Calculus (SMD001UM1) End-Sem Exam: Instructors: Rahul Kitture and Manmohan Vashisth
Calculus (SMD001UM1) End-Sem Exam: Instructors: Rahul Kitture and Manmohan Vashisth
Calculus (SMD001UM1) End-Sem Exam: Instructors: Rahul Kitture and Manmohan Vashisth
• You can refer to the Tutorial problems, class notes or book (write name with page
number), if you are using it. Otherwise, it will not be considered.
7
1. Let {an } be a sequence of real numbers such that |an+1 − an | < |an−1 − an |. Prove
8
that {an } is convergent (give details). (5 Marks)
3. For any integer n > 2, show that f (x) = xn − ax − b can have at most three distinct
roots in R. (5 Marks)
4. Determine the points a ∈ R at which the functions f (x) = | cos(x)| and g(x) = cos(|x|)
are differentiable (write details). (5 Marks)
7. Define f : [1, ∞) → R by
(
1
1, if x ∈ n, n + 2n
for n ∈ N
f (x) =
0, otherwise
Z ∞
Then show that the improper integral f (x) dx converges to 1 but the series
P∞ 1
n=1 f (n) is divergent. (5 Marks)
|xy|
9. Let f (x, y) = p .
x + y2 + 1
2
∂f ∂f
(a) Does (0, 0) and (0, 0) exist? (2.5 Marks)
∂x ∂y
(b) Is f differentiable at (0, 0)? (2.5 Marks)
11. Find the point on the line passing through (1, 0, 0) and (0, 1, 0) that is closest to the
line through (0, 0, 0) and (1, 1, 1). (5 Marks)
12. Find the extreme values of the function f (x, y) = x3 − x + y 2 − 2y on the closed
triangular region with vertices at (−1, 0), (1, 0) and (0, 2). (5 Marks)