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1. Match the functions in Column A with the corresponding (signed) area between its graph and the interval $[−1,1]$ on the X-axis in column B and the images of their graphs and the highlighted region corresponding to the area computed in Column C, given in Table M2W3G1.

*100 point*

2. A cylinder of radius $x$ and height $2h$ is to be inscribed in a sphere of radius $R$ centered at O as shown in Figure M2W3G1

The volume of such a cylinder is given by $V=2πx_{2}h$ and the surface area of the outer curved surface is given by $S=4πxh$. Choose the set of correct options.

*1000 point*

3. Which of the curves in the following figures enclose a negative area on the $X$ axis in the interval $[0,1]$?

Suppose $f_{1}(x)=x_{3}$ and $f_{2}(x)=x$ denote the profits of Company A and Company B, respectively, throughout 1 year (the beginning of the year is denoted by $x=0$ and the ending denoted by $x=1$). The predicted profits of Company A and Company B in the same year are given by the functions $g_{1}(x)=x$ and $g_{2}(x)=e_{x}$, respectively. The curves represented by the functions $f_{1}$ and $g_{1}$ are shown in Figure M2W3G2, and the curves represented by the functions $f_{2}$ and $g_{2}$ are shown in Figure M2W3G3.

4. What will the absolute difference between the minimum values of $f_{2}$ and $g_{2}$ in the interval $[0,1]$ be?

### Accepted Answers:

*167 point*

*1866 point*

### Accepted Answers:

6. Let $f(x)=x_{3}−3x+11$. What is the local minimum value of $f$ attained at a critical point?

### Accepted Answers:

*1999 point*

7. Let $f(x)=38x_{2}+640$, $0≤x≤6$. The estimated area obtained by dividing the interval into 3 sub-intervals of equal length and the left end points of the sub-intervals for height of the rectangles is (in square units)

### Accepted Answers:

*199 point*

Let

$f(x)={−2x+9x_{2}0≤x≤1010<x≤20$

8. What is the global minimum of $f$ on $[0,20]$

### Accepted Answers:

*1 point*

9. If $x−y=10$, find the least value of $2xy$.

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