Math 111 Exam 2, March 9, 2011
Read each problem carefully. Show ALL your work. You do NOT
need to simplify. No calculators.
1. (15 points) Differentiate the following functions:
(a)y=et−7
t(b)y= tan(x) tan−1(x) (c)y=2x2/3−5
3x2+x
2. (12 points) Differentiate the following functions (use logarithmic differ-
entiation in part(a)):
(a)y= (3x+ 2)3/2(2x+ 3)2/3(b)y= ln(ln(ln(sin x)))
3. (14 points) Find the derivatives of the following functions:
(a)y= (x+ 1)2x(b)y= sec( 1
x2) + xe(c)y= sin−1(x2+ 1)
4. (14 points)
(a) Find dy
dx if: x2+xy = 1 + y2and find the equation of the tangent
line to the curve at (-2, 1).
(b) Find dy
dx for cos2y=xsin y.
5. (9 points) Let f(x) = x3+ 3
(a) Find the inverse function f−1(x)
(b) Find the derivative of the inverse function, df−1(x)
dx
(c) Find the equation of the normal line to y=f−1(x) at x= 2.
6. (18 points)
(a) A ball’s height (in meters) is given by h(t) = 24 + 10t−5t2. Find
the total distance the ball travels from time t= 0 to t= 3.
(b) Find the value of dz/dr at r=π/3 if z(s) = ln(s2+ 1) and s= tan r
7. (18 points)
(a) Bob jogs around a circular track of radius 5 around the origin in the
(x, y) plane. If he runs such that dx
dt =−2 when x= 3 and y=−4, find dy
dt
at that instant.
(b) Consider the function y=x2−10. Find L(x), the linearization of y,
at x= 3 and find where this line (y=L(x)) crosses the x-axis.
