Question

the table above gives values of the differentiable functions f and g and their derivatives at x = 0. if $h(x)=\frac{f(x)}{g(x)}$, what is the value of h(0)? a $\frac{8 - 3pi}{4}$ b $\frac{3pi - 8}{4}$ c $\frac{4}{pi}$ d $\frac{2 - 3pi}{4}$ e $\frac{8 - 3pi}{4}$

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $h(x)=\frac{f(x)}{g(x)}$, then $h^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{[g(x)]^{2}}$.

Step2: Substitute $x = 0$

We know that $f(0) = 3$, $f^{\prime}(0)=4$, $g(0)=2$, and $g^{\prime}(0)=\pi$. Substituting these values into the quotient - rule formula for $h^{\prime}(x)$ at $x = 0$, we get $h^{\prime}(0)=\frac{f^{\prime}(0)g(0)-f(0)g^{\prime}(0)}{[g(0)]^{2}}$.

Step3: Calculate the value

$h^{\prime}(0)=\frac{4\times2 - 3\times\pi}{2^{2}}=\frac{8 - 3\pi}{4}$.

Answer:

A. $\frac{8 - 3\pi}{4}$