Recent Question

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}
Powered by upGrad Education Pvt. Ltd.
c Copyright. UpGrad Education Pvt. Ltd. All rights reserved
1
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.
Powered by upGrad Education Pvt. Ltd.
c Copyright . UpGrad Education Pvt. Ltd. All rights reserved
2

Disclaimer : LiveWebExperts.com provides assignment and homework help for guidance and reference purpose only. These papers are not to be submitted as it is. These papers are intended to be used for research and reference purposes only.