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

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

For the area under the curve f(x) = x+2 from x = 2 to x = 4 with mid points approximations for n = 1, what will be the value of xk* ?

• A) 3
• B) 3.5
• C) 2
• D) 2.5

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

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

$${\text{The integral }}\int {\frac{{4x}}{{{{(2{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}$$

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

If ‘n’ goes from 1 to 3 and the summation of ‘na’ = definite integral of ‘1’ on closed interval [0,1], then the value of ‘a’=------------

• A) -6
• B) 1/6
• C) 6
• D) -1/6

$${\text{The integral }}\int {\sqrt {4x - 3} \,dx\,} {\text{will be equal to ?}}$$

• A) $$\frac{{{{(4x - 3)}^{\frac{3}{2}}}}}{6} + c$$
• B) $$\frac{{{{(4x + 3)}^{\frac{3}{2}}}}}{6} + c$$
• C) $${\text{None of these}}$$
• D) $$\frac{{{{(4x - 3)}^{\frac{3}{2}}}}}{3} + c$$

summation of (6) ;where ( j varies from 1 to 8) indicates to add ‘6’ to itself ……….. times.

• A) 8
• B) 11
• C) 9
• D) 10

\[\int {\tan x} dx = \_\_\_\_\_\_\_\_\_\_\_\_\_\_.\]

• A) \[{\sec ^2}x + C\]
• B) \[\ln \left| {\sec x} \right| + C\]
• C) \[\ln |\cos x| + C\]
• D) \[\ln \left| {\sin x} \right| + C\]

$$\int {\tan x} dx = \_\_\_\_\_\_\_\_\_\_\_\_\_\_.$$

• A) $$\ln \left| {\sin x} \right| + C$$
• B) $$\ln |\cos x| + C$$
• C) $${\sec ^2}x + C$$
• D) $$\ln \left| {\sec x} \right| + C$$

The approximate solution is possible to generate using Newton’s Method if ___________.

• A) the tangent line(at approximated points) must crosses the x - axis.
• B) the slope of tangent line(at any approximated point) is zero.
• C) None of these.
• D) the tangent line(at any approximated point) is parallel to x- axis.

If f(x) = x^5 + x, then which of the following is true about it.

• A) Its anti – derivative is 5x^4 + 1.
• B) Its anti – derivative is x^5/5 + 1.
• C) Its anti – derivative is x^6/6 + x^2/2 + 6.
• D) None of these.

If $f''(x) > 0\,\forall \,x \in I$, then the graph of $f$ is___________ on $I$.

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

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

• A) 6, 9 and 12
• B) 0, 6 and 12
• C) 2, 7 and 12
• D) 4, 8 and 12

f(x)=x^3-3x, is increasing on the interval (0 ,infinity)

• A) True
• B) False
• C)
• D)

Integration of 5 with respect to x is…………

• A) 5x^2
• B) 5x
• C) 5
• D) x

Absolute minimum of the function f(x)=x in the semi open interval (0,2] is-------

• A) 0
• B) Undefined
• C) 1
• D) 2

Newton’s method is not applicable when ‘f’ is a …………

• A) Trignometric function
• B) Exponential function.
• C) Polynomial
• D) Constant function

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

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

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

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

If we subdivide the interval [1,6] into n equal parts, then what will be the length ∆x of each part?

• A) 1/n
• B) 6/n
• C) 5/n
• D) 7/n