WORKSHOP 1·13
Handout
This workshop is meant to help you review the main topics covered so far in the course, in preparation for
the upcoming December exam. All of these problems come from past MATH 110 exams.
Work out each of the following problems on the board. Make sure you discuss your solution with your TAs
before you erased the board and start a new problem.
1. Define what it means for a function f to be continuous at a point a. Find the domain of the
function g(x) = sin1 x . Is g(x) = sin1 x continuous on the interval (−∞, ∞). Why or why not?
2. True or False: The limit lim
x→2
f (x)
does not exist if g(2) = 0. Why or why not?
g(x)
3. Show that there is at least one point on the curve y = x4 + 2x2 − 4x − 1 where the tangent line
to the curve is horizontal.
4. Differentiate the following functions
(a) y = (1 + x2 )50 ,
(b) f (x) = e
(c) g(t) =
√
3x
,
ln(3t−4)
.
t
5. Let f (x) = x2 − 6x. Find the equations of all tangent lines to the curve y = f (x) which pass
through the point (4, −12).
6. Tsunamis are large ocean waves produced by underwater seismic activity. In deep water, they
can travel at extremely high speeds.
A tsunami wave spreads outward in a widening circle. Suppose the radius of the circle is increasing
at a rate of 900 km/h. Calculate how fast the area of the circle is increasing when the radius is
equal to 100 km.
7. Let f (x) =
1
. Find f 0 (x) using the limit definition of derivative.
x+1