For the area under the curve f(x) = x+1 from x = 1 to x = 5 with mid points approximations for n = 2, what will be the values of xk* ?

• A) 1 and 3
• B) 2 and 4
• C) 3 and 4
• D) 2.5 and 3.5

The derivative of the area under the continuous function f(x)= 2+3Sinx in the interval[-pi,pi] is---------

• A) 2+3Sinx
• B) 2-3Sinx
• C) 2-3Cosx
• D) 2+3Cosx

If a function f has a relative extrema at a point c, then c is a critical point for f .

• A) False
• B) True
• C)
• D)

For the area under the curve f(x) = 2x from x = 0 to x = 12 with mid points approximations for n = 3, what will be the values of xk* ?

• A) 3, 7 and 11
• B) 2, 6 and10
• C) 2, 5 and 8
• D) 1, 5 and 9

\[{\text{The integral }}\int {\cos (5x)\,dx\,} {\text{will be equal to ?}}\]

• A) \[\frac{{\sin (5x)}}{5} + c\]
• B) \[5\sin (5x) + c\]
• C) \[{\text{None of these}}\]
• D) \[ - \frac{{\sin (5x)}}{5} + c\]

\[\text{If a function }f\text{ has a relative extrema at }{{x}_{0}}\text{, then}\]

• A) \[\text{either}\,{f}'\left( {{x}_{0}} \right)=0\,\text{or }f\text{ is not differentiable at }{{x}_{0}}\]
• B) \[{f}'\left( {{x}_{0}} \right)\le 0\]
• C) \[f\text{ is differentiable at }{{x}_{0}}\]
• D) \[{f}'\left( {{x}_{0}} \right)>0\]

For the application of mean value theorem on f( x ) =x^3-3x^2-2x ; [0,2], which of the following is true?

• A) f( x ) is continuous over [ 0, 2] and f(x) is differentiable over [ 0,2 )
• B) f( x ) is continuous over [ 0, 2) and f(x) is differentiable over ( 0,2 )
• C) f( x ) is continuous over [ 0, 2] and f(x) is not differentiable over ( 0,2 )
• D) f( x ) is continuous over [ 0, 2] and f(x) is differentiable over ( 0,2 )

1+2+3……….+1000 equals -------

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

\[{\text{For}}\,{\text{Rolle's}}\,{\text{Theorem,}}\,f\,{\text{is}}\,{\text{continuous}}\,{\text{on}}\,{\text{the}}\,{\text{interval __________}}{\text{.}}\]

• A) \[[a,b]\]
• B) \[(a,b)\]
• C) \[[a,b)\]
• D) \[(a,b]\]

\[{\text{The integral }}\int {\frac{{6x}}{{{{(3{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}\]

• A) \[ - \frac{1}{{{{(3{x^2} + 1)}^3}}} + c\]
• B) \[ - \frac{1}{{6x + 1}} + c\]
• C) \[ - \frac{1}{{3{x^2} + 1}} + c\]
• D) \[\frac{1}{{3{x^2} + 1}} + c\]

If x = 5 + 6 + . . . + 40, then x = ________.

• A) None of these.
• B) 810.
• C) 820.
• D) 850.

The critical value of the function y=100x – x^2 is

NOTE: x^n means ‘x’ to the power ‘n’

• A) x=0
• B) None of these
• C) x=50
• D) x=25

$${\text{What}}\,{\text{does}}\,{\text{the}}\,{\text{indefinite}}\,{\text{integral }}\int_{}^{} {f(x)} dx\,{\text{represent?}}$$

• A) $${\text{None}}\,{\text{of}}\,{\text{these}}$$
• B) $${\text{Curvature}}\,{\text{of}}\,{\text{the}}\,{\text{curve}}$$
• C) $${\text{Area}}\,{\text{under}}\,{\text{the}}\,{\text{curve}}$$
• D) $${\text{Families}}\,{\text{of}}\,{\text{antiderivative}}\,{\text{of}}\,{\text{the}}\,{\text{function }}f(x)$$

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

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

\[\begin{align} & \text{A function }f\text{ is said to have a relative minimum at }{{x}_{0}}\text{, if}\,\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\,\text{for all }x\text{ } \\ & \text{in some open interval containing }{{x}_{0}}. \\ \end{align}\]

• A) \[f\left( {{x}_{0}} \right)\le \text{ }f\left( x \right)\]
• B) \[f\left( {{x}_{0}} \right)\ge \text{ }f\left( x \right)\]
• C) \[f\left( {{x}_{0}} \right)=f\left( x \right)\]
• D) \[f\left( {{x}_{0}} \right)>\text{ }f\left( x \right)\]

Sum of n-terms of a series whose nth term is ‘n’ = 1/n+1.then what is the sum of the first two terms is -----

• A) 6
• B) 6/5
• C) 5/6
• D) 6/4

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

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

Why the equation: x^2 + 8 = 0 does not have approximate solution while using Newton's method?

• A) x^2 will always be nonnegative
• B) x^2 will always be negative
• C)
• D)

$${\text{The integral }}\int {\cos e{c^2}(3{x^2})\,.6x\,dx\,} {\text{will be equal to ?}}$$

• A) $$\sec (3{x^2}) + c$$
• B) $$- \cot (3{x^2}) + c$$
• C) $${\text{cot(3}}{{\text{x}}^2}) + c,$$
• D) $${\text{None of these}}$$

Maximum of the function f(x)=2x+7 occurs at

• A) None of these
• B) x=-2/7
• C) x=-7/2
• D) x=0