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

If ‘k’ goes from zero to ‘n’ in the summation of ‘Sink’ then which of the following will be other expression for this given summation such that its lower limit changes to ‘5’?

• A) Summation ‘Sin(k-5)’ while ‘k’ goes from ‘5’ to ‘n-5’
• B) Summation ‘Sin(k+5)’ while ‘k’ goes from ‘5’ to ‘n+5’
• C) Summation ‘Sin(k-5)’ while ‘k’ goes from ‘5’ to ‘n+5’
• D) Summation ‘Sin(k+5)’ while ‘k’ goes from ‘5’ to ‘n-5'

If f(x)=Cot(x) then mean value theorem can be applied to it on the interval (0,2pi)

• A) True
• B) False
• C)
• D)

If f(x) = Cos(x) + Sin(x) + x, then which of the following is NOT true about it.

• A) Its anti – derivative is Sin(x) – Cos(x) + x^2/2 + 4.
• B) None of these.
• C) Its anti – derivative is Sin(x) – Cos(x) + x^2/2 + 5.
• D) Its anti – derivative is -Sin(x) – Cos(x) + x^2/2 + 4.

3(12)+3(22)+3(32)+…+3(152) equals _____.

• A) 15(16)(31)/2
• B) 15(16)(30)/2
• C) 15(16)(29)/2
• D) 15(16)(17)/2

Summation of 2 where sum ranges from 0 to 10 equals 20.

• A) False
• B) True
• C)
• D)

Integral of 5^2 is

NOTE: x^n means ‘x’ to the power ‘n’

• A) None of these
• B) 25x
• C) (1/3)5^3
• D) 10

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) 1
• C) 0
• D) 2

If ‘n’ goes from 1 to any large ODD number then the summation of ‘(-1)^n’ = ---------

• A) -1
• B) that specific large ODD number
• C) 1
• D) Zero

What is the estimated area under f(x) = x from x = 0 to x = 3 with left end points for n = 3?

• A) 3
• B) 6
• C) 5
• D) 4

The area of a rectangle can be found by simply …………its dimensions.

• A) multiplying
• B) subtracting
• C) dividing
• D) adding

What will be the value of summation of k2 where k goes from 11 to 12

• A) 281
• B) 212
• C) 265
• D) 244

If x =(1+2+3+...+10)+{(1^3)+(2^3)+(3^3)+...+ (10^3)}, then x = _______.

• A) 55.
• B) 3025.
• C) 3080.
• D) 110.

Integral of (5/2)cos(x) is

• A) (5/2)tan(x)+C
• B) (5/2)sin(x)+C
• C) None of these
• D) (3/2)csc(x)+C

Which of the following is the sum of 2k+1 where k goes from 0 to 2?

• A) 8
• B) 14
• C) 12
• D) 10

If f(x) = Sin(x) + x, then which of the following is true about it

• A) Its anti – derivative is –Cos(x) + x^2/2 + 10.
• B) Its anti – derivative is Cos(x) + x^2/2 + 10.
• C) None of these.
• D) Its anti – derivative is Cos(x) + 1.

Which of the following is the absolute minima of the function: f(x)=-x in the interval [-1,1]?

• A) 0
• B) 1
• C) 0.5
• D) -1

Which of the following is the sum of (2k+1) where k goes from 3 to 5?

• A) 27
• B) 19
• C) 21
• D) 16

If ‘n’ goes from 1 to 4 and the summation of ‘na’ =Maxima of (e^x) in the interval[-e,0], then the value of ‘a’ is -----------

• A) 1/10
• B) -1/10
• C) -10
• D) 10

If f (x) = x^3 is defined on the interval [1, 3], then which of the following is true about it.

• A) None of these.
• B) Its absolute minimum value exists at the critical point.
• C) Its relative minimum value exists at the critical point.
• D) Its relative minimum value does not exist at the critical point