Chapter 14.4, Problem 13AYU

To determine

To find: The slope of the tangent to the function $y = f\left(x\right)$ at the indicated point.

Answer to Problem 13AYU

12

Explanation of Solution

Given:

$f\left(x\right) = 3x^2$

Point: $\left(2, 12\right)$

Formula used:

The slope of the tangent to the curve $y = f\left(x\right)$ is the derivative of $f\left(x\right)$ w.r.to $x$ ie $\frac{dy}{dx}$.

Calculation:

Here, $y = f\left(x\right) = 3x^2$

Slope of the tangent is given by;

$\frac{dy}{dx} = \frac{d}{dx}\left(3x^2\right)$

$\frac{dy}{dx} = \mathrm{3 . }\frac{d}{dx}\left(x^2\right)$

$\frac{dy}{dx} = \mathrm{3 . }\left(2x\right)$

$\frac{dy}{dx} = 6x$

At the point $\left(2, 12\right)$; $\frac{dy}{dx} = 6\left(2\right) = 12$

Hence the slope of the tangent to the curve $y = f\left(x\right) = 3x^2$ at $\left(2, 12\right)$ is 12.

The graph is as follows: