Question

1. find the equation of the tangent line to the graph of $f(x)=4 - x^{2}$ at the points (-1,3)

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=4 - x^{2}$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=-2x$.

Step2: Evaluate the derivative at the given x - value

Substitute $x=-1$ into $f'(x)$. So $f'(-1)=-2\times(-1) = 2$. The value of the derivative at $x = - 1$ is the slope $m$ of the tangent line, so $m = 2$.

Step3: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(-1,3)$ and $m = 2$.
Substitute these values: $y - 3=2(x+1)$.

Step4: Simplify the equation

Expand the right - hand side: $y - 3=2x + 2$.
Add 3 to both sides to get the equation in slope - intercept form: $y=2x+5$.

Answer:

$y = 2x + 5$