## Calculus

 Question 1
Consider the functions Which of the above functions is/are increasing everywhere in [0,1]?
 A II and III only B III only C II only D I and III only
Engineering-Mathematics       Calculus       GATE 2020
Question 1 Explanation: Question 2
Let f(x) = x -1(1/3)  and A denote the area of the region bounded bu f(x) and rhe X-axis, when x varies from -1 to1. Which of the followin statements is/are TRUE? I) f is continuous in [-1,1] II)f is not bounded in [-1,1] III) A is nonzero and finite
 A II only B III only C II and III only D I, II and III
Engineering-Mathematics       Calculus       GATE 2015 -(Set-2)
Question 2 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 3
If f(x) is defined as follows, what is the minimum value of f(x) for x ∊ (0, 2] ? A B C D Engineering-Mathematics       Calculus       Gate 2008-IT
Question 3 Explanation: Question 4
 A B C D E None of the above
Engineering-Mathematics       Calculus       Gate 2006-IT
Question 4 Explanation: Question 5
If the trapezoidal method is used to evaluate the integral obtained 01x2dx ,then the value obtained
 A is always > (1/3) B is always < (1/3) C is always = (1/3) D may be greater or lesser than (1/3)
Engineering-Mathematics       Calculus       Gate 2005-IT
Question 5 Explanation:
Note: Out of syllabus.
 Question 6
 A -1 B 1 C 0 D π
Engineering-Mathematics       Calculus       Gate 2005-IT
Question 6 Explanation: In the limits are be -π to π, one is odd and another is even product of even and odd is odd function and integrating function from the same negative value to positive value gives 0.
 Question 7
If f(1) = 2, f(2) = 4 and f(4) = 16, what is the value of f(3) using Lagrange’s interpolation formula?
 A B C D Engineering-Mathematics       Calculus       Gate 2004-IT
Question 7 Explanation:
Note: Out of syllabus.
 Question 8
Consider the following iterative root finding methods and convergence properties: Iterative root finding Convergence properties methods (Q) False Position                        (I) Order of convergence = 1.62 (R) Newton Raphson                 (II) Order of convergence = 2 (S) Secant                                         (III) Order of convergence = 1 with guarantee of convergence (T) Successive Approximation (IV) Order of convergence = 1 with no guarantee of convergence
 A Q-II, R-IV, S-II, T-I B Q-III, R-II, S-I, T-IV C Q-II, R-I, S-IV, T-III D Q-I, R-IV, S-II, T-III
Engineering-Mathematics       Calculus       Gate 2004-IT
Question 8 Explanation:
Note: Out of sylabus.
 Question 9
 A S > T B S = T C S < T and 2S > T D 2S ≤ T
Engineering-Mathematics       Calculus       Gate-2000
Question 9 Explanation:
S is continuously increasing function but T represent constant value so S>T.
 Question 10
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 10 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 11
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 11 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 12
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 12 Explanation:
The formula which is used to compute the second derivation of a function f at point X is Question 13
The solution of differential equation y'' + 3y' + 2y = 0 is of the form
 A B C D Engineering-Mathematics       Calculus       Gate-1995
Question 13 Explanation:
Note: Out of syllabus.
 Question 14
 A B C D Engineering-Mathematics       Calculus       Gate-1994
Question 14 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 15
 A linear B non-linear C homogeneous D of degree two
Engineering-Mathematics       Calculus       Gate-1993
Question 15 Explanation:
Note: Out of syllabus. In this DE, degree is 1 then this represent linear equation.
 Question 16
 A B C D Engineering-Mathematics       Calculus       Gate-1993
Question 16 Explanation:
Note: Out of syllabus.
 Question 17
Which of the following improper integrals is (are) convergent?
 A B C D Engineering-Mathematics       Calculus       Gate-1993
 Question 18
 A 1
Engineering-Mathematics       Calculus       Gate-1993
Question 18 Explanation:
Since the given expression is in 0/0 form, so we can apply L-Hospital rule. Question 20
 A 1/3
Engineering-Mathematics       Calculus       Gate-1993
Question 20 Explanation: There are 21 questions to complete.