WORKSHOP 1·7
Handout
This workshop is meant to help you review the main topics covered so far in the course, in preparation for
the upcoming midterm exam.
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. Find the domain and range of f (f (x)) where
f (x) =
1
.
x
2. Evaluate the following limits, if they exist.
2
(i) lim x
+ 3x
x→0
x
x2 − 2
(ii) lim+ 2
x→1 x − 4x + 3
3. Carefully prove there is at least one solution to the equation 3 = 2x+3 between x = −4 and
x = −1.
4. Determine whether the following functions are continuous. If they are not, state the location of
the discontinuity/discontinuities and explain why the function is not continuous at that point.
a.
f (x) =
b.
(
f (x) =
x2 + 3x + 2
x+1
x2 − 3x + 4, 0 ≤ x ≤ 2
8 − 3x,
2<x≤5
c.
f (x) =
p
(x − 2)(x − 3)
5. Sketch the parabolas y = x2 and y = x2 − 2x + 2. Do you think there is a line that is tangent to
both curves? If so, find its equation. If not, why not?