If $f$ has a local maximum or minimum at c, and if $f'(c)$ exists, then_________.

• A) $f'(c) < 0$
• B) $f'(c) = 0$
• C) None of these
• D) $f'(c) > 0$

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) 2.5
• B) 2
• C) 3.5
• D) 3

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

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

Subdivide the interval [0, 1] into 2 equal parts, and then the width of each sub-interval will have length ………

• A) 1/2
• B) 1
• C) 0
• D) -1/2

Critical point of function f(x) =4x^2-4x+1 on closed interval [0,1] is x = ………

• A) 2
• B) 1/2
• C) 0
• D) 1,2

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

• A) True
• B) False
• C)
• D)

12+22+32+…+192 equals _____.

• A) 19(20)(31)/6
• B) 19(20)(37)/6
• C) 19(20)(21)/6
• D) 19(20)(39)/6

1+2+3+…+100 equals _____.

• A) 5050
• B) 5500
• C) 5055
• D) 5005

The indefinite integral of 5sinx is ……….

• A) -5cosx+c
• B) cosx/5+c
• C) -cosx/5+c
• D) 5cosx +c

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

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

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

• A) There is a point in the interval (-2, 2) where f(x) has a horizontal tangent
• B) There is no such point in the interval (-2, 2) where f(x) has a horizontal tangent.
• C) It is discontinuous on the interval [-2, 2].
• D) None of these.

Subdivide the interval [a, b] into 4 equal subintervals then the width of each subinterval is ------

• A) (b-2a)/4
• B) (b-a)/4
• C) (b-a)/2
• D) (2b-a)/4

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

• A) 5
• B) 6
• C) 4
• D) 3

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

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

The polynomial function f(x)=6x^2-30x+36 has the critical point over the real line is ………

• A) 2/5
• B) 5/2
• C) 2
• D) 5

Let f(x)=x^3 and a=-3 ,b=3 and n = no. of partition of [-3,3]= 4
Let the widths of first, second, third and fourth intervals are 2, 1, 1, 2 respectively. Then “Mesh size” of this partition of [-3,3] is-------

• A) -1
• B) 2
• C) 1
• D) 0

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

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

If f(x) = x^4, then which of the following is Not true about it.

• A) Its anti – derivative is x^5/5.
• B) Its anti – derivative is x^5 + 5.
• C) Its anti – derivative is x^5/5 + 2.
• D) Its anti – derivative is x^5/5 + 10.

If x = (4^2) + (5^2) + (6^ 2) + . . . + (30^2), then x = ________.

• A) 465.
• B) 9455.
• C) 9441.
• D) 400.

What is the estimated area under f(x) = 12 / x from x = 1 to x = 3 with mid points for n = 2?

• A) 12.8
• B) 18.0
• C) 13.1
• D) 10.5