1. Consider a function ( f(x) ) which is continuous for all real values. It is concave for ( x < 0 ) and convex for ( x \geq 0 ). Given ( f'(x) = x^2 ) for ( x > 0 ), which of the following is the correct form of ( f(x) )?(A) ( x^3 - x ) for ( x > 0 ) (B) ( -x^3 ) for ( x < 0 ) (C) ( |x|^3 ) (D) None of the above 2. Given ( f(x) = x^3 - 3x^2 + 2x ), find the inflection points.(A) ( x = 0, 2 ) (B) ( x = 1 ) (C) ( x = 2 ) (D) None of the above 3. For ( f(x) = x^3 - 3x^2 + 2x ), at the inflection points, what is the curvature?(A) 0 (B) 1 (C) 2 (D) None of the above 4. Consider ( f(x) = x^2 ) on [0,1]. The partition ( P = {0, 1/4, 1/2, 1} ) is given. What is the norm of the partition?(A) ( 1/2 ) (B) ( 1/4 ) (C) ( 1/3 ) (D) 15. For the partition ( P = {0, 1/4, 1/2, 1} ) of ( f(x) = x^2 ), find the upper sum.(A) 0.5 (B) 0.375 (C) 0.75 (D) 0.31256. For ( f(x) = \sin(x) ) over [0, ( \pi )], what is the limit of the difference between the upper and lower sums as the norm of the partition tends to 0?(A) 0 (B) ( \pi ) (C) 1 (D) 27. For a convex function ( f(x) = e^x ) over [0,1], with partition ( P = {0, 1/n, 2/n, ..., 1} ), what is the norm of the partition?(A) ( 1/n ) (B) ( 1/(n+1) ) (C) 1 (D) ( n )8. For ( f(x) = e^x ) over [0, 1], as ( n ) tends to infinity, what is the limit of the upper sum?(A) ( e ) (B) 1 (C) 2 (D) ( e - 1 )9. Given the curve ( f(x) = \sqrt{x} ) from ( x = 0 ) to ( x = 1 ), what is the volume of the solid generated by revolving it around the x-axis?(A) ( \pi/4 ) (B) ( \pi/2 ) (C) ( \pi/8 ) (D) ( \pi )10. Given ( f(x, y) = x^2 + y^2 ) over [0, 1] × [0, 1], find the exact value of the double integral.(A) ( 2/3 ) (B) 1 (C) ( 4/3 ) (D) ( 1/2 )