Question

f(x)=\begin{cases}x + k, &\text{if }x>4end{cases}
a) k = 12
b) k = 20
find an equation for the tangent to the curve at the given point.

1. y=x^{2}-x, (4, 12)

a) y = 7x+20
b) y = 7x + 16

Explanation:

Step1: Find the derivative of the function

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

Step2: Evaluate the derivative at the given x - value

Substitute $x = 4$ into $y'$. So $y'(4)=2\times4 - 1=7$. The slope of the tangent line $m = 7$.

Step3: Use the point - slope form of a line

The point - slope form is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(4,12)$ and $m = 7$.
$y - 12=7(x - 4)$.

Step4: Simplify the equation

$y-12 = 7x-28$.
$y=7x - 16$.

Answer:

The equation of the tangent line is $y = 7x-16$.