5 mark for review let g be the function with first derivative g(x) = √(x³ + x) for x > 0. if g(2) = -7, what…

Explanation:

Step1: Use the fundamental theorem of calculus

By the fundamental theorem of calculus, $g(5)-g(2)=\int_{2}^{5}g^{\prime}(x)dx$. So $g(5)=g(2)+\int_{2}^{5}g^{\prime}(x)dx$, where $g^{\prime}(x)=\sqrt{x^{3}+x}$ and $g(2)= - 7$.

Step2: Calculate the definite - integral

We need to calculate $\int_{2}^{5}\sqrt{x^{3}+x}dx$. Using a calculator (such as a graphing calculator with integral - calculation capabilities) to evaluate the definite integral $\int_{2}^{5}\sqrt{x^{3}+x}dx\approx18.402$.

Step3: Find the value of $g(5)$

Substitute the values into the formula $g(5)=g(2)+\int_{2}^{5}g^{\prime}(x)dx$. Since $g(2)=-7$ and $\int_{2}^{5}g^{\prime}(x)dx\approx18.402$, then $g(5)=-7 + 18.402=11.402$.

Answer:

B. 11.402