Calculus AB Independent Study Chain Rule Practice Study Sheet Note: The point here is just to practice with the chain rule. You need to have each part in the derivative perfect in order to count it as correct-there are no little forgivable errors here. On the other hand, there may be times when you make different algebraic choices from ours. As long as your answer is equivalent to ours (you can use algebra to get from one answer to the other), it's fine. Do not simplify these; simplification is a waste of time and a great place to make errors. Keeping track of your units is a really important part of these problems. The units can help you set up a problem or check your answer. Plus, you may have points taken off for neglecting your units on an exam question. 1. Air is being pumped into a spherical weather balloon. The radius (in feet) of the balloon at any time t 2 0 is given by the (differentiable) function r(t), where t is in minutes. The volume (in cubic feet) of the balloon is a function of the radius, denoted by V(r). A. Explain what each of the derivatives dr , du, ar , and represent. Be sure to include the units for each of the derivatives. B. Express du in terms of dr and r. 2. The length-mass relationship for Pacific halibut is well represented by the equation M = 10.375L', where M is mass in kilograms and L is the length in meters. In addition, the formula .18(2 -L) represents the rate of growth in length, , w AL , where t is time in years. A. Describe what each of the derivatives am and aM represent. Be sure to include the units for each of the derivatives. B. Find AM in terms of L. C. Find the rate of growth (with respect to time) of the mass of a Pacific halibut at the instant that it has a mass of 15 kilograms. 3. For each of the following, express both y and @ in terms of x. For example, given y = u, u = 2x - 5, you'd write y = u2 = (2x -5)2, and dy du dx dy du = (2u) (2) = 4u = 4(2x -5). A. y = u' ; u = 3x2+2x B. y = u 2 ; u = sinx C. y = sinu; u = -2cosx D. y = usecu; u = 3x