Chapter 4 I Real Analysis 281 51. Disprove the claim: If lim f (x) L, then either lim f (x) L or x a | | = x a = lim f (x) L. → → x a = − 52. If→ lim f (x) and lim g(x) , then lim f g . x a = ∞ x a = ∞ x a + = ∞ → → → f (x) 53. If lim f (x) and lim g(x) L R∗, then lim . x a x a x a g(x) → = ∞ → = ∈ → = ∞ 54. If lim f (x) L, then lim (f (x) L) 0. x a = x a − = 55. Disprove→ the two claims:→ (a) If lim f (x) L, then f (a) L. x a = = (b) If f→(a) L, then lim f (x) L. x a = → = 56. The squeeze theorem (theorem 4.3.4). In exercises 57–68, prove each mathematical statement about continuity. 57. The constant function is continuous (from theorem 4.3.5). 58. The sum of two continuous functions is continuous (from theorem 4.3.5). 59. The difference of two continuous functions is continuous (from theorem 4.3.5). 60. The product of two continuous functions is continuous (from theorem 4.3.5). 61. If f is continuous at x a, then f is continuous at x a. = − = 62. If f is continuous at x a, then f is continuous at x a. = | | = 63. Prove that f (x) xn is continuous for all n N via induction (and = ∈ theorem 4.3.5). 64. If lim f (a h) f (a), then f is continuous at x a. h 0 + = = 65. Disprove→ the claim: If f and g are not continuous at x a, then f g is not = + continuous at x a. = 66. Disprove the claim: If f is continuous at x a, then f is continuous at x a. | | = = 67. Disprove the claim: If the composite function f (g(x)) is continuous, then f (x) and g(x) are both continuous. 68. The following variation on the characteristic function of Q is continuous at x 0: = x if x Q f (x) ∈ = ½ 0 if x Q 6∈ In exercises 69–70, state both an intuitive description and a definition of each limit. 69. lim f (x) 70. lim f (x) L x a x → = −∞ →∞ = 4.4 The Derivative Calculus is the study of change. While Sir Isaac Newton and Gottfried Leibniz are both credited for independently developing calculus in the late 1600s, mathematicians had already been working with derivatives for nearly a half century. The study of change as expressed by the derivative was motivated by a sixteenth and seventeenth century European reflection on and ultimate rejection of ancient Greek astronomy and physics. The European astronomers Nicolaus Copernicus, Tycho Brahe, and Johannes Kepler 282 A Transition to Advanced Mathematics each had insights that challenged the theories of the ancient Greeks, setting the stage for the ground-breaking work of the Italian scientist Galileo Galilei in the early 1600s. Many of the questions about a moving object (that is, an object changing position and velocity) that these scientists were studying are readily answered by considering lines tangent to curves. A number of mathematicians from many different countries made important contributions to the question of finding the equation of a tangent line. Pierre de Fermat studied maxima and minima of curves via tangent lines, essentially using the approach studied in contemporary calculus courses. This work prompted fellow French mathematician Joseph-Louis Lagrange to assert that Fermat should be credited with the development of calculus! The English mathematician Isaac Barrow, who was Newton’s teacher and mentor, corresponded regularly with Leibniz on these mathematical ideas. As we have mentioned, neither Newton nor Leibniz thought of the derivative as a measure of change in terms of our contemporary definition involving limits. Our study of the derivative follows more closely the work of Fermat and Barrow from the early 1600s, in which we think of a tangent line as a limit of secant lines. Naturally, the contemporary presentation is informed by an understanding of Cauchy’s notion of the limit from the early 1800s. The derivative enables the determination of the equation of a line tangent to a given curve at a given point. Given a function y f (x), the slope of a secant line = joining two points ( c, f (c)) and ( c h, f (c h)) is + + rise 1y f (c h) f (c) f (c h) f (c) m + − + − . = run = 1x = (c h) c = h + − In this context, the symbol “ m” is the first letter in the French word montrer which translates as “to climb.” To find the slope of the line tangent to a function f at the point ( c, f (c)), we take a limit of the slopes of secant lines, letting h approach 0. Figure 4.13 illustrates why this limit process makes intuitive sense; you can see that the slopes of the secant lines get closer and closer to the slope of the tangent line as the point ( c h, f (c h)) gets + + closer and closer to ( c, f (c)). The definition of the derivative reflects these ideas. The following definition expresses the real number c as a variable quantity x to identify a general formula for the derivative, enabling us to determine the slope of the line tangent to f (x) whenever this slope is defined. Definition 4.4.1 Let f (x) be a function with domain D. Then the derivative of f (x) is f (x h) f (x) f 0(x) lim + − , = h 0 h → whenever this limit exists. We say that f (x) is differentiable at x c when f (c) = 0 exists for c D, and that f (x) is differentiable when f (x) exists for all x D. ∈ 0 ∈ The ratio f (x h) f (x) + − h is called the difference quotient of the derivative. Chapter 4 I Real Analysis 283 800 600 400 200 0 Figure 4.13 A tangent line at x 2 as a 2 3 4 5 limit of secant= lines x Recall from calculus that many different notations are used for the derivative of a function y f (x), including = df d dy f 0(x) (f ) y0 Dx(y) ˙y . = dx = dx = = dx = = Various phrases also refer to the derivative, including “ f (or y) prime,” “the derivative of f (or y) with respect to x,” the letters “d f d x” spoken individually, and “d y d x” spoken individually. Most of this notation for the derivative is attributable to Leibniz, who gave considerable thought to carefully identifying a useful symbolism and is recognized as a genius in developing notation to make subtle concepts understandable. The alternate definition of the derivative is sometimes helpful; if the limit exists, then f (t) f (x) f 0(x) lim − . = t x t x → − The proof of the equivalence of this alternate definition and the one given in definition 4.4.1 is left for exercise 50 at the end of this section. Example 4.4.1 We use the two definitions of the derivative to determine the equation of a line tangent to f (x) x2 at (2 , 4). = Applying the definition, (x h)2 x2 x2 2xh h2 x2 2xh h2 f 0(x) lim + − lim + + − lim + lim 2x h 2x. = h 0 h = h 0 h = h 0 h = h 0 + = → → → → Hence the slope of the tangent line at (2 , 4) is f (2) 2 2 4, and the equation 0 = · = of the line tangent to f (x) x2 at (2 , 4) is given by y 4 4( x 2). = − = − Applying the alternate definition produces the same result: t2 x2 (t x)( t x) f 0(x) lim − lim + − lim t x 2x. t x t x t x t x t x = → = → = → + = − − I When using the definition (as in example 4.4.1), we often algebraically manipulate the difference quotient so that h appears as a factor in the numerator. This factor 284 A Transition to Advanced Mathematics then cancels the denominator, simplifying the difference quotient so the limit can be evaluated. If the original function f (x) is a rational function, then finding a common denominator will simplify the difference quotient in this way. If f (x) contains a square root, multiplying by the conjugate square root function will simplify the difference quotient. Example 4.4.2 We use the definition of the derivative to find the derivative of f (x) 5√x 1. = + Multiplying both the numerator and the denominator of the difference quotient by the conjugate square root function and then simplifying yields the following calculation. 5√x h 1 5√x 1 f 0(x) lim + + − + = h 0 h → 5√x h 1 5√x 1 5√x h 1 5√x 1 lim + + − + + + + + = h 0 h · 5√x h 1 5√x 1 → + + + + 25 (x h 1) (x 1) 5h lim [ + + − + ] lim = h 0 5h(√x h 1 √x 1) = h 0 h(√x h 1 √x 1) → + + + + → + + + + 5 = 2√x 1 + I Question 4.4.1 Using the definition of the derivative, differentiate each function. 1 (a) f (x) 2x 1 (c) s(x) = + = x 5 1+ (b) g(x) 7x3 (d) t(x) = = 3√x I While we can use the formal definition of the derivative to compute derivatives of a given function, theoretical applications of the definition are more important. Using the definition, we can prove general theorems that hold for all derivatives, making it easy to differentiate many familiar functions without explicitly applying the definition one function at at time. Many functions are so complicated in structure that directly using the difference quotient becomes unwieldy or impossible.