ap calculus bc mcq practice test b\n22. the table above gives values of f, f’, g, and g’ for selected values…

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int_{a}^{b}u\mathrm{d}v=uv|{a}^{b}-\int{a}^{b}v\mathrm{d}u$. Let $u = f(x)$ and $\mathrm{d}v=g^{\prime}(x)\mathrm{d}x$, then $\mathrm{d}u = f^{\prime}(x)\mathrm{d}x$ and $v = g(x)$. So $\int_{0}^{1}f(x)g^{\prime}(x)\mathrm{d}x=f(x)g(x)|{0}^{1}-\int{0}^{1}f^{\prime}(x)g(x)\mathrm{d}x$.

Step2: Evaluate $f(x)g(x)|_{0}^{1}$

$f(x)g(x)|_{0}^{1}=f(1)g(1)-f(0)g(0)$. From the table, $f(0) = 2$, $g(0)=-4$, $f(1) = 4$, $g(1)=3$. Then $f(1)g(1)-f(0)g(0)=4\times3 - 2\times(-4)=12 + 8=20$.

Step3: Substitute the known integral value

We know that $\int_{0}^{1}f^{\prime}(x)g(x)\mathrm{d}x = 5$. Substitute into the formula from Step1: $\int_{0}^{1}f(x)g^{\prime}(x)\mathrm{d}x=f(1)g(1)-f(0)g(0)-\int_{0}^{1}f^{\prime}(x)g(x)\mathrm{d}x$. So $\int_{0}^{1}f(x)g^{\prime}(x)\mathrm{d}x=20 - 5=15$.

Answer:

E. 15