Chapter 4: Applications of derivative Extreme Values (SBS 4.1): Optimization Optimization is a whole eld in mathematics of nding the best (or worst) of something. Examples: UPS or Fedex wants to nd the best routes for drivers: minimize fuel usage or driving time (cost), maximize their prots. On a shuttle ight, NASA wants to minimize weight (i.e., fuel usage) while being able to accomplish the current mission. Airplane design: maximize performance, minimize cost or minimize weight 1 Absolute Extrema (Max and Min) Let f (x) be a function on some interval I containing the number c. 2 f (c) is an absolute maximum of f on I if f (c) ≥ f (x) for all x in I. 2 f (c) is an absolute minimum of f on I if f (c) ≤ f (x) for all x in I. 2 The absolute maximum and minimum are called extreme values or absolute extrema of f on I. 2 A function does not necessarily have a maximum or a minimum on a given interval. 2 Theorem. (Extreme Value Theorem) A function f has both an absolute maximum and an absolute minimum on any closed, bounded interval [a;b] where it is continuous. Note: if f is discontinuous or if the interval is not both closed and bounded, the EVT says nothing. We cannot conclude that f has or does not have an absolute max and min. 3 Relative Extrema (Max and Min) Denitions: 2 A function f has a relative maximum at a point c if f (c) ≥ f (x) for all x in an open interval containing c. 2 A function f has a relative minimum at a point d if f (d) ≤ f (x) for all x in an open interval containing d. 2 The relative maxima and minima are called relative extrema of f on I. 4 Suppose f is dened at c and either f 0(c) = 0 or f 0(c) does not exist. Then c is called a critical number of f , and the point P(c; f (c)) on the graph is called a critical point. Theorem. (Critical number theorem) If a continuous function f has a relative extremum at c, then c must be a critical number of f . (I.e., either the derivative is 0 or it does not exist at c.) 5 Procedure for nding absolute extrema of a continuous function f on [a;b]: 1. Compute f 0(x) and nd all critical numbers of f on [a;b]. (a) f 0(x) = 0 (b) f 0(x) does not exist 2. Evaluate f at the endpoints a and b and at each critical number c. 3. Compare the values from (2). (a) Largest is the absolute maximum of f on [a;b]. (b) Smallest is the absolute minimum of f on [a;b]. 6 Examples: f (x) = x5 − x4 on [−1;1] f (x) = sin2 x + cosx on [0;2] 7 The Mean Value Theorem (SBS 4.2) Theorem. (Rolle's Theorem) Suppose f is continuous on the closed interval [a;b] and dierentiable on the open interval (a;b). If f (a) = f (b), then there exists at least one number c between a and b such that f 0(c) = 0. 8 More generally, we have the MVT: Theorem. (Mean Value Theorem for Derivatives) If f is continuous on the closed interval [a;b] and dier- entiable on the open interval (a;b), then there exists in (a;b) at least one number c such that f (b) − f (a) = f 0(c) b − a That is, there is at least one point c at which the average value equals the instantaneous change in f . 9 Or, for example, if f represents the position of a moving object at a given time t: 2 There is at least one point t = c at which the aver- age velocity equals the instantaneous velocity. 2 If s(t) represents position at time t, then for a trip from time ta to time tb; average velocity is vave = s(tb)−s(ta) . There is at least one time on the trip when tb−ta my instantaneous velocity 0 ds equals my s (t) = dt = vinst average velocity vave. 10 Example: Verify that f (x) satises hypotheses of the MVT on [a,b] and nd the c guaranteed by the theorem. 1 f (x) = on [0,2] x + 1 11 Example: Show if Rolle's theorem applies, and if so, use it to conclude that there is a c such that f 0(c) = 0. f (x) = sin(x) on [0,2p] 12 Theorem. (Zero-derivative Theorem) If f is continuous on the closed interval [a;b] and dier- entiable on the open interval (a;b), with f 0(x) = 0 for all x on (a;b). Then the function f is constant on [a;b]. Theorem. (Constant Dierence Theorem) If f and g are continuous on the closed interval [a;b] and dierentiable on the open interval (a;b). Then if f 0(x) = g0(x) for all x in (a;b), there exists a constant C such that f (x) = g(x) +C for all x on [a;b]. 13 Derivatives for graph sketching (SBS 4.3) Recall: for x1and x2 in some interval I 2 A function f is strictly increasing on I if f (x1) < f (x2) whenever x1 < x2 2 A function f is strictly decreasing on I if f (x1) > f (x2) whenever x1 < x2 2 f is said to be strictly monotonic on an interval I if it is either strictly increasing on all of I or strictly decreasing on all of I. 14 Theorem. (Monotone function theorem) Let f be dierentiable on the open interval (a;b). If f 0(x) > 0 on (a;b), then f is strictly increasing on (a;b). If f 0(x) < 0 on (a;b), then f is strictly decreasing on (a;b). 15 1st Derivative Test for Relative Extrema of a Continu- ous Function f 1. Find all critical numbers c of f . 2. Classify each critical point (c; f (c)) as follows: 2 (c; f (c)) is a relative maximum if f 0(x) > 0 for all x in an open interval (a;c), and f 0(x) < 0 for all x in an open interval (c;b) 2 (c; f (c)) is a relative minimum if f 0(x) < 0 for all x in a open interval (a;c), and f 0(x) > 0 for all x in an open interval (c;b) 2 (c; f (c)) is not an extremum if the derivative f 0(x) has the same sign for all x in the open intervals (a;c) and (c;b) on each side of c. 16 Denition. If the graph of a function f lies above all of its tangents on an interval I, then it is said to be concave up on I. Denition. If the graph of a function f lies below all of its tangents on an interval I, then it is said to be concave down on I. 17 The graph of a function f is concave up on any open interval I where f 00(x) > 0, and concave down where f 00(x) < 0. 18 Denition. A point P(c; f (c)) on a curve is called an in- ection point if the graph is concave up on one side of P and concave down on the other side. Generally, concavity changes where f 00(x) = 0 or f 00(x) does not exist. These are second order critical numbers (i.e., critical numbers of f 0). 19 2nd Derivative Test for Relative Extrema Let f be a function such that f 0(c) = 0 and the second derivative exists on an open interval containing c. 2 If f 00(c) > 0 (concave up), there is a relative mini- mum at x = c. 2 If f 00(c) < 0 (concave down), there is a relative max- imum at x = c. 2 If f 00(c) = 0, the 2nd derivative test fails. (The point could be either a max, a min, or neither.) 20 Curve sketching examples: 21 Curve sketching with asymptotes (SBS 4.4) Limits to Innity lim f (x) = L x!+¥ means f (x) approaches L as x increases without bound. lim f (x) = M x→−¥ means f (x) approaches M as x decreases without bound. 22 Formal denition of limits to innity Denition. The limit statement limx!+¥ f (x) = L means that for any number e > 0, there is a number N1 such that j f (x) − Lj < e whenever x > N1 for x in the domain of f . Similarly, limx→−¥ f (x) = M means that for any number e > 0, there is a number N2 such that j f (x) − Mj < e whenever x < N2 for x in the domain of f . 23 Limit Rules If limx!+¥ f (x) and limx!+¥ g(x) exist, then for constants a and b: Power rule: lim [ f (x)]n = [ lim f (x)]n x!+¥ x!+¥ Linearity rule: lim [a f (x) + bg(x)] = a lim f (x) + b lim g(x) x!+¥ x!+¥ x!+¥ 24 Product rule: lim [ f (x)g(x)] = [ lim f (x)][ lim g(x)] x!+¥ x!+¥ x!+¥ Quotient rule: f (x) lim f (x) lim = x!+¥ if lim g(x) 6= 0 x!+¥ g(x) limx!+¥ g(x) x!+¥ Similarly for limx→−¥ f (x) and limx→−¥ g(x) if they exist. 25 More special limits If A is a real number and r is a positive rational number, then A lim = 0 x!+¥ xr If r is such that xr is also dened for x < 0, then A lim = 0 x→−¥ xr 26 Examples: limits to ± innity 27 Innite Limits lim f (x) = +¥ x!c means that f increases without bound as x approaches c (from either side).