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Calculas

This document contains a mathematics exam with 5 questions covering calculus topics. Question 1 involves limits, derivatives, and continuity. Question 2 covers derivatives, Rolle's theorem, the mean value theorem. Question 3 is about critical points and increasing/decreasing intervals for a function. Question 4 covers limit proofs and properties of convergent sequences. Question 5 defines a recursive sequence and asks to prove properties about it.

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kaveen danika
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70 views2 pages

Calculas

This document contains a mathematics exam with 5 questions covering calculus topics. Question 1 involves limits, derivatives, and continuity. Question 2 covers derivatives, Rolle's theorem, the mean value theorem. Question 3 is about critical points and increasing/decreasing intervals for a function. Question 4 covers limit proofs and properties of convergent sequences. Question 5 defines a recursive sequence and asks to prove properties about it.

Uploaded by

kaveen danika
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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UNIVERSITY OF SRI JAYEWARDENEPURA FACULTY OF APPLIED SCIENCES

B.Sc. Degree First Year Second Semester Course Unit Examination - May 2022
DEPARTMENT OF MATHEMATICS
MAT 122 2.0 Calculus

Time : Two hours Number of Pages: 2/ Number of Questions: 5 Total Marks: 100
Answer 4 questions(Question No 4 is complimentary)

1. (a) The graph of f (x) is given below

Fill in the following table


c lim f (x) lim f (x) lim f (x) f (c) f is cts at c
x→c− x→c+ x→c
0
2
4
(b) 

 a2 x + 5a + 11, x < 2

f (x) = 8 x=2
2
 3a
x + 5, x>2


2
i. Find all admissible value(s) for a such that lim f (x) is defined.
x→2
ii. Find all admissible value(s) for a such that f is continuous at 2.
(c) Evaluate each of the following limits
x2 − 5x + 6
i. lim 2
x→3
√ x −9
x−2
ii. lim
x→4 x − 4

(25 points)
MAT 122 2.0 - Page 2 of 2

2. (a) Use the definition to find the derivative of f (x) = sin(x) at x = x0 .


(b) State and prove Roller’s Theorem
(c) i. State Mean Value Theorem
ii. Show that for any x > 4, there is a number ω between 4 and x such that
√
x−2 1
=
x−4 2ω
iii. Use this fact to show that if x > 4 , then
√ x
x>1+
4

(25 points)

3. Given
x3 5x2
f (x) = − + 6x
3 2
(a) Find all critical numbers.
(b) Find where the function is increasing and decreasing.
(c) Find the critical points and identify each as a relative maximum, relative minimum,
or neither.

(25 points)

4. (a) Using the definition prove that


1
i. lim = 0
n→∞ n
2n − 1 2
ii. lim =
n→∞ 3n + 1 3
(b) Let {an } and {bn } be convergent real sequences, and tn = an + bn for all but finitely
many n show that
lim tn = lim an + lim bn
n→∞ n→∞ n→∞

(c) show that if {sn } and {tn } are convergent real sequences, and sn ≥ tn for all n, then
lim sn ≥ lim tn .
n→∞ n→∞

(25 points)

n
5. Let s1 = 1 and let sn+1 = (sn )2 for n ≥ 2.
n+1
(a) Prove that sn ≥ 0 for all n.
(b) Prove that sn+1 ≤ sn for all n.
(c) Show that sequence {sn } is convergent.
(d) Show that lim sn = 0
n→∞

(25 points)

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