Lecture 3: Derivatives and extremes of functions

Transcription

1 Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16

2 Outline Derivatives Derivatives Lejla Batina Version: spring 2011 Wiskunde 1 2 / 16

3 Chain rule Derivatives (g f ) (x) = g (f (x))f (x) Example Consider f (x) = a x f (x) =(a x ) = lim h 0 a x+h a x =lim h 0 a x ah 1 h h = a x f (0). f (0) = 1 for a = e = So, it follows: (e x ) = e x (e f (x) ) = e f (x) f (x) (a x ) = (e lna x ) = e lna x lna = a x lna. = lim h 0 a x (a h 1) h = (1) Lejla Batina Version: spring 2011 Wiskunde 1 4 / 16

4 Logarithmic differentiation Definition According to the chain rule: (lnf ) = f f. This is called the logarithmic derivative of f and this method is called logarithmic differentiation. Example Differentiate the following functions: y = (sinx) x y = 3 x 2 1 x 1+x 2 sin 3 x cos 2 x Lejla Batina Version: spring 2011 Wiskunde 1 5 / 16

5 L Hôpital s rule Theorem (L Hôpital s rule) Suppose that lim x a f (x) = 0 and lim x a g(x) = 0 (or both f and f g approach ), and suppose also that lim (x) x a g (x) exists. Then f (x) lim x a g(x) exists, and: lim x a f (x) g(x) = lim x a f (x) g (x). Example ([ ] or [ 0 0 ]) lim x + lnx x ([0 ]) lim x 0 xlnx ([ ]) lim x 0 ( 1 x 1 e x 1 ) ([0 0 ],[ 0 ], [1 ]) lim x 0+ x 3 4+lnx Lejla Batina Version: spring 2011 Wiskunde 1 6 / 16

6 Higher Derivatives Let y = f (x) a real function. f (x) = df dx f (x) = d dx f (x) = d2 f dx 2 f (n) (x) = d dx f (n 1) (x) Example f (x) = x n, find f (n) (x). Lejla Batina Version: spring 2011 Wiskunde 1 7 / 16

7 Monotonicity How related to derivatives If f (x) 0, x [a, b] f is increasing on [a, b]. If f (x) 0, x [a, b] f is decreasing on [a, b]. Lejla Batina Version: spring 2011 Wiskunde 1 9 / 16

8 Absolute vs local extreme Definition A real function f : D R has in a D absolute minimum (maximum) if f (a) f (x) (f (a) f (x), x D. A real function f : D R has in a D a local minimum (maximum) if δ > 0 such that f (a) f (x) (f (a) f (x)), x (a δ, a + δ). Lemma Let f : D R and f is differentiable in a. If f has a local minimum(maximum) in a then f (a) = 0. A function can have a local extreme in a point in which the first derivative does not exist. (Example: f (x) = x has a minimum in 0.) Lejla Batina Version: spring 2011 Wiskunde 1 10 / 16

9 Necessary conditions for extremes Definition A critical point of a function f : D R, is a point a D such that f (a) = 0. The value f (a) is called a critical value of f. Theorem Let f : D R, f is differentiable in a, and f has a local min/max in a. Then f (a) = 0. In order to find the maximum and minimum of f : D R three kinds of points must be considered: the critical points of f in D, points x in D such that f is not differentiable at x, points on the edge of D (x is on the edge of D if [x δ, x) D = or (x, x + δ] D = ). Lejla Batina Version: spring 2011 Wiskunde 1 11 / 16

10 Sufficient conditions for extremes Lemma A differentiable function f (x) has a local minimum in a if δ > 0 such that f (a) = 0, f (x) 0 for x (a δ, a) and f (x) 0 on (a, a + δ). Especially, if f (a) = 0 and f (a) > 0. A differentiable function f (x) has a local maximum in a if δ > 0 such that f (a) = 0, f (x) 0 for x (a δ, a) and f (x) 0 on (a, a + δ). Especially, if f (a) = 0 and f (a) < 0. Example Consider the function f (x) = x 4 2x 2. Critical points are -1,0,1. Which of them are min/max? Lejla Batina Version: spring 2011 Wiskunde 1 12 / 16

11 Convexity and Concavity Definition A function f is convex on an interval if for all a and b in the interval, the line segment joining (a, f (a)) and (b, f (b)) lies above the graph of f. A function f is concave on an interval if for all a and b in the interval, the line segment joining (a, f (a)) and (b, f (b)) lies under the graph of f. A point of inflection on a curve y = f (x) is a point at which f changes from concave to convex or vice versa. Theorem If f (x) > 0, x (a, b) Γ f is convex on (a, b). If f (x) < 0, x (a, b) Γ f is concave on (a, b). If f has an inflection point at x and f exists in (x δ, x + δ), where δ > 0 = f (x) = 0. Lejla Batina Version: spring 2011 Wiskunde 1 14 / 16

12 Asymptotes Derivatives Definition A vertical line x = a is called a vertical asymptote of the graph of f (x) if lim x a =. A horizontal line y = b is called a horizontal asymptote of the graph of f (x) if lim x = b. A line y = kx + l is called a slant asymptote of the graph of f (x) if f (x) the following limits exist: k = lim x a x and l = lim x a (f (x) kx). Lejla Batina Version: spring 2011 Wiskunde 1 15 / 16

13 Curve Sketching The following points should be investigated (for a function f (x)): 1 domain of f 2 parity i.e. is f even or odd 3 points of intersection with x-axis and y-axis 4 behaviour of f on the edges of the domain 5 asymptotes 6 monotonicity and min/max 7 concavity/convexity and points of inflection Lejla Batina Version: spring 2011 Wiskunde 1 16 / 16