August 20, Review of Integration & the. Fundamental Theorem of Calculus. Introduction to the Natural Logarithm.

Transcription

1 to Natural Natural to Natural August 20, 2017

2 to Natural Natural Natural 4

3 Incremental Accumulation of Quantities to Natural Natural Integration is a means of understanding and computing incremental accumulation of oriented quantities such as: displacement velocity work Although it might surprise you at first, area and volume are also best understood as oriented quantities. To see this, it might help to visualize volume as liquid in a tank: it can go up or down.

4 Canonical Example: Area Bounded by Graph of a Function to Natural As anor example, let f be continuous on an interval. It seems evident that region bounded by graph of f and x-axis along this interval has a well-defined area. y=f(x) Natural I To see this area as a quantity that accumulates with changes in x, we must view it an oriented quantity: just as with displacement, any accumulation can be reversed. We will now justify that this region has a well-defined oriented area. n we will see how to compute it with ease for many functions by first understanding its derivative.

5 Area Bounded by Graph of a Constant Function to Natural Natural First consider a constant function f (x) = c. oriented area bounded by graph of f from a to b is (b a)c. Note that this quantity is: a 0 if a = b or c = 0. Positive if a < b and c > 0 Negative if a < b and c < 0 Negative if a > b and c > 0 Positive if a > b and c < 0 + b a _ b b _ a b + a x-axis magnitude of oriented area is to oriented area as distance is to displacement and as speed is to velocity.

6 Area Bounded by Graph of a Function to Natural Natural Now consider any function that is continuous on an interval, along with two points a and b in that interval. Let us focus first on case that a < b and f is positive and increasing on [a, b]. For each positive integer n, consider partition of [a, b] into n subintervals of equal length x = b a n. ( x clearly depends on n, but it is cumbersome to incorporate n into notation.) Dx = (b-a)/n y=f(x) Dx Dx Dx Dx Dx Dx... a=x0 x1 x x =b n 2

7 Region Bounded by n th Lower Piece-wise Constant Approximation to Natural Natural On each interval [x i, x i+1 ], consider minimum value of f ; in this case, it will be f (x i ). oriented area bounded by graph of constant function g i (x) = f (x i ) is f (x i ) x. region bounded from a to b by graph of piecewise constant function g(x) = g i (x) for x [x i, x i+1 ] is contained in region bounded by graph of f. Dx = (b-a)/n y=f(x) Dx Dx Dx Dx Dx Dx... a=x0 x1 x x =b n 2

8 Area bounded by n th Lower Piece-wise Constant Approximation to Natural Natural Dx = (b-a)/n y=f(x) Dx Dx Dx Dx Dx Dx... a=x0 x1 x x =b n 2 Its oriented area is L n = f (x 0 ) x + f (x 1 ) x + f (x 2 ) x + + f (x n 1 ) x n 1 = f (x i ) x. i=0

9 Region Bounded by n th Upper Piece-wise Constant Approximation to Natural Natural On each interval [x i, x i+1 ], consider maximum value of f ; in this case, it will be f (x i+1 ). oriented area bounded by graph of constant function h i (x) = f (x i+1 ) is f (x i+1 ) x. region bounded from a to b by graph of piecewise constant function h(x) = h i (x) for x [x i, x i+1 ] clearly contains region bounded by graph of f. Dx = (b-a)/n y=f(x) Dx Dx Dx Dx Dx Dx... a=x0 x1 x x =b n 2

10 Area Bounded by n th Upper Piece-wise Constant Approximation to Natural y=f(x) Natural Dx = (b-a)/n Dx Dx Dx Dx Dx Dx... a=x x x x =b 0 1 n 2 Its oriented area is U n =?

11 Area Bounded by n th Upper Piece-wise Constant Approximation to Natural Natural Dx = (b-a)/n y=f(x) Dx Dx Dx Dx Dx Dx... a=x0 x1 x x =b n 2 Its oriented area is U n = f (x 1 ) x + f (x 2 ) x + f (x 3 ) x + + f (x n ) x = n f (x i ) x. i=1

12 Limits as n to Natural Natural As we let n, L n increases and U n decreases. (Why?) Clearly L m U n for any m and n. By continuity of real number system, lim n L n and lim n U n must exist. In fact, se limits are same, as we will see in a moment. Since region bounded by function f contains region bounded by each lower piece-wise constant approximation and is contained in region bounded by each upper piece-wise constant approximation, its area must be this common limit.

13 lim n L n = lim n U n = Area of Region bounded by f to Natural Natural f(b) f(a) Dx Dx = y=f(x) Dx Dx Dx Dx Dx Dx (b-a)/n a=x x x... x =b 0 1 n 2 difference between L n and U n, in this case, is (f (b) f (a)) x n. As n, x n 0; hence this difference goes to 0.

14 Area in General to Natural Natural A similar argument shows this is case on intervals on which f is decreasing. It also causes no difficulty if f (x) 0 for some inputs x or if b < a. Some of approximating constant pieces may bound negative areas, but all of relationships and reasoning extend to general case. re is also no need to subdivide interval from a to b into equal subintervals; any subintervals will do as long as ir maximum length goes to 0 as n. For less tractable but continuous functions, such as f (x) = x sin( π x ) on an interval containing 0, domain cannot be divided into finitely many intervals of increase or decrease. For such functions, more refined arguments are needed. Noneless, all of results above hold.

15 Or Approximations to Natural Natural Dx = (b-a)/n y=f(x) Dx Dx Dx Dx Dx Dx... a=x0 x1 x x =b n 2 Furrmore, all or approximations, such as midpoint and trapezoidal approximations, are squeezed between L n and U n as well. se or approximations (especially trapezoidal) converge more quickly than L n and U n. But we can often calculate area exactly.

16 Extension of se Methods to Or Measures; of Measure to Natural methods we used to define oriented area bounded by a curve extend naturally to or measures such as oriented volume bounded by a surface. y also extend more generally to physical quantities such as work. Natural

17 Area as a Variable Quantity to Natural Natural Consider oriented area bounded by a continuous function f from an initial input a to x. This oriented area is a function of x: call it A = F (x). We will first compute F (x) and use it to compute F (x). Note that to do this we must view oriented area bounded by f as a quantity that varies with ending point of interval, x, just as we viewed velocity as a quantity that varied with time. Recall that F (x) = A lim x 0 x.

18 { Bounding Derivative of Area Let f (ˇx) be minimum value of f on interval form x to x + x, and let f (ˆx) be maximum value of f on interval form x to x + x. to Natural A=F(x) DA Natural a x x x x+dx n f (ˇx) x A f (ˆx) x; hence, Dx f (ˇx) A x f (ˆx).

19 Computing Derivative of Area Using Squeeze orem to Natural Natural Since ˇx and ˆx are between x and x + x, ˇx x and ˆx x as x 0. Since f is continuous, limˇx x f (ˇx) = limˆx x f (ˆx) = f (x). Combining two previous observations, we obtain lim f (ˇx) = lim f (ˆx) = f (x). x 0 x 0 Thus, by Squeeze orem, F (x) = A lim x 0 x = f (x)! fact that F = f and initial condition F (a) = 0 completely determine function F.

20 : Formal Definition to Natural Natural general concept that captures all of se examples and more is Riemann : Definition Let f be a function defined on interval from a to b. For each positive integer n, let x = b a n. Let x i be any point between x i 1 and x i integral of f from a to b, denoted by b a f (x)dx, is following limit, if it exists and has same value for all choices of xi : b a f (x)dx = lim n n f (xi ) x Remark: interval from a to b is directed: a can be greater than b; x can be negative. i=1

21 Existence of Riemann to Natural Natural If Riemann integral of f from a to b exists, n f is Riemann integrable on interval from a to b. A sum n i=1 f (x i ) x is called an nth Riemann Sum. If f (ˇx i ) is minimum value of f (x) between x i 1 and x i, n n i=1 f (ˇx i) x is n th lower Riemann Sum. If f (ˆx i ) is maximum value of f (x) between x i 1 and x i, n n i=1 f (ˆx i) x is n th upper Riemann Sum. For a function to be Riemann integrable on an interval, it suffices that upper and lower Riemann sums converge to same limit as n. Continuity of f on an interval is sufficient to guarantee that f is Riemann integrable on this interval. (Weaker but more complicated conditions also suffice.)

22 of to Natural Natural value of Riemann integral is clearly oriented area bounded by graph of f from a to b. But it provides a general concept and notation that is valuable in situations where we don t necessarily want to picture this quantity as an oriented area. following important properties of Riemann integral are consequences of its definition: b 1 a f (x)dx + c b f (x)dx = c a f (x)dx. ( a f (x)dx = 0.) a a 2 b f (x)dx = b a f (x)dx. 3 If m f (x) M for a x b, n m(b a) b f (x)dx M(b a). a 4 For real numbers λ and µ, b a [λf (x) + µg(x)]dx = λ b a f (x)dx + µ b a g(x)dx. Make sure you can explain m using pictures!

23 to Natural Natural Using properties just mentioned and translating our reasoning about area into modern language of integration, we state our most significant orem: orem ( ) Let f be a function that is continuous on [a, b]. 1 Let F (x) = x a f (t)dt, a x b. n F is continuous on [a, b] and differentiable on (a, b), and F (x) = f (x). 2 If G be any anti-derivative for f, n b a f (x)dx = G(b) G(a). Please take note of hyposis: result does not apply unless function f is continuous on [a, b].

24 Proof of to Natural Natural 1 F F (x+ x) F (x) (x) = lim x 0, by definition of x F (x+ x) F (x) x = derivative, and definition of function F. x+ x a f (t)dt x a f (t)dt x R x+ x a = f (t)dt R x a f (t)dt x x+ x x f (t)dt x, by by Property (1) of Riemann integral, as listed previously. Since f is continuous on [x, x + x], Extreme Value orem applies, giving points ˇx and ˆx such that f (ˇx) and f (ˆx) are absolute minimum and maximum values on [x, x + x], respectively. By Property (2) of Riemann integral, as listed previously, f (ˇx) x x+ x x f (t)dt f (ˆx) x

25 Proof of orem, Continued to Natural Natural Hence, f (ˇx) x+ x x f (t)dt x f (ˆx). As x 0, clearly ˇx x and ˆx x (since x ˇx, ˆx x + x); furrmore, since f is continuous, as ˇx x and ˆx x, f (ˇx) f (x) and f (ˆx) f (x) (by definition of continuity). Thus, by Squeeze orem, F (x) = lim x 0 x+ x x f (t)dt x = f (x) 2 By definition of F, b a f (x)dx = F (b). Since G = F, F = G + C, where C is a constant. Since F (a) = 0, C = G(a). Thus b a f (x)dx = F (b) = G(b) G(a). Q.E.D.

26 An Important Function: Anti-Derivative of f (x) = 1 x to Natural Natural Consider function f (x) = 1 x for x > 0. (On interval (0, ), f is continuous.) Is re a function F such that F (x) = f (x) = 1 x? x 1 Yes, of course! It is given by F (x) = dt, oriented 1 t area under curve y = 1 t between t = 1 and t = x. We choose 1 for starting point because we want F (1) = 0 (for reasons that will become apparent shortly). Any or fixed positive number a would do, and would simply give a function that differs from ours by a constant. That constant would be 1 1 a t dt, oriented area under curve between x = a and x = 1. Since y > 0, F (x) = x 1 1 t dt is positive if x > 1 and negative for x < 1.

27 Calculating F (x) = x 1 1 t dt. to Natural Natural A harder question: Is re a formula for F constructed by adding, subtracting, multiplying, dividing, or composing algebraic and trigonometric functions? No! But we can calculate values of F (x), for any positive input x, to any desired degree of accuracy using Riemann Sums to estimate Riemann. First let us see why function F is so important! We now turn to Assignment 1, Section 1.

28 Function F is Natural to Natural Natural transcendental function F is important enough to have a name, just like trigonometric functions. F is called natural logarithm, abbreviated ln. Why? Consider: by definition, ln(ab) = ab 1 1 t dt = a 1 1 ab t dt + a 1 ab 1 dt = ln a + t a t dt. For second integral, substitute u = t a, obtaining ln(ab) = ln(a) + ln(b) F is a logarithmic function! It follows that for any rational number r, ln(a r ) = r ln a. (Why?)

29 Natural to Natural Natural natural logarithm is differentiable and increasing, hence injective, on interval (0, ). natural logarithm is surjective onto R. Why? Thus ln has a differentiable inverse exp : R (0, ). inverse of a logarithmic function is an exponential function: exp(x + y) = exp(x) exp(y). It follows that for any rational number r, exp(rx) = exp(x) r. (Why?) Let e = exp(1). n for any rational number r, exp(r) = e r. More generally, since e r agrees with exp(r) for any rational number r, it is natural to define e x = exp(x) for any real number x (wher rational or irrational).

30 Estimating value of e to Natural Natural Using a little trick with derivative of natural logarithm, along with continuity and computational properties of exponential function, we can express number e as limit: e = e 1 = e ln (1) = e lim h 0 ln(1+h) ln(1) h = lim h 0 (1 + h) 1 h ( = lim n n = lim h 0 e 1 h ln(1+h) ) n Using this limit we can estimate e to any degree of accuracy that we wish. (We will do this in class.)

31 Derivative of Natural Function to Natural Natural To calculate derivative of exponential function, we use implicit differentiation, which works whenever we know derivative of inverse of a function. Observe that ln(e x ) = x. Let Let u = e x. Taking derivative of each side, applying chain rule to composition, we obtain 1 u du dx = 1. Thus, du dx = u = ex. exponential function is its own derivative! This means that e x increases at an extraordinary rate: bigger it gets, faster it increases; faster it increases, faster its rate of increase increases; faster its rate of increase increases, faster rate of increase of its rate of increase increases,...! Its growth compounds upon itself.

32 General Real Exponents and s to Natural Natural For every positive real number a and every rational number r, a r = ( e ln a) r = e r ln a. Thus it is natural to define a x = e x ln a for any positive real number a and any real number x (wher rational or irrational). function exp a : R (0, ) defined by exp a (x) = a x is differentiable and bijective, except in special case a = 1. (Why?) refore, exp a has a differentiable inverse log a. It is easy to compute that log a = ln ln a. (That is, log a(x) = ln x ln a ).

33 When a Rabbit Meets Anor Rabbit and y Fall in Love to Natural Natural exponential function is crucial for modeling quantities whose rates of growth are proportional to ir sizes. For example, it has many applications to population biology Suppose re is a population of, say, 100, 000 rabbits with an unlimited food supply and no predators. We wish to study how population will grow over time. Obviously, population changes one rabbit at a time. But with so many rabbits, both birth of new rabbits and death of old rabbits will be very frequent, and step of adding one rabbit is very small compared to total population. So we can learn a lot by approximating population of rabbits as a smooth function of time, p = F (t). What differentiable function F best models rabbit population?

34 Rabbits & More Rabbits & More & More Rabbits to Natural Natural It is reasonable to assume that, given ir unlimited food supply and absence of predators, rabbits have a constant birth rate and a constant death rate. Subtracting death rate from birth rate gives a constant rate of increase. If we measure time in months, this rate will be in rabbits per rabbit per month. Thus, rate of change in rabbit population (in rabbits per month) at any instant is proportional to number of rabbits at that instant. For example, let us suppose birth rate is 2 rabbits per rabbit per month and death rate is.1 rabbit per rabbit per month. This gives a rate of increase of 1.9 rabbits per rabbit per month. So when re are p rabbits, rate of increase will be 1.9p rabbits per month.

35 & More & More & More Rabbits to Natural Natural Translating into a differential equation, we obtain dp dt = 1.9p Rearranging this equation, we obtain dp p = 1.9dt Anti-differentiating, we obtain ln p = 1.9t + C At t = 0, p = 100, 000, so C = ln(100, 000); hence p = e 1.9t+ln(100,000) = e ln(100,000) e 1.9t = 100, 000e 1.9t. p = 100, 000e 1.9t

36 Doubling Time to Natural How long does it take before population of rabbits doubles to 200, 000? Solving for t in equation 200, 000 = 100, 000e 1.9t yields t = ln Let s use Riemann sums to calculate ln 2. (We will do this in class.) Natural