let $g$ be the function with first derivative $g(x)=sqrt{x^{3}+x}$ for $x > 0$. if $g(2)=-7$, what is the…

let $g$ be the function with first derivative $g(x)=sqrt{x^{3}+x}$ for $x > 0$. if $g(2)=-7$, what is the value of $g(5)$?\n(a) 4.402\n(b) 11.402\n(c) 13.899\n(d) 20.899

let $g$ be the function with first derivative $g(x)=sqrt{x^{3}+x}$ for $x > 0$. if $g(2)=-7$, what is the value of $g(5)$?\n(a) 4.402\n(b) 11.402\n(c) 13.899\n(d) 20.899

Answer

Answer:

B. 11.402

Explanation:

Step1: Use the fundamental theorem of calculus

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

Step2: Calculate the definite - integral

We use a calculator or software to evaluate $\int_{2}^{5}\sqrt{x^{3}+x}dx$. Using a graphing utility or integral - calculating software (such as a TI - 84 Plus calculator's numerical integration function fnInt), $\int_{2}^{5}\sqrt{x^{3}+x}dx\approx18.402$.

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

Since $g(5)=g(2)+\int_{2}^{5}\sqrt{x^{3}+x}dx$ and $g(2)=-7$, then $g(5)=-7 + 18.402=11.402$.