Answer
The equation of the tangent line is $y=2x+5$.
Work Step by Step
The slope $m$ of the tangent line of the curve $f(x)$ at $A(a,b)$ is also the slope of the curve at that point, which is calculated by $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$y=f(x)=4-x^2\hspace{1cm}A(-1,3)$$
1) Find the slope $m$ of the tangent: $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$m=\lim_{h\to0}\frac{4-(a+h)^2-(4-a^2)}{h}=\lim_{h\to0}\frac{-(a+h)^2+a^2}{h}$$
$$m=\lim_{h\to0}\frac{-a^2-2ah-h^2+a^2}{h}=\lim_{h\to0}\frac{-2ah-h^2}{h}$$
$$m=\lim_{h\to0}(-2a-h)=-2a-0=-2a$$
$$m=-2\times(-1)=2$$
2) Find the equation of the tangent line at $A(-1,3)$:
The tangent line would have this form: $$y=2x+m$$
Substitute $A(-1,3)$ here to find $m$: $$2\times(-1)+m=3$$ $$-2+m=3$$ $$m=5$$
So the equation of the tangent line at $A$ is $y=2x+5$.
