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

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

If the function is continuous on the closed interval [a,b], then the function has …………

• A) Only minimum value on the [a,b]
• B) Only minimum value on the [a,b]
• C) Only maximum value on the [a,b]
• D) Both maximum and minimum values on the [a,b]

\[{\text{The integral }}\int {{{\left( {{x^2} + 1} \right)}^{\frac{5}{2}}}.\,x\,dx\,} {\text{will be equal to ?}}\]

• A) \[{\left( {{x^2} + 1} \right)^{\frac{7}{2}}} + c\]
• B) \[\frac{{{{\left( {{x^2} + 1} \right)}^{\frac{7}{2}}}}}{7} + c\]
• C) \[{\text{None of these}}\]
• D) \[\frac{{2{{\left( {{x^2} + 1} \right)}^{\frac{7}{2}}}}}{7} + c\]

Which of the following is the sum of (2k-1) where k goes from 0 to 2?

• A) 4
• B) -2
• C) -1
• D) 3

If we subdivide the interval [2,4] into n equal parts, then what will be the length ∆x of each part?

• A) 8/n
• B) 1/n
• C) 2/n
• D) 6/n

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

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

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

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

Integral of 3sec(x)tan(x) is

NOTE: x^n means ‘x’ to the power ‘n’

• A) 3sec(x)+C
• B) None of these
• C) (3/2)sec^2(x)+C
• D) 3tan(x)+C

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

• A) 0, 5 and 10
• B) 0, 3 and 6
• C) 0, 6 and 12
• D) 0, 4 and 8

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

\[\text{The vertical asymptotes of the function }f(x)=\frac{{{x}^{2}}-2x+1}{x(x-2)}\text{ are}\]

• A) 0, 2
• B) 0, 1
• C) 1, 2
• D) 1, -1

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

• A) 24
• B) 8
• C) 18
• D) 10

If [-8,8] is subdivided into ‘16’ equally spaced subintervals, then the MIDDLE point of 8th
sub-interval will be--------.

• A) 0.5
• B) 1.5
• C) -0.5
• D) 2.5

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

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

Which of the following will be left end points if the interval [-2,2] is divided into 4 equal subintervals.

• A) -2,-1,0,1
• B) None of these
• C) -1,0,1,2
• D) -2,-1,1,2

The dimensions of a rectangle are given to be 8ft by 12ft. The perimeter of rectangle will be…………..

• A) 30 feet
• B) 60 feet
• C) 40 feet
• D) 50feet

$$\text{If }{f}'(x)={{x}^{2}}-1.\text{ Then the critical points of the function }f\text{ are}$$

• A) 1, 2
• B) 0, 1
• C) 1, -1
• D) 0, 2

\[{\text{The integral }}\int {{{\sec }^2}(2{x^2})\,.4x\,dx\,} {\text{will be equal to ?}}\]

• A) \[\tan (2{x^2}) + c\]
• B) \[\sec (2{x^2}).\tan (2{x^2}) + c\]
• C) \[\tan (2x) + c\]
• D) \[{\text{se}}{{\text{c}}^2}(2x) + c\]

\[{\text{The}}\,{\text{value}}\,{\text{of}}\,c\,{\text{in}}\,{\text{Rolle's}}\,{\text{Theorem}}\,{\text{for}}\,{\text{the}}\,{\text{function}}\,f(x) = {e^x}\sin x,\,\,x \in [0,\pi ]\,{\text{is}}\_\_\_\_\_\_\_\_\_\_.\]

• A) \[\frac{{3\pi }}{2}\]
• B) \[\frac{{3\pi }}{4}\]
• C) \[\frac{\pi }{4}\]
• D) \[\frac{\pi }{6}\]

\[{\text{In}}\,{\text{the}}\,{\text{notation:}}\,\int_a^b {f(x)} dx{\text{,}}\;f(x)\,{\text{is}}\,{\text{called___________}}{\text{.}}\]

• A) \[{\text{Integration}}\]
• B) \[{\text{None}}\,{\text{of}}\,{\text{these}}\]
• C) \[{\text{Integrand}}\]
• D) \[{\text{Differential}}\]