Question 1: What is the fourth derivative of [tex]y=2x^5−4x^4+3x^3[/tex]?

A. [tex]24(9x−4)[/tex]
B. [tex]36(6x−5)[/tex]
C. [tex]48(5x−2)[/tex]
D. [tex]60(4x−5)[/tex]
E. [tex]72(3x−8)[/tex]

Question 2: What is the best approximation of [tex]\int_{0}^{2}(e^{2x}+2)\,dx[/tex] found by MRAM with 4 subintervals?

A. 4.28
B. 4.36
C. 4.47
D. 4.53
E. 4.62

The fourth derivative of the given polynomial doesn't match with any of the provided options. For the approximation of integral using MRAM, specification of the intervals is essential for calculation.

Explanation:

The fourth derivative of y=2x^5−4x^4+3x^3 is achieved by differentiating four times. Starting off with the function, the first derivative is: y' = 10x^4 - 16x^3 + 9x^2. Differentiating the first derivative to obtain second derivative we have: y'' = 40x^3 - 48x^2 + 18x. Further, the third derivative, y''' = 120x^2 - 96x + 18. The fourth derivative, y'''' is 240x - 96. However, this doesn't match with any of the options provided.

For the second part of the question, using MRAM (Midpoint Rectangle Approximation Method) with 4 subintervals to find the approximation of ∫−2^0(e^2x+2)dx, we would need more information. MRAM requires specification of the intervals, and without this information, I can't provide an accurate approximation.

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