2.2. THE PRODUCT AND QUOTIENT RULES 179. P dv dt + V dp. dt.

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1 22 THE PRODUCT AND QUOTIENT RULES 179 Thus, using the Product Rule, we find that dt = 1 k P dv + V dp At t = t 0, we obtain dt = 1 [(100, 000)(0005) + (002)( 100)] = 6225 K/s t0 8 Hence, at time t 0, the temperature is increasing at a rate of 6225 K/s Example 2211 In Example 1413, we looked at Newton s 2nd Law of Motion: the net force acting on an object equals the (instantaneous) rate of change, with respect to time, of the momentum of the object, when the mass is constant or when mass is added or subtracted with zero velocity If F denotes the net force, m the mass, v the velocity, a the acceleration, and t the time, Newton s 2nd Law, combined with the Product Rule, tells us F = d dv (mv) = m + v dm = ma + v dm If the mass is constant, which is the usual case, then dm/ equals 0, and the formula above collapses to the familiar F = ma However, if the mass is changing with time, then the dm/ can be very important See, for instance, Exercise 44 in this section 221 Exercises Find the first and second derivatives of the rational functions in Exercises 1 through 8 1 f(x) = x x 1 2 f(x) = x 1 x

2 180 CHAPTER 2 BASIC RULES FOR CALCULATING DERIVATIVES 3 f(x) = 2x2 +15x 2 3x 3 +2x 4 f(x) = 2x 1 x 2x + 1 x 5 f(x) = 3x2 + x 4 + x x f(x) = x + a x + b 7 f(x) = 4 x 2 +2 (x 4)2 8 f(x) = (either expand the numerator and denominator, or rewrite each of them (x 3) 2 as a product) Find the critical points of each of the functions in Exercises 9 through 11, and describe the intervals on which the function is increasing and those on which it is decreasing 9 f(x) =(2x + 8)( 3x + 6) 10 f(x) = x2 2x 1 2x 5 11 f(x) =(x 1)(x 3 +3x 2 +7x + 19) In Exercises 12-15, determine if the statement is true or false If it is true, what theorem(s) supports the conclusion? If it is false, present a counterexample 12 If f and g are di erentiable at x, then the function fg is di erentiable at x 13 If f and g are functions such that the function fg is di erentiable at x, then at least one of f or g must be di erentiable at x 14 If f and g are di erentiable at x, then the quotient f/g is di erentiable at x 15 If f and g are di erentiable at x and g(x) 6= 0, then the quotient f/g is continuous at x 16 What technique (or trick ) is used in the proof of the Product Rule? 17 Generalize the Product Rule to the product of 3 functions, ie, if f = f 1 f 2 f 3,findf 0,in terms of f 1, f 2, f 3, and their individual derivatives Can you see how this generalizes to a product of n functions?

3 22 THE PRODUCT AND QUOTIENT RULES Use your formula for the derivative of a product of n functions from the previous problem to determine the derivative of the following function: f(x) =(2x + 5) n 19 Let k(x) = f(x) Assume f and g are di erentiable, and that f and g are everywhere g(x) f non-zero Show that k 0 0 g 0 = k f g 20 Suppose the cost of producing n cars is p n dollars a Write a formula for the cost per car of producing n cars b Use the Quotient Rule to determine the rate of change of the cost per car function In Exercises 21 through 23, give an equation for the tangent line to the graph of y = f(x) at the given point Leave your answers in point-slope form 21 f(x) = ax + b,(0, b/d) Assume d 6= 0 cx + d 22 f(x) = (x + 1)3 (x + 2) 3,( 1, 0) x 23 f(x) = 1,(1, 1/2) 1+xn 24 Let g n (x) = xn 1 x 1 What is g0 n(1/2)? What is the limit as n!1of gn(1/2)? 0 Is there another way to see this? 25 Assume that f and g are di erentiable Derive expressions for d 2 dx 2 [f(x)g(x)] and d 3 dx 3 [f(x)g(x)] 26 Assume that f and g are di erentiable, and that g is non-zero Derive an expression for d 2 apple f(x) dx 2 g(x) 27 Recall that the magnitude F of the force of gravity between two masses M and m is given by F = GMm/r 2,wherer is the distance between the two masses and G is the universal gravitational constant Suppose that one object is increasing in mass and that the distance between the two objects is increasing Let M = 1000(1 + t 3 ) and r(t) =t 2 Assume m and G are constant and calculate F 0 (t)

4 182 CHAPTER 2 BASIC RULES FOR CALCULATING DERIVATIVES 28 Assume f and g are di erentiable, and that g(x) > 0, for all x Suppose that, for all x, g 0 f(g 2 + 1) + f 0 g(g 2 1) = 0 Show that fg and f/g di er by a constant Calculate the derivatives in Exercises 29 through 33 Assume that f, g and h are all di erentiable and positive valued functions 29 y =(x a) 2 f(x) 30 y = 31 y = f(x) x f(x) g(x) f(x)+g(x) 32 y = f(x) x n,napositive integer 33 y = f(x)g(x) h(x) 34 Let f(x) be di erentiable and let a be any point in the domain of f Consider the AROC function: AROC(x) = f(x) f(a) Calculate d AROC(x) using the Quotient Rule x a dx In Exercises 35 through 38 you are given the velocity function of a particle at time t Calculate the acceleration and jerk functions 35 v(t) =(t 2 +3t + 1)(t 5 1) 36 v(t) = t +2 p t 37 v(t) = 38 v(t) = t7 1 t 1 t 4 1 (t 2 1)(t + 3) 39 A root a of a polynomial p(x) has multiplicity n if (x a) n is the highest power of (x a) that s a factor of p(x) For example, if p(x) =(x 7) 3 (x + 4) 2, then 7 is a root with multiplicity 3 and 4 is a root with multiplicity 2 Show that if x = a is a root of multiplicity n 2 of a polynomial p(x), then a is also a root of p 0 (x) 40 Suppose that f and g are di erentiable functions and that, for all x, f(x) 0, g(x) > 0, and g 0 (x) apple 0 Suppose also that h(x) =f(x) g(x) is an increasing function Show that f(x) must also be an increasing function

5 22 THE PRODUCT AND QUOTIENT RULES Suppose that f(x),g(x) are di erentiable functions whose common domain is I, an open interval Suppose further that neither function has a root in I (ie, for all x in I, f(x) 6= 0, and g(x) 6= 0) Suppose further that, for all x in I, f 0 (x) f(x) = g0 (x) g(x) Show that f(x) g(x) is constant, and conclude that f(x) is a multiple of g(x) 42 Work is defined to be the product of force and displacement Thus, if p(t) is the position, in meters, of an object, which is traveling in a straight line, at time t seconds, and the object is acted on by a constant net force F, in Newtons, then we say that the work done by the force, at time t, isw (t) =F (p(t) variable force requires Integral Calculus) p(0)), in Joules (Calculating work done by a a Write a formula describing the rate of change of work, as a function of time b If p(t) =t 2 +4t + 9 meters, and F = 16 Newtons, find the rate of change of work, with respect to time, at t = 20 seconds 43 Ohm s Law gives a relationship between current, resistance, and voltage in electrical circuits One formulation of Ohm s Law is the relationship I = V/R, where I is current measured in amperes, V is potential di erence (or voltage) across the circuit, measured in volts, and R is resistance, measured in ohms a Assuming that, in some circuit, both V and R are functions of time, find the rate of change of I, withrespecttothetime,t, in seconds, in terms of the rates of change of V and R, withrespecttotime b If V = t 2 2t+11 and R = t 2 2t+3, find the rate of change of current, with respect to time, at t = 1 c At what time does the current reach its maximum? d What are the voltage and resistance when the current is at its maximum? 44 A car is traveling along a road At a particular time t 0 seconds, the velocity of the car is 10 m/s, and its acceleration is 2 m/s 2 Attimet 0, the mass of the car is 2000 kg However, due to a gas leak, the mass of the car is decreasing at a rate of 1/60 kg/s Assume (somewhat strangely) that the leaking gas exits the car with zero velocity (relative to whomever measures the car s velocity as being 10 m/s) What is the rate of change of the momentum of this car, with respect to time, at time t 0? What does this tell you about the net force acting on the car at time t 0? 45 Suppose that the population of a country is modeled using the logistic equation This country can support a maximum population of 25 million people, and the proportionality constant is k =16

6 184 CHAPTER 2 BASIC RULES FOR CALCULATING DERIVATIVES a Write a formula for the rate of change of the population over time, dp b For what values of P is the population increasing? c For what values of P is the population decreasing? d Write a formula for d2 P 2 e For what values of P is the rate of increase of the population increasing? f For what values of P is the rate of increase of the population decreasing? 46 Given a pair (h 1,h 2 ) of di erentiable functions, define the derivative of the pair to be the pair of derivatives, ie, define (h 1,h 2 ) 0 = (h 0 1,h 0 2) Now, consider two pairs of di erentiable functions (f 1 (x),f 2 (x)) and (g 1 (x),g 2 (x)) Given such a pair, we define the dot product as follows: (f 1 (x),f 2 (x)) (g 1 (x),g 2 (x)) = f 1 (x)g 1 (x)+f 2 (x)g 2 (x) Prove that there is a Product Rule for the dot product, that is, prove that [(f 1 (x),f 2 (x)) (g 1 (x),g 2 (x))] 0 = (f 1 (x),f 2 (x)) (g 1 (x),g 2 (x)) 0 + (f 1 (x),f 2 (x)) 0 (g 1 (x),g 2 (x)) Generalize the definition of the dot product to pairs of n functions (pairs of ordered n- tuples), and show that the Product Rule is still satisfied 47 Verify that the Product Rule for dot products from the previous problem holds for the two triplets of functions (x 2 +1, 1/x, 5x + 7) and (4x +3, 1+1/x 2, 2x + 1) 48 Bob and Jane both know the Product Rule and, from Example 1418, they both know that x at not di erentiable at 0 Bob and Jane are having an argument about the di erentiability of f(x) = x x at 0 Bob says that f(x) is not di erentiable at 0, because the Product Rule tells us that f 0 (x) = x x 0 + x (x) 0, and x 0 is not defined at 0