 |
|
 | 1. The product and quotient rules give you the ability to symbolically calculate the
derivative of a wide range of functions. However, many calculators and almost every computer algebra system (CAS) can do this work for you. Discuss why you should learn these basic rules anyway. (Keep
Example 4.7 in mind.) |
| | 2.
Gottfried Leibniz is recognized (along with
Sir Isaac
Newton) as a co-inventor of calculus. Many of the fundamental methods and (perhaps more importantly) much of the notation of calculus are due to Leibniz. The product rule was worked out by Leibniz in 1675, in the form d(xy) = (dx)y+x(dy). His “proof, ” as given in a letter written in 1699, follows. “If we are to differentiate xy we write:
(x+dx)(y+dy)-xy = x dy+y dx+dx dy.
But here dx dy is to be rejected as incomparably less than x dy+y dx. Thus in any particular case the error is less than any finite quantity. ” Answer Leibniz' letter with one describingyour own “discovery” of the product rule for d(xyz). Use Leibniz' notation. |
| | 3. In
Example 4.2, we cautioned you against multiplying out the terms of the
derivative. To see one reason for this warning, suppose that you want to find solutions of the equation f ' (x) = 0. (In fact, we do this routinely in
Chapter 3.) Explain why having a factored form of f ' (x) is very helpful. Discuss the extent to which the product rule gives you a factored form. |
| | 4. Many students prefer the product rule to the quotient rule. Many computer algebra systems actually use the product rule to compute the
derivative of f (x)[g(x)]-1 instead of using the quotient rule on  (see exercise 26 below). Given the simplifications in problems like
Example 4.5, explain why the quotient rule can be preferable. |
| In exercises 5-24, find the derivative of each
function.
|
| 5. f (x) = (x2+3)(x3-3x+1)
|
| 6. f (x) = (x3-2x2+5)(x4-3x2+2)
|
| 7. f (x) = (3x+4)(x3-2x2+x)
|
| 8. f (x) = (x2-4)(x4+2x+1)
|
| 9. 
|
| 10. 
|
| 11.  | 12.  |
| 13.  | 14.  |
| 15.  | 16.  |
| 17.  | 18.  |
| 19.  | 20.  |
| 21.  | 22.  |
| 23.  | 24. 
|
| 25. Write out the product rule for the
function f (x)g(x)h(x). (Hint: Group the first two terms together.) Describe the general product rule: for n functions, what is the
derivative of the product f 1(x) f 2(x) f 3(x) f n(x) ? How many terms are there? What does each term look like? |
| 26. Use the quotient rule to show that the
derivative of (g(x))-1 is -g' (x)(g(x))-2. Then use the product rule to compute the
derivative of f (x)(g(x))-1. |
| In exercises 27-32, use the symbolic
differentiation feature on your CAS or calculator.
|
| 27. Repeat
Example 4.5 with your CAS. Does its answer have the same form as ours in the text, or is more like the form in exercise 26? Write out the “quotient rule” used by your CAS as completely as possible. |
| 28. Repeat
Example 4.6 with your CAS. If its answer is not in the same form as ours in the text, explain how the CAS computed its answer. |
| 29. Repeat exercise 17 with your CAS. If its answer is not in the same form as ours in the back of the book, explain how the CAS computed its answer. |
| 30. Repeat exercise 19 with your CAS. If its answer is not in the same form as ours in the back of the book, explain how the CAS computed its answer. |
| 31. Find the
derivative of  on your CAS. Compare its answer to  for x < 0. Explain how to get this answer and your CAS' answer, if it differs. |
| 32. Find the
derivative of  on your CAS. Compare its answer to 2. Explain how to get this answer and your CAS' answer, if it differs. |
| In exercises 33-36, find the
derivative of each function using the general product rule developed in exercise 25.
|
| 33. f (x) = x2/3(x2-2)(x3-x+1)
|
| 34. f (x) = (x+4)(x3-2x2+1)(3-2/x)
|
| 35. f (x) = (x+1)(x3+4x)(x5-3x2+1)
|
| 36. f (x) =  (x-2)(x+1)(x+5)
|
| 37. Suppose that for some toy the quantity sold, Q(t ) , decreases ata rate of 4%; explain why this translates to Q' (t ) = -0.04Q(t ). Suppose also that the price increases at a rate of 3%; write out a similar equation for P' (t ) in terms of P(t ).
The revenue for the toy is R(t ) = Q(t )P(t ). Substituting the expressions for Q' (t ) and P' (t ) into the product rule R' (t ) = Q' (t )P(t )+ Q(t )P' (t ) , show that the revenue decreases at a rate of 1%. Explain why this is “obvious. ” |
| 38. As in exercise 37, suppose that the quantity sold decreases at a rate of 4%. By what rate must the price be increased to keep a constant revenue? |
| 39. Suppose the price of an object is $20 and 20, 000 units are sold. If the price increases at a rate of $1.25 per year and the quantity sold increases at a rate of 2000 per year, at what rate will revenue increase? |
| 40. Suppose the price of an object is $14 and 12, 000 units are sold. The company wants to increase the quantity sold by 1200 units per year, while increasing the revenue by $20,000 per year. At what rate would the price have to be increased to reach these goals? |
| 41. A baseball with mass 0.15 kg and speed 45 m/s is struck by a baseball bat of mass m and speed 40 m/s (in the opposite direction of the ball's motion). After the collision, the ball has initial speed  m/s. Show that u'(m) > 0 and interpret this in baseball terms. Compare u'(1) and u'(1.2). |
| 42. In exercise 41, if the baseball has mass M kg at speed 45 m/s and the bat has mass 1.05 kg at speed 40 m/s, the ball's initial speed is  . Compute u'(M) and interpret its sign (positive or negative) in baseball terms. |
| 43. In
Example 4.8, it is reasonable to assume that the speed of the golf club at impact decreases as the mass of the club increases. If, for example, the speed of a club of mass m is  = 8.5/m m/s at impact, then the initial speed of the golf ball
is  . Showthat u'(m) < 0 and interpret this in golf terms. |
| 44. In
Example 4.8, if the golf club has mass 0.17 kg and strikes the ball with speed m/s, the ball has initial speed u(v) =  . Compute and interpret the
derivative u'(v). |
| 45. Suppose that F(x) = f (x)g(x) for infinitely differentiable functions f (x) and g(x) (that is, f ' (x) , f '' (x) etc. exist for all x). Show that F'' (x) = f '' (x)g(x)+2f ' (x)g' (x) + f (x)g'' (x). Compute F'' ' (x). Compare F'' (x) to the binomial formula for (a+b)2 and compare F'' ' (x) to the formula for (a+b)3. |
| 46. Given that (a+b)4 = a4+4a3b+6a2b2+4ab3+b4 , write out a formula for F(4)(x). (See exercise 45.) |
| 47. Use the product rule to show that if g(x) = [f (x)]2 and f (x) is differentiable, then g' (x) = 2f (x)f ' (x). This is an example of the chain rule, to be discussed in
Section 2.7. |
| 48. Use the result from exercise 47 and the product rule to show that if g(x) = [f (x)]3 and f (x) is differentiable, then g' (x) = 3[f (x)]2f ' (x). Hypothesize the correct chain rule for the
derivative of [f (x)]n. |
| 49. The relationship among the pressure P, volume V and temperature T of a gas or liquid is given by van der Waal's equation
|
|  for positive constants a, b, n and R. Solve the equation for P. Treating T as a
constant and V as the variable, find the critical point (Tc, Pc, Vc) such that P' (V) = P'' (V) = 0. (Hint: Don't solve either equation separately, but substitute the result from one equation into the other.) For temperatures above Tc , the substance can only exist in its gaseous form; below Tc , the substance is a gas or liquid (depending on the pressure and volume). For water, take R = 0.08206 l-atm/mo-K, a = 5.464 l 2-atm/mo2 and b = 0.03049 l/mo. Find the highest temperature at which n = 1 mo of water may exist as a liquid. (Note: your answer will be in degrees Kelvin; subtract 273.15 to get degrees Celsius.) |
 | 50. In many sports, the collision between a ball and a striking implement is central to the game. Suppose the ball has weight w and velocity before the collision, and the striker (bat, tennis racket, golf club, etc.) has weight W and velocity -V before the collision (the negative indicates the striker is moving in the opposite direction from the ball). The velocity of the ball after the collision will be  where the parameter c, called the coefficient of restitution, represents the “bounciness” of the ball in the collision. Treating W as the
independent variable (like x) and the other parameters as constants, compute the
derivative and verify that  since all parameters are non-negative. Explain why this implies that if the athlete uses a bigger striker (bigger W) with all other things equal, the speed of the ball increases. Does this match your intuition? What is doubtful about the assumption of all other things being equal? Similarly compute and interpret    and  (Hint: c is between 0 and 1 with 0 representing a dead ball and 1 the liveliest ball possible.) |
