The Area $A$ of the region $S$ that lies under the graph of the continuous function $f$ is the limit of the sum of the areas of approximating rectangles:

• A) $A = \mathop {\lim }\limits_{n \to \infty } {R_n} = \mathop {\lim }\limits_{n \to \infty } \left[ {f\left( {{x_1}} \right)\Delta x + f\left( {{x_2}} \right)\Delta x + \ldots + f\left( {{x_{n - 1}}} \right)\Delta x} \right]$
• B) $A = \mathop {\lim }\limits_{n \to \infty } {R_n} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{{\Delta x}}\;\left[ {f\left( {{x_1}} \right) + f\left( {{x_2}} \right) + \ldots + f\left( {{x_n}} \right)} \right]$
• C) $A = \mathop {\lim }\limits_{n \to \infty } {R_n} = \mathop {\lim }\limits_{n \to \infty } \left[ {f\left( {{x_1}} \right) + f\left( {{x_2}} \right) + \ldots + f\left( {{x_n}} \right)} \right]$
• D) $A = \mathop {\lim }\limits_{n \to \infty } {R_n} = \mathop {\lim }\limits_{n \to \infty } \Delta x\left[ {f\left( {{x_1}} \right) + f\left( {{x_2}} \right) + \ldots + f\left( {{x_n}} \right)} \right]$

If f(x) = Sec^ (2) x + x^3, then which of the following is NOT true about it.

• A) Its anti – derivative is Tan(x) + x^4/4 + 15.
• B) Its anti – derivative is Tan(x) + x^4/4 + 10.
• C) Its anti – derivative is Tan(x) + x^4/4 + 12.
• D) None of these.

What is the estimated area under f(x) = 2x from x = 0 to x = 4 with right end points for n = 2?

• A) 8
• B) 24
• C) 18
• D) 10

The area between f(x)=-x and the closed interval
[-1,0] on x-axis is ----------

• A) 1
• B) 1/2
• C) -1/2
• D) 2

The estimated area under f(x) = x from
x = 0 to x = 3 with right end points for
n = 3 is ________.

• A) 6.
• B) 7.
• C) None of these
• D) 5.

$${\text{The integral }}\int {\frac{{10x}}{{{{(5{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}$$

• A) $$- \frac{1}{{5x + 1}} + c$$
• B) $$- \frac{1}{{{{(5{x^2} + 1)}^3}}} + c$$
• C) $$- \frac{1}{{5{x^2} + 1}} + c$$
• D) $$\frac{1}{{5{x^2} + 1}} + c$$

Given f(x)= 1/(x-1) on the interval (0,3) , then ‘f ’ has no relative maxima or relative minima due to………

• A) Open interval
• B) Discontinuity in the interval
• C) Rational function
• D) None of these

Let f(x)=x^3 and a= -5 ,b=5 and n= no. of partition of [-3,3]=4
Let the widths of first, second, third and fourth intervals are 3,0,1,2 respectively. Then the “Mesh size” of partition is……………

• A) 3
• B) 0
• C) 1
• D) 2

1+2+3……….+1000 equals -------

• A) 500500
• B) 1000
• C) None of these
• D) 3000

Approximation to an area improves as number of partitions is decreased.

• A) True
• B) False
• C)
• D)

$\text{A line }y={{y}_{\text{0}}}\text{ is called a horizontal asymptote for the graph of function }f\text{ if}$

• A) $\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=0$
• B) $\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=\infty$
• C) $\underset{x\to +\infty }{\mathop{\lim }}\,f(x)={{y}_{0}}$
• D) $\underset{x\to 0}{\mathop{\lim }}\,f(x)={{y}_{0}}$

13+23+33+…+153 equals _____.

• A) 15(16)2/2
• B) [15(16)]2/2
• C) 15(16)/22
• D) [15(16)/2]2

If the function f(x)= (x^2)+1 satisfies all the conditions of Rolle’s theorem in the interval [-1,1] , then the value of ‘c’ will be……..

• A) 2
• B) 0
• C) 4
• D) 3

If a function f(x) is defined on an interval
[a, b] satisfying the conditions of the Rolle’s Theorem, then which of the following is true about it.

• A) It may or may not have a critical point.
• B) None of these.
• C) It must have at least one critical point.
• D) It would not have more than one stationary point.

If the graph of $f$ lies below all of its tangents on an interval $I$,then it is called___________ on $I$.

• A) constant function
• B) concave upward
• C) None of these
• D) concave downward

The estimated area under f(x) = x^2 + 2 from
x = 1 to x = 5 with right end points for
n = 2 is ________.

• A) 28.
• B) 76.
• C) 50.
• D) 30.

The indefinte integral of ‘cos (x) – sin (x)’ is………………

• A) cos (x) .sin (x) +c
• B) -cos (x) + sin (x) +c
• C) -cos (x) – sin (x)+c
• D) cos (x) +sin (x)+c

Which of the following is the regular partition of the interval [0,2]?

• A) [0,0.5],[0.5,1.25],[1.25,1.50],[1.50,2]
• B) [0,0.50],[0.50,1],[1,1.50],[1.50,2]
• C) [0,0.25],[0.25,1],[1,1.50],[1.50,2]
• D) [0,0.25],[0.25,0.75],[0.75,1.25],[1.25,2]

If f(x) =cos(x) then mean value theorem can be applied on this in the interval (pi/2, 3pi/2)

• A) True
• B) False
• C)
• D)

What will be the value of summation of k3 where k goes from 9 to 10

• A) 1729
• B) 1998
• C) 1081
• D) 1100