Final Exam Extra Practice 1
Make sure to practice writing full solutions like you will have to on the final exam. It is a
good idea at this stage to review were you have lost marks on quizzes and on the midterm
for things like missing justifications (for example, not saying a function is continuous), and
make sure that you practice writing these so that you won’t lose the same marks on the
final.
1. Sketch the graph of f (x) = (1 − x)ex . Use the format for curve sketching questions
provided on Learn.
2. Use the Inverse Function Theorem to prove that (arctan(x))0 =
1
.
1 + x2
x
3. Evaluate lim (1 − e−x )e .
x→∞
1
4. Prove or disprove: The function f (x) = x3 − x2 + 2x + 1 has at most one real root.
3
(
x2 + 2x, if x < 0
5. Let f (x) =
where a and b are constants. Find the values of
ax + b,
if x ≥ 0
a and b that make f differentiable at x = 0.