The result of the integral [tex]\int\limits^2_0 {(2\cdot x + 3)\cdot f''(x)} \, dx[/tex] based on the incomplete graph of [tex]f'[/tex] 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: [tex](x_{1}, y_{1}) = (0, 1)[/tex], [tex](x_{2}, y_{2}) = (1, 2.718)[/tex], [tex](x_{3}, y_{3}) = (2, 7.389)[/tex]
Then, we form the following system of linear equations:
[tex]c = 1[/tex] (1)
[tex]a + b + c = 2.718[/tex] (2)
[tex]4\cdot a + 2\cdot b + c = 7.389[/tex] (3)
The solution of this system is: [tex]a = 1.4765[/tex], [tex]b = 0.2415[/tex], [tex]c = 1[/tex]. Then, the expression of the first derivative is:
[tex]f'(x) = 1.4765\cdot x^{2} + 0.2415\cdot x + 1[/tex] (4)
And the second derivative is:
[tex]f''(x) = 2.953\cdot x + 0.2415[/tex] (5)
Then, we have the following integral equation:
[tex]I = \int\limits^{2}_{0} {(2\cdot x + 3)\cdot (2.953\cdot x + 0.2415)} \, dx[/tex]
[tex]I = \int\limits^{2}_{0} {(5.906\cdot x^{2}+9.342\cdot x +0.7245)} \, dx[/tex]
[tex]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[/tex]
[tex]I = \frac{5.906}{3}\cdot (2^{3}-0^{3}) + \frac{9.342}{2}\cdot (2^{2}-0^{2}) + 0.7245\cdot (2-0)[/tex]
[tex]I = 35.882[/tex]
This choice that is closest to this result is C. [tex]\blacksquare[/tex]
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