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Answer 1

The result of the integral $\int\limits^2_0 {(2\cdot x + 3)\cdot f''(x)} \, dx$ based on the incomplete graph of $f'$ is approximately 35.943. (Choice C)

Determination of an integral based on a graph and a given expression

Based on the information given on the figure, we have a set of points which resembles a second order polynomial, whose expression can be found by the fact that coefficients can be found by know three distinct points: $(x_{1}, y_{1}) = (0, 1)$, $(x_{2}, y_{2}) = (1, 2.718)$, $(x_{3}, y_{3}) = (2, 7.389)$

Then, we form the following system of linear equations:

$c = 1$ (1)

$a + b + c = 2.718$ (2)

$4\cdot a + 2\cdot b + c = 7.389$ (3)

The solution of this system is: $a = 1.4765$, $b = 0.2415$, $c = 1$. Then, the expression of the first derivative is:

$f'(x) = 1.4765\cdot x^{2} + 0.2415\cdot x + 1$ (4)

And the second derivative is:

$f''(x) = 2.953\cdot x + 0.2415$ (5)

Then, we have the following integral equation:

$I = \int\limits^{2}_{0} {(2\cdot x + 3)\cdot (2.953\cdot x + 0.2415)} \, dx$

$I = \int\limits^{2}_{0} {(5.906\cdot x^{2}+9.342\cdot x +0.7245)} \, dx$

$I = 5.906\int\limits^{2}_{0} {x^{2}} \, dx + 9.342\int\limits^{2}_{0} {x} \, dx + 0.7245\int\limits^{2}_{0} \, dx$

$I = \frac{5.906}{3}\cdot (2^{3}-0^{3}) + \frac{9.342}{2}\cdot (2^{2}-0^{2}) + 0.7245\cdot (2-0)$

$I = 35.882$

This choice that is closest to this result is C. $\blacksquare$

To learn more on definite integrals, we kindly invite to check this verified question: brainly.com/question/22655212