1. The function \(A\) with formula \( A(r) = \pi r^2\) returns the area of a circle given its radius. What is a formula for the derivative \(\frac{\mathrm{d}A}{\mathrm{d}r}?\) This formula should look familiar. Is there a reason for this, or is it just a coincidence?
2. The function \(V\) with formula \( V(r) = \frac{4}{3}\pi r^3\) returns the volume of a sphere given its radius. What is a formula for the derivative \(\frac{\mathrm{d}V}{\mathrm{d}r}?\) Based on the familiar formula for \(A'\) that appeared in the previous question, speculate what \(V'\) could be a formula for.
3. Consider a sphere with radius increasing at a constant rate of 2in/min (imagine a balloon). At what rate is the volume of the sphere increasing the moment that the radius is 3in?
4. Consider a sphere with volume increasing at a constant rate of 5in³/min. At what rate is the radius of the sphere increasing the moment that the volume is 7in³?
5. A magic cube of jello, 3 mm on each side, is beginning to grow before your eyes. If a side-length of the cube is growing at a constant rate of 7 mm/s, at what rate is the volume changing the moment a side is 5 mm long?
6. Two cars are sitting next to each other in the desert, one pointing north and the other pointing east. At the same moment, they each take off in the direction they're pointed, the first car at 50 mph and the second car at 32 mph. At what rate is the distance between them increasing ½ hour after they take off?
7. Given a triangle with side-lengths \(A\) and \(B\) with angle \(\theta\) between those sides, the formula \(\frac{1}{2}AB\sin(\theta)\) gives the area of the triangle.
   1. Suppose \(A = 5\) ft and \(B = 2\) ft and the angle \(\theta\) is opening (increasing) at a rate of \(\frac{1}{4}\) radian-per-second. How fast is the area changing the moment the angle is \(\frac{2\pi}{3}?\)
   2. Suppose \(A = 5\) ft and \(B = 2\) ft and the area of the triangle is increasing at a rate of 1 ft²/s. At what rate is \(\theta\) changing at the moment \(\theta = \frac{\pi}{6}?\)
   3. Suppose \(A = 5\) ft and \(B = 2\) ft and the area of the triangle is increasing at a rate of 1 ft²/s. At what rate is \(\theta\) changing at the moment the area is 4 ft²?
   4. Suppose now that \(A = 5\) ft, but that \(B\) is increasing at a rate of \(3\) ft/s while \(\theta\) is increasing at a rate of \(\frac{1}{4}\) radian-per-second. At a certain moment in time, \(\theta = \frac{\pi}{6}\) and \(B = 8.\) At what rate is the area changing at this moment?