Chapter 3a Topics in differentiation Lectures in Mathematical Economics L Cagandahan Abueg De La Salle University School of Economics Problems in differentiation Problems in differentiation Problem 1. Using the definition, find the derivative of the following functions: ( i) f( x) = 1 x 2 3 ( ii) g( x) = 5x ( iii) hx ( ) = x 1 chapter 3a: topics in differentiattion 1
Problems in differentiation Problem 2. Use L Hospital s Rule to give an alternate proof of the existence of Euler s number: 1 lim + = 1 e n n n Problems in differentiation Problem 10. Let f differentiable and real-valued with f( a + b) = f( af )( b), a, b R Suppose that f(0) = 1 and f (0) exists. Show that f ( x) = f (0)( f x), x R If in addition, f (0) = 1, what is f? The derivative Remar. It can be shown that i. if f(x) is differentiable, then f( x + h) f( x h) lim = f ( x) h 0 2h ii. if f (c) exists, then xf( c) cf( x) lim = f( c) cf ( c) x c x c chapter 3a: topics in differentiattion 2
Trigonometric functions Trigonometric functions Theorem 3a.1. [Derivatives of trigonometric functions] For any x in the real line, d i x x dx ( ) sin = cos d ii x x dx ( ) cos = sin Trigonometric functions d iii x x dx d iv cotx = csc x dx 2 ( ) tan = sec ( ) 2 d v x x x dx d vi x = x x dx ( ) sec = sec tan ( ) csc csc cot chapter 3a: topics in differentiattion 3
Trigonometric functions Remar. For any x in the real line, observe that 4 d i sinx = sinx 4 dx ( ) 4 d ii cosx = cosx 4 dx ( ) Trig. & hyp. functions Exercise. Find the derivative of the following functions: ( i) f( x) = cos(sin x) ( ii) g( x) = sin(exp x) ( iii) hx ( ) = tan(ln x) One-sided derivatives chapter 3a: topics in differentiattion 4
One-sided derivatives Recall the concept of a onesided limit: from the epsilondelta criterion for limit, it is possible that the direction of x in approaching a cluster point c may be only done in one direction. We review the corresponding criterion when x approaches c from the left [or from the right]. One-sided limits Definition. Let A R f: A R If c is a cluster point of { } A ( c, ) = x A: x > c then lim f( x) = L iff ε > 0 δ > 0 + r x c (0 < x c < δ f( x) L < ε) L r is the right-hand limit of f at c. One-sided limits Definition. Let A R f: A R If c is a cluster point of A (, c) = { x A: x < c} then lim f( x) = L iff ε > 0 δ > 0 l x c (0 < c x < δ f( x) L < ε) L l is the left-hand limit of f at c. chapter 3a: topics in differentiattion 5
One-sided limits L x c c x c + One-sided derivatives Definition. A function f is said to be differentiable from the left of x if f( x + h) f( x) lim h h 0 exists. We call this the left derivative of f at x, and we denote f( x + h) f( x) f ( x) = lim h 0 h One-sided derivatives Definition. A function f is said to be differentiable from the right of x if f( x + h) f( x) lim h + h 0 exists. We call this the right derivative of f at x, and we denote f( x + h) f( x) f + ( x) = lim + h 0 h chapter 3a: topics in differentiattion 6
One-sided limits Theorem 3a.3. We say that a function f is differentiable at x iff f ( x) = f ( x) = f ( x) + i.e., the right and left derivatives coincide. In mathematical literature, given a differentiable function f : I R on a closed and bounded interval I = [a,b] with < a < b < it is usual [and logical] that f is differentiable on (a,b) and continuous on [a,b]. chapter 3a: topics in differentiattion 7
Recall that f( x) = x, x R at the point x = 0, f is not differentiable since by the uniqueness of limits (implying a unique tangent line at a point on f), this unique tangent line does not exist. We call the point x = 0 in this case a sharp point. However, note that on the subintervals (,0), (0, ) the function does not have sharp points. Moreover, the absolute value function is differentiable on each of the subintervals above. y x f( x) = x, x R chapter 3a: topics in differentiattion 8
Definition. If a function f :[ a, b] R has a finite number of points of discontinuity given by x 1,x 2,,x, such that f is continuous on each subinterval ( a, x ),...,( x, x ),...,( x, b) 1 i i+ 1 for i = 1,2,, 1 and possibly defined on some (or all) points of discontinuity, we say that f is a piecewise continuous function. Definition. Consider a function If f :[ a, b] R f :( a, b) R exists, we say that f is a smooth function. If there is a finite number of points given by x 1,x 2,,x, such that f exists on each subinterval ( a, x ),...,( x, x ),...,( x, b) 1 i i+ 1 for i = 1,2,, 1, we say that f is a piecewise smooth function. Remar. If a function is piecewise smooth, it is also piecewise continuous (immediate from Theorem 3.1). The converse does not hold. chapter 3a: topics in differentiattion 9
Example. Consider the function 1 f( x) =, x R * x [i.e., f is defined on the nonzero reals]. Note that 1 f ( x) =, x R * 2 x y x 0 lim f( x) = + + x 0 + x 0 x lim f( x) = x 0 1 f( x) =, x R * x x 0 y + x 0 x lim f ( x) = x 0 lim f ( x) = + x 0 1 f ( x) =, x R * 2 x chapter 3a: topics in differentiattion 10
Definition. Consider a piecewise smooth function f and a finite number of sharp points [or points of discontinuity] x, x,..., x 1 2 If f is linear in each of the subintervals ( a, x ),...,( x, x ),...,( x, b) 1 i i+ 1 [possibly defined on some or all of the sharp points or points of discontinuity]. We then call f a piecewise linear function. Example. The absolute value function f( x) = x, x R is a piecewise linear function. Note that this not only piecewise continuous, but is continuous in the whole of the real line. chapter 3a: topics in differentiattion 11
Example. Consider the function 2 f( x) = x 1, x R By definition, 2 x 1 x (, 1] [1, ) f( x) = 2 1 x x ( 1,1) y (1,1) ( 1,0) (1,0) x f( x) = 1 x 2 Note that f is not differentiable at the points x = 1, 1 i.e., f is piecewise smooth. The derivatives of f on the particular intervals are, 2 x x (, 1) (1, ) f ( x) = 2 x x ( 1,1) chapter 3a: topics in differentiattion 12
y (2, 4) ( 1, 2) (1, 2) (0,0) x ( 2, 4) ( 1, 2) (1, 2) Since f is not differentiable at the points x = 1, 1, clearly the graph of f is not continuous at these points in the domain of f. Example. Consider a piecewise linear function given by f( x) = a + b x x [ x, x ] m m m 1 m m {1,2,..., + 1} with a fixed positive integer and x = a x = b, 0 + 1 Obtaining the derivative of f respective to subintervals, we have f ( x) = b x [ x, x ] m m 1 m Let g( x) = f ( x) m {1,2,..., + 1} The function g is a particular example of a class of functions important in real analysis. chapter 3a: topics in differentiattion 13
Definition. A function defined by bm x ( xm 1, xm) hx ( ) = cm x = xm 1 m {1,2,..., } where a fixed positive integer is called a step function. y b 1 c 5 c 2 b 3 c 0 b 2 b 5 b 4 c 4 x 0 c 1 c 3 x 1 x 2 x 3 x 4 x 5 a step function x Example. The function given by where ( x) = x x = n, x ( n 1, n], n Z is called the greatest integer function. chapter 3a: topics in differentiattion 14
y 5 4 3 2 1 0 1 2 3 4 5 ( x) = x A circle with shade indicates that only the rightmost endpoints are included in each step. x Example. Consider again the absolute value function f( x) = x, x R Since f above is piecewise smooth. By obtaining the derivative of f on respective subintervals, we get 1 x > 0 f ( x) = 1 x < 0 By defining f (x) = 0 [and naming this function g, we have 1 x > 0 g( x) = 0 x = 0 1 x < 0 chapter 3a: topics in differentiattion 15
1 y 0 x -1 Observe that this function g is precisely the signum function denoted sgn(x), which is also a step function. Example. [Weierstrass] There is a function f continuous on the real line but is not differentiable at any point in its domain. This function is given by 1 n f( x) = cos( b πx) n a n= 0 Proof. See Wade [2010], pp. 264-266. Remar. The function provided by Weierstrass in the previous example is a case of a class of functions called nowhere differentiable. Note that nowhere differentiable functions are continuous. This is an extreme case of showing that the converse of Theorem 3.1 is not true. chapter 3a: topics in differentiattion 16
Uniform derivatives Uniform derivatives Remar. Recall the definition of uniform continuity from 2a: Let We say that f is uniformly continuous on A iff ε > 0 δ > 0 A R f: A R ( x, c A 0 < x c < δ f( x) f( c) < ε) Analogously, we can define the concept of uniform continuity given in the next slide: ε Uniform derivatives Definition. A function f : I R on a closed and bounded interval I is said to be uniformly differentiable on I iff for every ε > 0, there is a δ > 0 such that if 0 < x y < δ, then f( x) f( y) f ( x) < ε x y chapter 3a: topics in differentiattion 17
Uniform derivatives Remar. From the definition of uniform differentiability, it requires that the function f is differentiable, as seen in the expression f( x) f( y) f ( x) < ε x y Thus, if a function is uniformly differentiable, it is also differentiable. Uniform derivatives Exercise. Prove or give a counterexample: the differentiability of a function f is sufficient for f to be uniformly differentiable. Uniform derivatives Theorem 3a.4. If f : I R is uniformly differentiable on I, then f is continuous on I. Proof. Obvious [immediately following from the previous remar, and Theorem 3.1]. chapter 3a: topics in differentiattion 18
Uniform derivatives Exercise. Let f : I R Show that if f is differentiable on I and f is bounded on I, then f satisfies the Lipschitz condition. Carathédory s theorem Carathédory's theorem Remar. Recall the mean value theorem (Theorem 3.19): Let f be a continuous function on [a,b], differentiable on (a,b), and let a < b. Then there exists a real number c in (a,b) such that f( b) f( a) f ( c) = b a chapter 3a: topics in differentiattion 19
Carathédory's theorem Reexpressing the mean value theorem, we have f ( c) b a = f( b) f( a) Note that the existence of f (c) is not always guaranteed to exist for any function f. The necessary conditions are stated in the next theorem, due to Carathédory. Carathédory's theorem Theorem 3a.5. [Carathédory] Let f be a function defined on I = [a,b] and let c be a point in [a,b]. Then f is differentiable at c iff there is a function ϕ :[ a, b] R that is continuous at c and satisfies ϕ( x) x c = f( x) f( c), x I Carathédory's theorem Example. Consider the function Note that f( x) = x f( c) = c From elementary algebra, 3 3 2 2 x c = ( x c)( x + cx + c ) 3 3 chapter 3a: topics in differentiattion 20
Carathédory's theorem Using Carathédory s theorem f( x) f( c) = x c 3 3 = + + 2 2 ( x c)( x cx c ) = ( x c) ϕ( c) Thus, ϕ = + + 2 2 ( x) ( x cx c ) Carathédory's theorem Observe that and f ( x) = 3 x f ( c) = 3c ϕ ( x) = x + cx + c 2 2 2 2 ϕ( c) = c + cc ( ) + c 2 2 = c + c + c 2 = 3c = f ( c) 2 2 2 Carathédory's theorem which is an expected result since f( x) = x is a continuously differentiable function for any real number c. 3 chapter 3a: topics in differentiattion 21
Carathédory's theorem Example. Using a similar argument, the function g( x) = x satisfies Carathédory s theorem with 2 ϕ ( x) = x + c Directional derivatives Directional derivative We first recall the definition of a partial derivative in 3 and the corresponding geometric interpretation, given in the next slides. chapter 3a: topics in differentiattion 22
Partial derivative Definition. Let f be a function of several variables, say x 1,x 2,..., x n. The partial derivative of f at x j is given as y f( x1,..., xj + h,..., xn) f( x1,..., xn) = lim x h 0 h j if this limit exists. If all partial derivatives with respect to the n variables exist, then the n-tuple Partial derivative y y y,,..., x x x 1 2 n is called the gradient of f at the point (x 1,x 2,...,x n ). This is also denoted as f read as del f. Partial derivative Remar. Given f be a function of several variables, say x 1,x 2,..., x n, we denote the corresponding partial derivative with respect to x j as y x j = f x j chapter 3a: topics in differentiattion 23
Geometric interpretation z f x (x,y) is the slope of the curve BDC on the surface above the line l parallel to the x-axis; B D f(x,y) x y A C t l Directional derivative Remar. The idea that the partial derivative y x j = f is the rate of change of f in the direction of the x j -axis may be generalized to any direction represented by a vector d θ. x j Directional derivative i.e., d = (,,..., ) (0,0,...,0) = d d d 1 2 n θ n Without loss of generality, we may consider a vector d whose length is 1, i.e., n 1= d = d = 1 2 chapter 3a: topics in differentiattion 24
Directional derivative We wish to determine the rate of change f in the direction d. To this end, we formally define the derivative of f at x = (x 1,x 2,,x n ) in the direction d by considering the values of f on the line x + pd, i.e., f(x + pd) for small values of p. Directional derivative Definition. Let f be a real valued function of n variables, i.e., n f : R R and let d be a vector such that its length is 1. The derivative of f at x in the direction d, defined by f( x + pd) f( x) f ( x) = lim d p 0 p Directional derivative if this limit exists. We call this derivative a directional derivative. Theorem 3a.6. If f is differentiable at x, then n f f ( x d ) = d x = 1 chapter 3a: topics in differentiattion 25
Directional derivative Remar. In the language of linear algebra, n f f ( x ) = d d x = 1 T = [ f( x)] d [ f( )] = x d i.e., a directional derivative is a dot product of the gradient of f and the vector d. Directional derivative Corollary 3a.7. If d = u = (0,0,...,0,1,0,...,0) i.e., d is the unit vector (such that the th entry is 1 and the rest are zeroes), then f [ f( x)] u =, = 1,2,..., n x Directional derivative Corollary 3a.8. If f( x) θn then the gradient of f is the direction of maximum rate of increase of f. chapter 3a: topics in differentiattion 26
Directional derivative Corollary3a.9. Let d be a vector such that its length is 1 and [ f( x)] d > 0 Then there exists a q > 0 such that f( x + pd) > f( x), 0 < p < q Directional derivative Remar. The above corollary says that if [ f( x)] d > 0 (i.e., d has the same general direction as the gradient of f in the sense that the angle θ formed by these two vectors is less than π/2 radians), then any small movement in the direction d will Directional derivative increase f. Thus, if the gradient f( x) θn then this points in the direction of increasing values of f(i.e., the gradient points uphill). chapter 3a: topics in differentiattion 27
Directional derivative Definition. Consider a relation z = f( x, y) given a fixed constant z. Let [a,b] be a subset of the graph of f. The set { : (, ), [, ]} C = y z = f x y x a b is called a curve in the Cartesian plane. Directional derivative We call ( xt yt ) f( t) = ( ), ( ) a parametric representation of f. If x and y are differentiable functions of t, we call the curve C a differentiable curve. Directional derivative Remar. Consider differentiable curve C in the Cartesian plane. The function z = f( x, y) may be represented implicitly by an equation 0 = g( x, y) chapter 3a: topics in differentiattion 28
Directional derivative At a point (x,y), the tangent to the curve has slope dy dy dt = dx dx dt dy / dt = dx / dt y ( t) = x ( t) and observe that ( x t y t ) f ( t) = ( ), ( ) Directional derivative which is just a ray containing the origin (0,0) and the point (x,y). Thus, the tangent to the curve C and f (t) have the same slopes and if we translate f (t) to (x,y), then f (t) will be tangent to C. Thus, we call f (t) a tangent vector to the curve C. Directional derivative By totally differentiating 0 = g( x, y) (i.e., applying chain rule with respect to t), we have g dx g dy 0 = + x dt y dt chapter 3a: topics in differentiattion 29
Directional derivative In the language of linear algebra, g dx g dy 0 = + x dt y dt dx g g dt = x y dy dt = [ g] f ( t) Directional derivative i.e., the gradient of g and the tangent vector f are orthogonal, at the point (x,y). It can be shown that this is true if n C R Homothetic functions chapter 3a: topics in differentiattion 30
Homothetic functions Recall the definition of a homogeneous function f: A function f is said to be homogeneous of degree iff f( tx) = t f( x), t > 0 If = 1, we then say that the function f is linearly homogeneous. Homothetic functions From a remar in implicit function theorems, if z = f( x, y) is a homogeneous function of degree, then the slopes of the level curves are the same at each point on the ray from the origin. Homothetic functions y f x ( x0, y0) f ( x, y ) y 0 0 ty 0 (tx 0,ty 0 ) y 0 (x 0,y 0 ) z = f( tx, ty ) t 0 0 z = f( x, y ) 0 0 0 (0,0) x 0 tx 0 x chapter 3a: topics in differentiattion 31
Homothetic functions This property is possessed by a larger class of functions that contain the homogeneous functions as a subclass. These are called homothetic functions, which we formally define in the next slide. Before stating this definition, recall the class of monotone functions in 2.1. Monotone functions Definition. A function f is said to be wealymonotone iff f is either nondecreasing or nonincreasing. A function f is said to be monotone (or strictly monotone) iff f is either increasing or decreasing. Monotone functions The flat portions of the graph of f maes it nondecreasing a nondecreasing function chapter 3a: topics in differentiattion 32
Monotone functions The flat portions of the graph of f maes it nonincreasing a nonincreasing function Homothetic functions Definition. A function f : D R, D R is said to behomothetic iff there is a monotone (increasing) function h: R R and a homogeneous function g: D R n Homothetic functions such that f( x) = hg( x) = h g( x) where x = (,,..., ) x x x 1 2 n chapter 3a: topics in differentiattion 33
Homothetic functions Remar. In the definition of a homothetic function f (and the corresponding homogeneous function g), if n g: D R, D R then D satisfies the condition { x 0} D t > tx D Homothetic functions This translates to the usual problem of production functions (that are homogenous of some degree) in microeconomics: the problem of feasibility. Homothetic functions Example. Consider the function given by Since 2 f : ++ R R 3 3 ( 1 2 ) f( x) = ln x + 4x g( x) = x + 4x 3 3 1 2 chapter 3a: topics in differentiattion 34
Homothetic functions is homogeneous of degree 3 and h( z) = ln( z) is monotone (i.e., increasing), then f is homothetic. But observe that f is not homogeneous: f t t f t f 3 ( x) = 3ln + ( x) ( x) Homothetic functions Theorem 3a.10. Every homogeneous function is homothetic. Remar. The above theorem stresses the fact that the class of homogeneous functions is a subset of the class of homothetic functions. Homothetic functions Theorem 3a.11. A nonnegativevalued function f and homogeneous of degree > 0 can be expressed as a homothetic function f = hg where g is linearly homogeneous. chapter 3a: topics in differentiattion 35
Homothetic functions Remar. Production functions belong to the class of functions that are homogeneous of positive degree. From 3, we have noted that the degree of homogeneity of a production is its returns to scale; thus, it maes economic sense if this degree is positive. Homothetic functions Thus, by the previous theorem, any production function can be expressed as a composition of two functions h and g, where the function g is linearly homogeneous. Homothetic functions Theorem 3a.12. The slopes of level curves of a homothetic function are the same at every point of a ray emanating from the origin. chapter 3a: topics in differentiattion 36
Homothetic functions Remar. In consumer theory, given a utility function U(x) and ain increasing function g(x), then the composition h given by ( ) hx ( ): = g U( x) preserves ordering of preferences (i.e., ordinal utility). Homothetic functions Moreover, if in addition U is homogeneous of degree [with respect to the consumption bundle x], then ( ) hx ( ): = g U( x) is a homothetic function. IN microeconomics, we call h a monotonic transformation of U. Homothetic functions Theorem 3a.13. If f is a homothetic function with then implies 1 2 x, x D R f 1 2 ( x ) = f( x ) f α α α 1 2 ( x ) = f( x ), > 0 n chapter 3a: topics in differentiattion 37
Homothetic functions Remar. In the previous theorem, if f is a production function, then if 1 2 x, x D R are two input vectors that produce the same output, i.e., q = f = 1 2 ( x ) f( x ) n Homothetic functions then i.e., q = f α = α α > 1 2 ( x ) f( x ), 0 αx, αx 1 2 will produce the same output for every positive α. Functions from C n [a,b] chapter 3a: topics in differentiattion 38
Functions from C n [a,b] Recall that a differentiable function y = f(x) may be again differentiable [under some (necessary) conditions], which may give rise to a new differentiable function. This leads us to the most commonly used terminologies for characterizing functions in economics. Functions from C n [a,b] Recall that in 3, we have the following definition: If f is a differentiable function and f exists, we say that f is differentiable. If f is a continuous function, we say that f is continuously differentiable. If f is again differentiable, and f exists, we say that f is twice differentiable. If f is a continuous function, we say that f is twice continuously differentiable. Functions from C n [a,b] It is always of interest in economics [which is also desired] that functions depicting economic behavior are twice continuously differentiable. We formally define a phrase that characterizes economic behavior pertaining to a particular variable of interest given a function. chapter 3a: topics in differentiattion 39
Functions from C n [a,b] Definition. We say that a function y = f(x) is well-behaved whenever ( i) f ( x) 0, x domf ( ii) f ( x) < 0, x domf Functions from C n [a,b] Definition. We say that a function z = g(x 1,x 2,,x n ) is well-behaved [with respect to x 0 ] whenever 0 ( ) x i f( ) θ 2 0 0 ( ) x = x ii f( ) H f ( ) is negative definite Functions from C n [a,b] 2 2 2 f f f 2 x x 1 2 x1 xn x1 2 2 2 f f f 2 Hg( x) = x x x x 1 2 x 2 n 2 2 2 2 f f f 2 x x x x 1 n 2 n xn chapter 3a: topics in differentiattion 40
Functions from C n [a,b] Definition. A function y = f( x) is said to be n th -differentiable iff n d y ( n ) ( ) n dx =f x exists. If g is a function having the following properties: Functions from C n [a,b] ( i) ( ii) n d g dx n n d g dx n exists on ( a, b) continuous on [ a, b] then we say g belongs to the class of n th -continuously differentiable functions on [a,b], i.e., n g C [ a, b] Functions from C n [a,b] We also have the following special classes of functions: i. If n = 0, then C 0 [a,b] is the class of continuous functions on [a,b]. ii. If n = 1, then C 1 [a,b] is the class of continuously differentiable functions on [a,b]. chapter 3a: topics in differentiattion 41
Functions from C n [a,b] iii. If n = 2, then C 2 [a,b] is the class of twice continuously differentiable functions on [a,b]. Remar. Observe that most of the properties of differentiable functions in 3 follow the desired properties of functions in C n [a,b]: differentiable in (a,b) and continuous in [a,b]. Functions from C n [a,b] Theorem 3a.14. If n y = f( x) C [ a, b] then f has a Taylor approximation of the form n ( j) f ( x0) 0 x x0 j= 1 j! j f( x) f( x ) + ( ) x domf 0 Functions from C n [a,b] Corollary 3a.15. If n y = f( x) C [ a, b] and f and its derivatives up to the n th order are defined at zero, then f has the Maclaurin approximation f( x) f(0) n + j= 1 ( j) f (0) j x j! chapter 3a: topics in differentiattion 42
Functions from C n [a,b] Exercise. Prove the Leibniz Rule for the n th derivative of a product: n = = 0 ( ) n n ( n ( ) ( ) ) ( ) ( fg x f x g ) ( x) [Real] analytic functions Analytic functions Definition. A real-valued function f is said to be [real] analytic on a nonempty, open interval (a,b) iff given x 0 (a,b), there is a power series centered at x 0 which converges to f near x 0 ; i.e., iff there exist coefficients { a } N chapter 3a: topics in differentiattion 43
Analytic functions and points c, d (a,b), such that c < x 0 < d and f( x) = a ( x x ), x ( c, d) = 0 0 Remar. We apply some conventions from the previous definition: that f (0) = f, and from factorials, 0! = 1. Analytic functions Exercise. Let f(x) be an infinitely differentiable function such that for all reals, we have f (x) = f(x) and that f(0) = 1. What is f? Analytic functions Theorem 3a.16. Let c, d be extended real numbers with c < d, let x 0 (c,d), and suppose that f :( c, d) R If f( x) = a ( x x ), x ( c, d) = 0 0 chapter 3a: topics in differentiattion 44
Analytic functions then and a f C ( c, d) ( ) f ( x0) =, N {0}! Analytic functions Remar. The previous theorem yields the Taylor expansion [or Taylorseries representation] of f centered at x 0. Note also that this implies that f C ( a, b) This is in contrast to the version in 3: differentiation of f is only up to the (n + 1)-order. Analytic functions Definition. Let f C (a,b). The Taylor series expansion of fabout x 0 (a,b) is the infinite sum ( ) f x0 = 1 ( ) ( x x )! 0 If x 0 = 0, the above infinite sum is called a Maclaurin series expansion of f. chapter 3a: topics in differentiattion 45
Analytic functions Remar. [A.L. Cauchy] The function f( x) = 0 x = 0 2 exp( x ) x 0 has no Taylor approximation around the point x = 0. Analytic functions Definition. Let f C (a,b) and x 0 (a,b). The remainder term of order n of the Taylor expansion of f centered at x 0 is the function R x R x f, x0 n( ) = n ( ) n 1 ( ) f ( x0) : = f( x) ( x x )! = 1 0 Analytic functions Remar. From Theorem 3a.16 and the definition of the remainder, a function f C (a,b) is analytic on (a,b) iff for each x 0 (a,b) there is an interval (c,d) containing x 0 such that, 0 lim R f x n ( x) = 0, x ( c, d) n chapter 3a: topics in differentiattion 46
Analytic functions Recall that from Theorem 3.17, for every pair of points x, x 0 in (a,b), there is a p between x, x 0 such that R f ( p) ( ) ( n + 1)! ( n+ 1) = n 1 n 1 x + + x0 where n is shifted to n + 1. Analytic functions Theorem 3a.17. Let f C (a,b). If there is an M > 0 such that ( n ) n f ( x) M, x ( a, b), n N then f is analytic on (a,b). In fact, for each x 0 (a,b), f( x) = a ( x x ), x ( a, b) = 0 0 Analytic functions Example. The sine, cosine, and natural exponential functions are analytic on the whole of the real line and have the following Maclaurin series representations: ( i) expx = = 0 x! chapter 3a: topics in differentiattion 47
Analytic functions ( ii) ( iii) cosx sinx = = = 0 = 0 ( 1) x (2 )! 2 ( 1) x (2 + 1)! 2+ 1 Analytic functions Theorem 3a.18. Let I be an open interval and let c be the center of this interval. Suppose further that f( x) = a ( x c), x I = 0 If x 0 I and r > 0 satisfy ( x r, x + r) I 0 0 Analytic functions then ( ) f ( x0) f( x) = ( x x0),! = 0 x ( x r, x + r) 0 0 In particular, if f C whose Taylor expansion converges to f on some open interval J, then f is analytic on J. chapter 3a: topics in differentiattion 48
Analytic functions Example. The natural logarithmic function has Taylor series representation + 1 ( 1) ln x = ( x 1), x (0,2) = 1 Analytic functions Example. The function a x is analytic on the whole real line and has a Taylor series representation x (ln a) a = x, a > 0! = 1 Analytic functions Theorem 3a.19. [Bernstein] If f C (a,b) and f (n) (x) > 0 for all x (a,b) and for all natural numbers n, then f is analytic on (a,b). In fact, if x 0 (a,b) and f (n) (x) > 0 for all x [x 0,b) and for all natural numbers n, ( ) f ( x0) f( x) = ( x x0), x [ x0, b)! = 1 chapter 3a: topics in differentiattion 49
Analytic functions Lemma 3a.20. Suppose that f and g are analytic on the open interval (c,d) and let x 0 (c,d). If f( x) = g( x), x ( c, x ) then there is a δ > 0 such that f( x) = g( x), x ( x δ, x + δ) 0 0 0 Analytic functions Theorem 3a.21. [Analytic continuation] Suppose that I and J are open intervals, that f is analytic on I, that g is analytic on J, and that a, < b are points in the intersection of I and J. If f( x) = g( x), x ( a, b) then f( x) = g( x), x I J To end... Pure mathematics is on the whole distinctively more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. GH Hardy [1877-1947] chapter 3a: topics in differentiattion 50