Graded Assignment
1. Construct a truth table for each of the following statements.
(a) p =? ¬q
(b) (p ? (p =? q)) =? q
(c) (p =? (q ? ¬q)) ?? ¬p
2. Is Q an ordered field? Prove your answer?
3. If a x b and a y b, show that |x - y| b - a. Interpret it
geometrically.
4. Find the infimum and supremum, if they exist of the following sets. Justify
every step in the argument.
(a) A := {x ? R : 3x + 8 0}
(b) B := {x ? R : 2x - 1 = x
2}
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5. Show that every polynomial of odd degree with real coefficients has at
least one real root.
6. Let f(x) = x
3 - 3x
2 + 2x for x ? [-1, 3]
(a) Find where f is strictly increasing and where it is strictly decreasing.
(b) Find the maximum and minimum of f on [-1, 3].
7. Let f and g be differentiable on R. Suppose that f(0) = g(0)
and f
0
(x) = g
0
(x) for all x = 0. Show that f(x) = g(x) for all x = 0.
8. Find the primitive of the following functions.
(a) e
x
.sin(x)
(b) 1
x
2 + 6x + 8
9. Calculate e correct to seven decimal places. (Hint: Use Taylor’s Theorem)
10. Let f : [0, 3] ? R be defined by
f(x) =
?
?????????
?????????
x if 0 = x 1
1 if 1 = x 2
3 - x if 2 = x = 3
Obtain formulas for F(x) = R x
0
f and sketch the graphs for f and F. Calculate the points where
F is differentiable.
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