Question

find (a) the slope of the curve at the given point p, and (b) an equation of the tangent line at p

$y = 9x^{2}+1$, $p(4,145)$

(a) the slope of the curve at $p(4,145)$ is
(type an integer or a decimal )

Explanation:

Step1: Find the derivative of the function

The derivative of $y = 9x^{2}+1$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $y'=\frac{d}{dx}(9x^{2}+1)=18x$.

Step2: Evaluate the derivative at the x - coordinate of point P

We want to find the slope of the curve at $x = 4$. Substitute $x = 4$ into $y'$. So $y'(4)=18\times4 = 72$.

Step3: Find the equation of the tangent line

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(4,145)$ and $m = 72$.
Substitute these values into the formula: $y-145 = 72(x - 4)$.
Expand the right - hand side: $y-145=72x-288$.
Add 145 to both sides to get the equation of the tangent line: $y=72x - 143$.

Answer:

(a) 72
(b) $y = 72x-143$