Question

if $h(x)=4 - 2x^{3}$, find $h(2)$. use this to find the equation of the tangent line to the curve $y = 4 - 2x^{3}$ at the point $(2,-12)$. the equation of this tangent line can be written in the form $y=mx + b$ where $m$ is: and where $b$ is:

Explanation:

Step1: Find the derivative of $h(x)$

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, if $h(x)=4-2x^{3}$, then $h'(x)=\frac{d}{dx}(4)-2\frac{d}{dx}(x^{3})$. Since $\frac{d}{dx}(4) = 0$ and $\frac{d}{dx}(x^{3})=3x^{2}$, we have $h'(x)=-6x^{2}$.

Step2: Evaluate $h'(2)$

Substitute $x = 2$ into $h'(x)$. So $h'(2)=-6\times2^{2}=-24$.

Step3: Find the slope $m$ of the tangent line

The slope $m$ of the tangent line to the curve $y = h(x)$ at a point is equal to the value of the derivative at that point. Since $h'(2)=-24$, the slope $m=-24$.

Step4: Find the y - intercept $b$ of the tangent line

The equation of the tangent line is $y=mx + b$. We know that the line passes through the point $(2,-12)$ and $m=-24$. Substitute $x = 2$, $y=-12$ and $m=-24$ into $y=mx + b$: $-12=-24\times2 + b$. Solve for $b$:
\[

$$\begin{align*} -12&=-48 + b\\ b&=-12 + 48\\ b&=36 \end{align*}$$

\]

Answer:

$h'(2)=-24$
$m=-24$
$b = 36$