Calculus

Question 1
A
II only
B
III only
C
II and III only
D
I, II and III
       Engineering-Mathematics       Calculus       GATE 2015 -(Set-2)
Question 1 Explanation: 
Since f(0)→∞
∴ f is not bounced in [-1, 1] and hence f is not continuous in [-1, 1].

∴ Statement II & III are true.
Question 2
 
A
B
C
D
       Engineering-Mathematics       Calculus       Gate 2008-IT
Question 2 Explanation: 
Question 3
 
A
S > T
B
S = T
C
S < T and 2S > T
D
2S ≤ T
       Engineering-Mathematics       Calculus       Gate-2000
Question 3 Explanation: 
S is continuously increasing function but T represent constant value so S>T.
Question 4
Consider the function y = |x| in the interval [-1,1]. In this interval, the function is  
A
continuous and differentiable
B
continuous but not differentiable
C
differentiable but not continuous
D
neither continuous nor differentiable
       Engineering-Mathematics       Calculus       Gate-1998
Question 4 Explanation: 
The given function y = |x| be continuous but not differential at x= 0.
→ The left side values of x=0 be negative and right side values are positive.
→ If the function is said to be differentiable then left side and right side values are to be same.
Question 5
What is the maximum value of the function f(x) = 2x2 - 2x + 6 in the interval [0,2]?
A
6
B
10
C
12
D
5.5
       Engineering-Mathematics       Calculus       Gate-1997
Question 5 Explanation: 
For f(x) to be maximum
f'(x) = 4x - 2 = 0
⇒ x = 1/2
So at x = 1/2, f(x) is an extremum (either maximum or minimum).
f(2) = 2(2)2 - 2(2) + 6 = 10
f(1/2) = 2 × (1/2)2 - 2 × 1/2 + 6 = 5.5
f(0) = 6
So, the maximum value is at x=2 which is 10 as there are no other extremum for the given function.
Question 6
The formula used to compute an approximation for the second derivative of a function f at a point X0 is
A
B
C
D
       Engineering-Mathematics       Calculus       Gate-1996
Question 6 Explanation: 
The formula which is used to compute the second derivation of a function f at point X is
Question 7
The solution of differential equation y'' + 3y' + 2y = 0 is of the form
A
B
C
D
       Engineering-Mathematics       Calculus       Gate-1995
Question 7 Explanation: 
Note: Out of syllabus.
Question 8
   
A
B
C
D
       Engineering-Mathematics       Calculus       Gate-1994
Question 8 Explanation: 

With initial value y(x0) = y0. Here the function f and the initial data x0 and y0 are known. The function y depends on the real variable x and is unknown. A numerical method produces a sequence y0, y1, y2, ....... such that yn approximates y(x0 + nh) where h is called the step size.
→ The backward Euler method is helpful to compute the approximations i.e.,
yn+1 = yn + hf(x n+1, yn+1)
Question 9
       
A
linear
B
non-linear
C
homogeneous
D
of degree two
       Engineering-Mathematics       Calculus       Gate-1993
Question 9 Explanation: 
Note: Out of syllabus.

In this DE, degree is 1 then this represent linear equation.
Question 10
   
A
B
C
D
       Engineering-Mathematics       Calculus       Gate-1993
Question 10 Explanation: 
Note: Out of syllabus.
Question 11
Which of the following improper integrals is (are) convergent?
A
B
C
D
       Engineering-Mathematics       Calculus       Gate-1993
Question 12
   
A
1
       Engineering-Mathematics       Calculus       Gate-1993
Question 12 Explanation: 
Since the given expression is in 0/0 form, so we can apply L-Hospital rule.
Question 13
       
A
Out of syllabus.
       Engineering-Mathematics       Calculus       Gate-1993
Question 14
 
A
1/3
        Engineering-Mathematics       Calculus       Gate-1993
Question 14 Explanation: 
Question 15
 
A
Out of syllabus.
        Engineering-Mathematics       Calculus       Gate-1993
There are 15 questions to complete.