
sin xdx
x3
1
(b) limx!0 ( 2 )sin x
x
2. Find the derivative of the function
p
y
(a) de…ned implicitly by the equation arctan( ) = ln( x2 + y 2 );
x
1
ln(1+ )
p
x :
(b) de…ned as y = arcsin x + (2 + sin x)
3. Compute the following integrals:
R p
(a) x3 1 + x2 dx
R ln 2 dx
(b) 0
1 + ex
4. Let f (x) = x2 e2x .
(a) Find the critical points and local extrema of the function f .
(b) Find the intervals of increase and decrease of f .
(c) Find the equation of the tangent line to the graph of f at x = 1:
5. Find the radius of the cylinder of maximal volume which can be inscribed in a right circular cone with
radius R and height H.
1