defined, lim x→ξ y(x) exists and lim x→ξ y(x) = y(ξ). A function is continuous if it is continuous at each point in its domain. A function is continuous on the closed interval [a, b] if the function is continuous for each point x ∈ (a, b) and lim x→a + y(x) = y(a) and lim x→b − y(x) = y(b). Discontinuous Functions. If a function is not continuous at a point it is called discontinuous at that point. If lim x→ξ y(x) exists but is not equal to y(ξ), then the function has a removable discontinuity. It is thus named because we could define a continuous function z(x) =  y(x) for x = ξ, lim x→ξ y(x) for x = ξ, to remove the discontinuity. If both the left and right limit of a function at a point exist, but are not equal, then the function has a jump discontinuity at that point. If either the left or right limit of a function does not exist, then the function is said to have an infinite discontinuity at that point. Example 3.2.1 sin x x has a removable discontinuity at x = 0. The Heaviside function, H(x) =      0 for x < 0, 1/2 for x = 0, 1 for x > 0, has a jump discontinuity at x = 0. 1 x has an infinite discontinuity at x = 0. See Figure 3.3. Properties of Continuous Functions. Arithmetic. If u(x) and v(x) are continuous at x = ξ then u(x) ± v(x) and u(x)v(x) are continuous at x = ξ. u(x) v(x) is continuous at x = ξ if v(ξ) = 0. Function Composition. If u(x) is continuous at x = ξ and v(x) is continuous at x = µ = u(ξ) then u(v(x)) is continuous at x = ξ. The composition of continuous functions is a continuous function. 54 Figure 3.3: A Removable discontinuity, a Jump Discontinuity and an Infinite Discontinuity Boundedness. A function which is continuous on a closed interval is bounded in that closed interval. Nonzero in a Neighborhood. If y(ξ) = 0 then there exists a neighborhood (ξ − , ξ + ),  > 0 of the point ξ such that y(x) = 0 for x ∈ (ξ −, ξ + ). Intermediate Value Theorem. Let u(x) be continuous on [a, b]. If u(a) ≤ µ ≤ u(b) then there exists ξ ∈ [a, b] such that u(ξ) = µ. This is known as the intermediate value theorem. A corollary of this is that if u(a) and u(b) are of opposite sign then u(x) has at least one zero on the interval (a, b). Maxima and Minima. If u(x) is continuous on [a, b] then u(x) has a maximum and a minimum on [a, b]. That is, there is at least one point ξ ∈ [a, b] such that u(ξ) ≥ u(x) for all x ∈ [a, b] and there is at least one point ψ ∈ [a, b] such that u(ψ) ≤ u(x) for all x ∈ [a, b]. Piecewise Continuous Functions. A function is piecewise continuous on an interval if the function is bounded on the interval and the interval can be divided into a finite number of intervals on each of which the function is continuous. For example, the greatest integer function, x, is piecewise continuous. (x is defined to the the greatest integer less than or equal to x.) See Figure 3.4 for graphs of two piecewise continuous functions. Uniform Continuity. Consider a function f(x) that is continuous on an interval. This means that for any point ξ in the interval and any positive  there exists a δ > 0 such that |f(x) − f(ξ)| <  for all 0 < |x − ξ| < δ. In general, this value of δ depends on both ξ and . If δ can be chosen so it is a function of  alone and indepen dent of ξ then 55 Figure 3.4: Piecewise Continuous Functions the function is said to be uniformly continuous on the interval. A sufficient cond ition for uniform continuity is that the function is continuous on a closed interval. 3.3 The Derivative Consider a function y(x) on the interval (x . . . x + ∆x) for some ∆x > 0. We define the increment ∆y = y(x + ∆x) − y(x). The average rate of change, (average velocity), of the function on the interval is ∆y ∆x . The average rate of change is the slope of the secant line that passes through the points (x, y(x)) and (x + ∆x, y(x + ∆x)). See Figure 3.5. y x ∆ y ∆ x Figure 3.5: The increments ∆x and ∆y. 56 If the slope of the secant line has a limit as ∆x approaches zero then we call this slope the derivative or instantaneous rate of change of the function at the poin t x. We denote the derivative by dy dx , which is a nice notation as the derivative is the limit of ∆y ∆x as ∆x → 0. dy dx ≡ lim ∆x→0 y(x + ∆x) − y(x) ∆x . ∆x may approach zero from below or above. It is common to denote the derivative dy dx by d dx y, y  (x), y  or Dy. A function i s said to be differentiable at a point if the derivative exists there. Note that differentiability implies continuity, but not vice versa. Example 3.3.1 Consider the derivative of y(x) = x 2 at the point x = 1. y  (1) ≡ lim ∆x→0 y(1 + ∆x) − y(1) ∆x = lim ∆x→0 (1 + ∆x) 2 − 1 ∆x = lim ∆x→0 (2 + ∆x) = 2 Figure 3.6 shows the secant lines approaching the tangent line as ∆x approaches zero from above and below. Example 3.3.2 We can compute the derivative of y(x) = x 2 at an arbitrary p oint x. d dx  x 2  = lim ∆x→0 (x + ∆x) 2 − x 2 ∆x = lim ∆x→0 (2x + ∆x) = 2x 57 0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 3.5 4 Figure 3.6: Secant lines and the tangent to x 2 at x = 1. Properties. Let u(x) and v(x) be differentiable. Let a and b be constants. Some fundamental properties of derivatives are: d dx (au + bv) = a du dx + b dv dx Linearity d dx (uv) = du dx v + u dv dx Product Rule d dx  u v  = v du dx − u dv dx v 2 Quotient Rule d dx (u a ) = au a−1 du dx Power Rule d dx (u(v(x))) = du dv dv dx = u  (v(x))v  (x) Chain Rule These can be proved by using the definition of differentiation. 58 Example 3.3.3 Prove the quotient rule for derivatives. d dx  u v  = lim ∆x→0 u(x+∆x) v(x+∆x) − u(x) v(x) ∆x = lim ∆x→0 u(x + ∆x)v(x) − u(x)v(x + ∆x) ∆xv(x)v(x + ∆x) = lim ∆x→0 u(x + ∆x)v(x) − u(x)v(x) − u(x)v(x + ∆x) + u(x)v(x) ∆xv(x)v(x) = lim ∆x→0 (u(x + ∆x) − u(x))v(x) − u(x)(v(x + ∆x) − v(x)) ∆xv 2 (x) = lim ∆x→0 u(x+∆x)−u(x) ∆x v(x) − u(x) lim ∆x→0 v(x+∆x)−v(x) ∆x v 2 (x) = v du dx − u dv dx v 2 59 Trigonometric Functions. Some derivatives of trigonometric functions are: d dx sin x = cos x d dx arcsin x = 1 (1 − x 2 ) 1/2 d dx cos x = −sin x d dx arccos x = −1 (1 − x 2 ) 1/2 d dx tan x = 1 cos 2 x d dx arctan x = 1 1 + x 2 d dx e x = e x d dx ln x = 1 x d dx sinh x = cosh x d dx arcsinh x = 1 (x 2 + 1) 1/2 d dx cosh x = sinh x d dx arccosh x = 1 (x 2 − 1) 1/2 d dx tanh x = 1 cosh 2 x d dx arctanh x = 1 1 − x 2 Example 3.3.4 We can evaluate the derivative of x x by using the identity a b = e b ln a . d dx x x = d dx e x ln x = e x ln x d dx (x ln x) = x x (1 · ln x + x 1 x ) = x x (1 + ln x) Inverse Functions. If we have a function y(x), we can consider x as a function of y, x(y). For example, if y(x) = 8x 3 then x(y) = 3 √ y/2; if y(x) = x+2 x+1 then x(y) = 2−y y−1 . The derivative of an inverse function is d dy x(y) = 1 dy dx . 60 Example 3.3.5 The inverse function of y(x) = e x is x(y) = ln y. We can obtain the derivative of the logarithm from the derivative of the exponential. The derivative of the exponential is dy dx = e x . Thus the derivative of the logarithm is d dy ln y = d dy x(y) = 1 dy dx = 1 e x = 1 y . 3.4 Implicit Differentiation An explicitly defined function has the form y = f (x). A implicitly defined function has the form f(x, y) = 0. A few examples of implicit functions are x 2 + y 2 −1 = 0 and x + y + sin(xy) = 0. Often it is not possible to write an implicit equation in explicit form. This is true of the latter example above. One can calculate the derivative of y(x) in terms of x and y even when y(x) is defined by an implicit equation. Example 3.4.1 Consider the implicit equation x 2 − xy −y 2 = 1. This implicit equation can be solved for the dep en dent variable. y(x) = 1 2  −x ± √ 5x 2 − 4  . We can differentiate this expression to obtain y  = 1 2  −1 ± 5x √ 5x 2 − 4  . One can obtain the same result without first solving for y. If we differentiate the implicit equation, we obtain 2x − y −x dy dx − 2y dy dx = 0. 61 We can solve this equation for dy dx . dy dx = 2x − y x + 2y We can differentiate this expression to obtain the second derivative of y. d 2 y dx 2 = (x + 2y)(2 − y  ) − (2x − y)(1 + 2y  ) (x + 2y) 2 = 5(y −xy  ) (x + 2y) 2 Substitute in the expression for y  . = − 10(x 2 − xy −y 2 ) (x + 2y) 2 Use the original implicit equation. = − 10 (x + 2y) 2 3.5 Maxima and Minima A differentiable function is increasing where f  (x) > 0, decreasing where f  (x) < 0 and stationary where f  (x) = 0. A function f (x) has a relative maxima at a point x = ξ if there exists a neighborhood around ξ such that f(x) ≤ f(ξ) for x ∈ (x −δ, x + δ), δ > 0. The relative minima is defined analogously. Note that this definition does not require that the function be differentiable, or even continuous. We refer to relative maxima and minima collectively are relative extrema. 62 [...]... − (x − 1) 2 (x − 1) 3 (x − 1) 4 (x − 1) n (x − 1) n +1 1 + − + · · · + ( 1) n 1 + ( 1) n 2 3 4 n n + 1 ξ n +1 72 2 1 -1 0.5 1 1.5 2 2.5 3 -2 -3 -4 -5 -6 2 1 -1 0.5 1 1.5 2 2.5 3 -2 -3 -4 -5 -6 2 1 -1 0.5 1 1.5 2 2.5 3 -2 -3 -4 -5 -6 2 1 -1 0.5 1 1.5 2 2.5 3 -2 -3 -4 -5 -6 Figure 3 .15 : The 2, 4, 10 and 50 Term Approximations of ln x Below are plots of the 2, 4, 10 and 50 term approximations Note that the approximation... ( 1) n/2 cos x for even n, (n +1) /2 ( 1) sin x for odd n Since cos(0) = 1 and sin(0) = 0 the n-term approximation of the cosine is, x2 x4 x6 x2(n 1) x2n 2(n 1) cos x = 1 − + − + · · · + ( 1) + cos ξ 2! 4! 6! (2(n − 1) )! (2n)! Here are graphs of the one, two, three and four term approximations 1 0.5 1 0.5 -3 -2 -1 -0.5 -1 1 2 3 -3 -2 -1 -0.5 -1 1 0.5 1 2 3 -3 -2 -1 -0.5 -1 1 0.5 1 2 3 -3 -2 -1 -0.5 -1 1... = 1 Taylor’s theorem thus states that xn xn +1 ξ x2 x3 e, ex = 1 + x + + + ··· + + 2! 3! n! (n + 1) ! for some ξ ∈ (0, x) The first few polynomial approximations of the exponent about the point x = 0 are f1 (x) = 1 f2 (x) = 1 + x x2 2 x2 x3 + f4 (x) = 1 + x + 2 6 f3 (x) = 1 + x + The four approximations are graphed in Figure 3 .11 2.5 2 1. 5 1 0.5 -1 -0.5 0.5 1 2.5 2 1. 5 1 0.5 -1 -0.5 2.5 2 1. 5 1 0.5 -1. .. the 71 1 0.5 -10 -5 5 10 -0.5 -1 -1. 5 -2 Figure 3 .14 : Ten Term Taylor Series Approximation of cos x point x = 1 The first few derivatives of f are f (x) = ln x 1 f (x) = x 1 f (x) = − 2 x 2 f (x) = 3 x 3 f (4) (x) = − 4 x The derivatives evaluated at x = 1 are f (0) = 0, f (n) (0) = ( 1) n 1 (n − 1) !, for n ≥ 1 By Taylor’s theorem of the mean we have, ln x = (x − 1) − (x − 1) 2 (x − 1) 3 (x − 1) 4 (x − 1) n... lim 1 + x→∞ x→∞ x where a and b are constants Hint, Solution 86 bx , 3. 9 Hints Hint 3 .1 Apply the , δ definition of a limit Hint 3. 2 Set y = 1/ x Consider limy→∞ Hint 3. 3 √ Write n 5 in terms of the exponential function Hint 3. 4 The composition of continuous functions is continuous Apply the definition of continuity and look at the point x = 0 Hint 3. 5 Note that for x1 = 1 (n 1/ 2)π and x2 = 1 (n +1/ 2)π... = 1 x→0 80 3. 8 3. 8 .1 Exercises Limits of Functions Exercise 3 .1 Does 1 x lim sin x→0 exist? Hint, Solution Exercise 3. 2 Does lim x sin x→0 1 x exist? Hint, Solution Exercise 3. 3 Evaluate the limit: lim √ n n→∞ 5 Hint, Solution 3. 8.2 Continuous Functions Exercise 3. 4 Is the function sin (1/ x) continuous in the open interval (0, 1) ? Is there a value of a such that the function defined by f (x) = sin (1/ x)... /20! is plotted in Figure 3 . 13 1 0.8 0.6 0.4 0.2 2 4 6 8 10 Figure 3 . 13 : Plot of x20 /20! Note that the error is very small for x < 6, fairly small but non-negligible for x ≈ 7 and large for x > 8 The ten term approximation of the cosine, plotted below, behaves just we would predict The error is very small until it becomes non-negligible at x ≈ 7 and large at x ≈ 8 Example 3. 6 .3 Consider the function... (x) = sin (1/ x) for x = 0, a for x = 0 81 is continuous on the closed interval [0, 1] ? Hint, Solution Exercise 3. 5 Is the function sin (1/ x) uniformly continuous in the open interval (0, 1) ? Hint, Solution Exercise 3. 6 √ Are the functions x and Hint, Solution 1 x uniformly continuous on the interval (0, 1) ? Exercise 3. 7 Prove that a function which is continuous on a closed interval is uniformly continuous... Solution 3. 8.4 Implicit Differentiation Exercise 3 . 13 (mathematica/calculus/differential/implicit.nb) Find y (x), given that x2 + y 2 = 1 What is y (1/ 2)? Hint, Solution Exercise 3 .14 (mathematica/calculus/differential/implicit.nb) Find y (x) and y (x), given that x2 − xy + y 2 = 3 Hint, Solution 3. 8.5 Maxima and Minima Exercise 3 .15 (mathematica/calculus/differential/maxima.nb) Identify any maxima and minima... accurate method Equation 3. 2 is called the forward difference scheme for calculating the first derivative Figure 3 .17 shows a plot of the value of this scheme for the function f (x) = sin x and ∆x = 1/ 4 The first derivative of the function f (x) = cos x is shown for comparison 1 0.5 -4 -2 2 4 -0.5 -1 Figure 3 .17 : The Forward Difference Scheme Approximation of the Derivative Another scheme for approximating the . ( 1) 2(n 1) x 2(n 1) (2(n − 1) )! + x 2n (2n)! cos ξ. Here are graphs of the one, two, three and four term approximations. -3 -2 -1 1 2 3 -1 -0.5 0.5 1 -3 -2 -1 1 2 3 -1 -0.5 0.5 1 -3 -2 -1 1 2 3 -1 -0.5 0.5 1 -3. are f 1 (x) = 1 f 2 (x) = 1 + x f 3 (x) = 1 + x + x 2 2 f 4 (x) = 1 + x + x 2 2 + x 3 6 The four approximations are graphed in Figure 3 .11 . -1 -0.5 0.5 1 0.5 1 1.5 2 2.5 -1 -0.5 0.5 1 0.5 1 1.5 2 2.5 -1 -0.5. ( 1) n 1 (n − 1) !, for n ≥ 1. By Taylor’s theorem of the mean we have, ln x = (x − 1) − (x − 1) 2 2 + (x − 1) 3 3 − (x − 1) 4 4 + ··· + ( 1) n 1 (x − 1) n n + ( 1) n (x − 1) n +1 n + 1 1 ξ n +1 . 72