\[{\text{The}}\,{\text{value}}\,{\text{of}}\,c\,{\text{in}}\,{\text{Rolle's}}\,{\text{theorem}}\,{\text{for}}\,{\text{the}}\,{\text{function}}\,f(x) = {x^3} - 3x\,{\text{in}}\,{\text{the}}\,{\text{interval}}\,\left[ {0,\sqrt 3 } \right]{\text{.}}\]

• A) \[\frac{2}{3}\]
• B) \[ - 1\]
• C) \[\frac{1}{3}\]
• D) \[1\]

The estimated area under f(x) = 12 / x from
x = 1 to x = 3 with right end points for
n = 2 is ________.

• A) 10
• B) 18.
• C) None of these.
• D) 20.

For the area under the curve f(x) = x+1 from x = 1 to x = 3 with mid points approximations for n = 1, what will be the value of xk* ?

• A) 2.5
• B) 2
• C) 1.5
• D) 1

12+22+32+…+152 equals _____.

• A) 15(16)(30) / 6
• B) 15(16)(29) / 6
• C) 15(16)(17) / 6
• D) 15(16)(31) / 6

13+23+33+…+153 equals _____.

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

If f (x) = x^4 is defined on the interval
[-2, 2], then which of the following is true about it.

• A) Its relative maximum value exists at 2.
• B) None of these.
• C) Its absolute minimum value exists at -2
• D) Its absolute maximum value exists at 2.

The anitderivative of the function f(x)=2-7Cosx, is -------

• A) 2x-7Sinx+100
• B) 2x-7Sinx-100
• C) All choices are true
• D) 2x-7Sinx+C

For the area under the curve f(x) = x+2 from x = 2 to x = 8 with left end points approximations for n = 3, what will be the values of xk* ?

• A) 2, 6 and 8
• B) 3, 5 and 7
• C) 2, 4 and 6
• D) 4, 6 and 8

If the closed interval [-10,x] is divided into ‘20’ equally spaced subintervals each of which having the width equals to ‘1’ unit then the value of ‘x’ is --------

• A) 20
• B) 0
• C) 10
• D) 30

Summation of ‘kx’ where k goes from 1 to 5 equals

• A) 15k
• B) 15x
• C) None of these
• D) 55

$\begin{align} & \text{If a function }f\text{ is twice differentiable at a stationary point }{{x}_{0}}\text{ and }{f}''({{x}_{0}})>0, \\ & \text{then }f\text{ has relative }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ at }{{x}_{0}}. \\ \end{align}$

• A) b. Minimum
• B) d. None of these
• C) a. Maximum
• D) c. Both a and b

If f (x) = 2 x + 7 is defined on the interval
[2, 4), then which of the following is true about it.

• A) It has only absolute minimum value.
• B) It has both absolute maximum and minimum values.
• C) It has only absolute maximum value.
• D) None of these.

What will be the value of summation of k3 where k goes from 3 to 3?

• A) 9
• B) 3
• C) 1
• D) 27

In [0,pi],if the area under the curve f(x)=Sinx is divided into ‘3’ equally spaced rectangles of width ‘pi/3’ then by choosing middle point for height, the area of middle rectangle is ---------.

• A) pi/4
• B) pi/6
• C) pi/3
• D) pi/2

What will be the sigma notation for 32+52+72+92?

• A) summation of (2k-1)2 where (k varies from 1 to 4)
• B) summation of (2k2+1) where (k varies from 3 to 6)
• C) summation of (k2) where (k varies from 3 to 9)
• D) summation of (2k+1)2 where (k varies from 1 to 4)

If $f'$ changes from negative to positive at c, then $f$ has a _______________ at c.

• A) constant
• B) local maximum
• C) local minimum
• D) None of these

While solving the equation, 10x-(e^x)=0 by using Newton method, if x=0 is the initial approximate solution then the next approximation will be-----

• A) -1/9
• B) -1/10
• C) 1/9
• D) 1/10

Newton’s method uses the …………… to approximate the root.

• A) Tangent line
• B) Normal line
• C) Secant line
• D) None of these

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

• A) 24
• B) 10
• C) 8
• D) 18

2(13)+2(23)+2(33)+…+2(153) equals _____.

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