91Ó°ÊÓ

Let \(f(x)=m x+b,\) where \(m\) and \(b\) are constants. If \(\lim _{x \rightarrow 1} f(x)=-2\) and \(\lim _{x \rightarrow-1} f(x)=4\) , find \(m\) and \(b\)

Use the alternate definition \(\lim _{x \rightarrow 0} \frac{f(x)-f(a)}{x-a}\) to calculate the instantaneous rate of change of \(f(x)\) at the given point or value of \(x\) a. \(f(x)=-x^{2}+2 x+3,(-2,-5)\) b. \(f(x)=\frac{x}{x-1}, x=2\) c. \(f(x)=\sqrt{x+1}, x=24\)

A circular oil spill on the surface of the ocean spreads outward. Find the approximate rate of change in the area of the oil slick with respect to its radius when the radius is \(100 \mathrm{m}\).

Show that the rate of change in the volume of a cube with respect to its edge length is equal to half the surface area of the cube.

Show that, at the points of intersection of the quadratic functions \(y=x^{2}\) and \(y=\frac{1}{2}-x^{2},\) the tangents to the functions are perpendicular.