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SCNC1111 Scientific Method and Reasoning Part 2 a Lecture 8 Calculus 21st, 24th October, 2014 Dr. Rachel Lui Truth Table for OR Rain? No No Yes Yes Coffee? No No Yes Yes UV index > 6? No Yes No Yes Tea? No Yes No Yes Bring umbrella? No Yes Yes Yes Coffee OR Tea? No Yes Yes Yes 2 We introduced him in Lectures 1 & 2 Who is he? 3 Surely You’re Joking, Mr. Feynman! • Released in 1985 • Title: Cream or Lemon in his tea • His fascination with Safe-cracking • Studied various languages • Ventures into art and Samba music • The Manhattan Project • His slightly nervous presentation of his graduate work in front of Einstein New York Times Bestseller 4 what determines that curve? • When I was in high school, I’d see water running out of a faucet growing narrower, and wonder if I could figure out what determines that curve. I found it was rather easy to do. —Richard Feynman 5 water running out of a faucet growing narrower • Position of a certain water particle from the faucet? • Assuming the water is falling smoothly and the horizontal cross-section is a circle, what is the rate of change of the radius of the stream? • Rate of water running out of the faucet? Gallons per minute? Liters per fortnight? • ….it was rather easy to do. 6 How do you call it? 7 Evacuation slides • Slides must deploy in six seconds in temperatures ranging from –54 to 71 degrees Celsius and unfurl in winds up to 12.8 m/s. • Slides inflate with an initial boost from a canister of compressed carbon dioxide and nitrogen. Gas from the canister accelerates. • If we want the slides to be ready in six seconds, how fast should we inflate them? 8 COOL Pilot In Command 9 What does the pilot in command need to know? • The change of static pressure with respect to altitude • The change of temperature with respect to altitude • The change of air density with respect to pressure • Calculus is the mathematics of change. Whenever we have situations of dynamical (and continuous) changes, we would consider using calculus to study and solve the problem. 10 How do all these things related to SCNC1111? 11 Pioneers of modern Calculus • Leibniz introduced the symbol ∫ for integral and wrote the derivative of a function y of the variable x as . 12 What is Calculus About Calculus differential calculus (differentiation) integral calculus (integration) rate of change accumulation inverse process of differentiation 13 Rate of change = finding slope? y = 100 • Change = ? • Rate of change =? • Slope = ? y = 2x+5 • Change = ? • Rate of change =? • Slope =? Slope = Δ Δ = rise run • Rate of Change = finding slope 14 How to find the slopes of complex curves? 15 Slope of the tangent at a point 16 Slopes keep changing 17 Let’s zoom in… 18 Let’s zoom in… 19 Derivative of a Function Definition Let () be a function. As ℎ approaches 0, the derivative of , denoted by () or . In other words, ′ = ≔ lim ℎ→0 +ℎℎ−(). 20 Example Example For the function = 2 ′ + ℎ − () = lim ℎ→0 ℎ = lim +ℎ 2−2 ℎ→0 ℎ = lim 2+2+ℎ2 −2 ℎ→0 ℎ = lim 2+ℎ2 ℎ→0 ℎ = lim 2 + ℎ ℎ→0 = 2 21 Example (-2,4) = 2 ′ = 2 (1,1) 22 Differentiation • Using the definition of the derivative to find the () can be very tedious. • E.g., if = cos3 2+1 − +4+32+2 then according to the definition we have ′ + ℎ − () = lim ℎ→0 ℎ cos3 + ℎ 2 + 1 − + ℎ cos3 2 + 1 − = +ℎ lim + ( + ℎ)4 + 3( + ℎ)2+2 − + 4 + 32 +2 ℎ→0 ℎ which is difficult to compute directly. 23 Derivative of Some Simple Functions Powers of • If = is a constant function, then ′ = 0 • If = where is a nonzero integer, then ′ = −1 • More generally, if = where is a nonzero real number, then ′ = −1. – Note: in some cases when is not an integer then the domain of needs to be restricted to the set of nonnegative . • Examples: • 2 = 2 • 1 5 = −5 = −5−6 = −5 6 • = 1 2 = 1 2 −12 = 1 2 24 Derivative of Some Simple Functions (cont.) Trigonometric functions • sin = cos , • cos = −sin y =sin x 25 Exponential function • = 2 Exponent 1 2 3 4 0 −1 −2 − −0.5 2 4 8 16 1 1 1 1 1 248 2 Characteristics of this function? 1. When x>0? 2. When x<0? 3. Will it touch the x-axis? 26 Derivative of Some Simple Functions (cont.) Natural exponential function • The Euler number, named after the mathematician Leonhard Euler, is a special number. • ≈ 2.7182818284590... • Taking exponents of gives the natural exponential function . 27 Derivative of Some Simple Functions (cont.) logarithmic functions • The inverse function of is the natural log function, denoted as ln (= log ). • = if and only if ln = . 28 Derivative of Some Simple Functions (cont.) = • More generally, if is a positive real number, then = (ln ) . 1 ln = • More generally, if is a positive real number, then • log =1 (ln ) 29 Algebra of Derivatives Theorem Let , be differentiable functions. Then • + ′() = ′() + ′() (i.e., + = + ) • ′ = ′ , where is any real constant number – log = 1 (ln ) • (product rule) ′ = ′ + () • (quotient rule) if ≠ 0, then ′ = ′ − 2 ′ 30 Examples • (2 + sin ) = • 2 sin = • ln 2+1 = • tan = 31 The Chain Rule Theorem Let , be differentiable functions, and let ( ∘ ) denote the composite function . Then • ∘ ′ = = ′ () Example • sin 3 = cos 3 32 = 32 cos 3 Exercises • ln 2 +1 =? • ln sin (2 + 1) =? • Find () if = cos3 2+1 − +4+32+2 32 Finding derivatives using Wolfram Alpha • d/dx ((cos (x^2+1))^3 - x^(1/2))/(e^x+x^4+3x^2+2)) 33 Some Remarks • In a rigorous calculus course, the materials are usually built on the concept of the limit of functions. In our course we can only introduce calculus using a quick approach and skip the details of the limit of functions. • The notations or look like fractions. But they are not fractions! (They are in fact the limit of the difference quotient Δ Δ when Δ approaches 0.) Therefore you should never use cancellation of factors in quotients to “simplify” the terms. 34 Some Remarks • So far we have used as the variable of the functions. One may use other letters or symbols to denote variables, and the rules on derivatives still apply. E.g., one will have: 2 − 3 = 2 − 3, = , etc. 35 Inputting math expressions to Wolfram Alpha • d/dx [function in x] or d/dy [function in y] : find derivative E.g.: d/dx x^3+cos(x) (although we do not put brackets around x^3+cos(x), wolfram alpha will interpret it as ( 3 + cos ) , not 3 + cos • lim [function in x] as x->a : find limit of function as approaches a constant E.g.: lim ((x+h)^n-x^n)/h as h->0 or lim ((x+h)^a-x^a)/h as h->0 Exercise The number of subscribers to Cable TV in Hong Kong is approximated by the function = 1500 2 + 5 where denotes the number of subscribers to the service in the month. Find the rate of change of the number of subscribers in the 8th month. Please correct your answer to the nearest integer. Rate of change = ′ 8 = 750 × 5 × 2 + 5 × 8 −12 = 579. 37 Exercise At noon, a ship A is 50km due west of a ship B. A is sailing due north at 25km/h while B is sailing at 20km/h due west. What is the rate of change in the distance between the two ships at 2:30pm? Answer: the rate of change in the distance between the ships is 25 km/h. 38 Higher Order Derivatives Definition For a function , its derivative function ′() = is also a function in , and we may differentiate it to obtain the second derivative of (), which is denoted as () or 2 2 or 2 2 (). • In other words, ′ = = ′′ = 2 2 • Examples: • If = , then ′ = −1, ′′ = − 1 −2 • If = sin 2, then = 2 cos 2 , 2 2 = −4 sin 2 39 Higher Order Derivatives Definition More generally, we differentiate a function times to obtain the -th derivative of , which is denotes as () or • Finding higher order derivatives with Wolfram Alpha: input d^2/dx^2 [function in x] or d^3/dy^3 [function in y] 40 Interpretation of the First and Second Derivatives • Because ′ is the slope of the tangent line to the graph of , if ′ > 0 then () is increasing; if ′ < 0 then () is decreasing. 41 Interpretation of the First and Second Derivatives • As () is the rate of change of , ′′ is the rate of change of (). • If we interpret () as the slope of the tangent lines to the graph = (), then ′′() is the rate of change of the slope of the tangent lines. ′ < 0 ′ > 0 ′′ > 0 ′′ < 0 42 Interpretation of the First and Second Derivatives • if ′′ > 0 at = then the slope of the tangent lines is increasing as increases around the point , . We say the graph concaves up at = • if ′′ < 0 at = then the slope of the tangent lines is decreasing as increases around the point , . We say the graph concaves down at = Graph Concaves down Graph concaves up 43 Interpretation of the First and Second Derivatives 44 Interpretation of the First and Second Derivatives • Suppose an object moves along a straight line, and denotes the distance travelled (measure from a fixed point O) along a certain direction, where denotes the time elapsed. • Then the derivative () is the velocity of the object, and the second derivative ′′ is the acceleration of the object. () O 45 Shape of Graphs by Derivatives • For functions which have smooth graphs, their first and second derivatives can help us to locate the points , at which () is a local extremum. local maximum local minimum local and global maximum 46 Shape of Graphs by Derivatives • If ′ = 0 and ′′ > 0, then the tangent line at , to the graph is horizontal, and the graph concaves down at = . Hence attains a relative minimum at = • Similarly if ′ = 0 and ′′ < 0, then attains a relative maximum at = Note that the converse of the above is not true. E.g., = 4 has a local (and global) minimum at = 0, yet ′′ 0 = 0. 47 Finding Extrema with Wolfram Alpha • Input: maximize [function in x] or minimize [function in x] 48 Optimization examples • Bracken and McCormick (1968): Determine the equilibrium composition of compound 1 2 24 + 1 2 2 at temperature = 3500 and pressure = 50 . • Laws of Thermodynamics • Quantum Theory 49 Optimization Problem Problem An epidemic is spreading through a town. Let denotes the total population and is the number of infected people. It is estimated that the epidemic spreads at a rate that is proportional to the number of uninfected people and to the square of the number of infected people. Then the rate is = 2( − ) where is a constant. When will this rate be greatest, in terms of the number of infected people? 50 Optimization Problem Solution • To solve this problem, we may first find the value(s) = 0 such that ′ 0 = 0, and test whether ′′ 0 < 0. Such a value of 0 will give local maximum of . – It may require additional argument to conclude that it actually gives a global maximum. • Another method is to use Wolfram Alpha. However, because , have unspecified values, Wolfram Alpha might treat them as variables and hence cannot give a useful result right away. For this particular problem, there is no loss of generality to put = 1. We may input maximize x^2(p-x), 0<=x<=p so that Wolfram Alpha will find the maximum value of the function 2( − ) for the range 0 ≤ ≤ . 51 Optimization Problem Solution (cont.) The following is the output of Wolfram Alpha • We conclude that () is maximum when = 23. 52 Exercise It is estimated that there are 4,000,000 passengers taking MTR everyday. Assume that the fare per ride is $8 for all people. MTR company is considering raising the fare in order to generate a larger profit. However, a marketing research shows that for each $0.5 increase in fare, the number of passengers taking MTR would be reduced by 30,000 per day. What should be the fare per person in order to maximize the profit? Answer: ≈ $37.3 53 Exercise The amount of sulphur dioxide, a major source of air pollution, present in the atmosphere on a certain day in Hong Kong is estimated by = 140 1+0.16(−6.4)2 + 20 where is measured in pollutant standard index (PSI) and is measured in hours, with = 0 corresponding to 6am. Find the time of day when the pollution is at its highest level. Answer: 12:24pm. 54 Method of exhaustion • To compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. 55 How to find the areas of complex objects? • Algebra deals with areas of triangles and rectangles. • Calculus deals with areas of complex objects. 56 Integration • Integration is about accumulation, summing up, finding area (or finding volume), etc. • It is the reverse process of differentiation • There are two kinds of integration: – indefinite integration – definite integration 57 A Simple Example of Anti-Derivatives () () 2 2 + 1 2 − 3.141592653589 2 + 2.718281828 2 + 10100 • All functions of the form 2 + , where is an arbitrary constant, are anti-derivatives of 2. • We call the most general form 2 + of the anti- derivative of 2 the indefinite integral of 2, and denote this as ∫ 2 = 2 + 58 Indefinite Integral Definition If () and () are such that = (), then we call an anti-derivative of (). In this case all anti-derivatives of must be in the form of + , where is any arbitrary constant. We write ∫ = + and call ∫ the indefinite integral (or simply integral) of (). 59 Some Simple Integral Formulae Differentiation If (≠ −1) is a real constant, 1 + 1 +1 = 1 ln || = = If (> 0) is a real constant, ln = sin = cos (−cos ) = sin Integration ∫ = 1 +1 +1 + 1 ∫ = ∫ = ln + ∫ = + ∫ = ln + � cos = sin + � sin = − cos + 60 Algebra of Indefinite Integrals • ∫ ± = ∫ ± ∫ • ∫ = ∫ where is any constant real number 61 Examples •∫ 1 = ∫ 2 = 1 12+1 12+1 + = 2 3 3 2 + •∫ + 2 = 1 2 2 + 2 ln + • ∫ = ∫ 1 = ∫ 0 = 1 0+1 0+1 + = + 62 Some Remarks on Indefinite Integrals Remarks • Because anti-derivatives of a function is not unique, finding indefinite integrals may result in (seemingly) very different answers. But these different answers always differ by only some constants. • E.g., ∫ 1+ 2 = −1 1+ + , also ∫ 1+ 2 = 1+ + . Although −1 1+ and 1+ look quite different, actually −1 1+ + 1 = 1+ . 63 Some Remarks on Indefinite Integrals Remarks • While finding derivatives by hand calculation is quite straightforward (by using the product rule, quotient rule, chain rule, etc.), finding anti-derivatives can be very tricky and difficult. • Some common functions (e.g., −2) even do not have antiderivatives that can be expressed in closed forms in terms of elementary functions! • So far we used as the variable. Actually we can use any other symbol as the variable. E.g., we have ∫ 2 = 1 3 3 + , ∫ = + , ∫ = ln + 64 Definite Integral Let be a function defined over [, ]. Consider the area bounded by the vertical lines = and = , and between the graph = () and the horizontal -axis. Source: Wikimedia Commons Definition The definite integral of over [, ] (or the definite integral of from to ) is denoted and defined as ∫ = (area of the portion above the -axis) −(area of the portion below the -axis) • In the above, and are respectively called the lower limit and the upper limit of the integration. 65 Algebra of Definite Integrals • ∫ ± = ∫ ± ∫ • ∫ = ∫ where is any constant real number • For convenience, we also define ∫ = − ∫ (here we interchange the lower and upper limits of integration). • We have � = � + � for any real numbers , , (here it is not required that ≤ or is between and ) 66 Fundamental Theorem of Calculus Theorem (Fundamental Theorem of Calculus) a) If () is a function which is differentiable over [, ], and = () (i.e., is an anti-derivative of ), then � = − () b) If () is a “well-behaved” function defined over [, ], then � = () 67 Example Problem A person can memorize words at a rate of 4/ words per minute. a) Find a formula for the total number of words that can be memorized in minutes (Hint: At time = 0, 0 words have been memorized). b) Use your result in (a) to find the total number of words that can be memorized in 25 minutes Solution (a) Let denote the total number of words that can be memorized at time min. Because = 4 = 4−12, we have = ∫ 4−12 = 4 1 −12+1 −12+1 + = 8 + At = 0, we have 0 = 0, i.e., 0 = 8 0 + = 0 + , which gives = 0. Therefore the formula is = 8 . b) When = 25 min, the total number of words that can be memorized is 25 = 8 25 = 40. 68 Example Problem a) Show that 1 2 ( + sin cos ) is an anti-derivative of the function cos2 . b) Evaluate the definite integral ∫04 cos2 . Solution a) To verify that 1 2 ( + sin cos ) is an anti- derivative of cos2 , we differentiate 1 2 ( + sin cos ) to see whether the result is equal to cos2 or not (continued next slide) : 69 Example (Solution continued) a) 1 2 ( + sin cos ) = 1 2 + (sin cos ) =1 2 1+ sin cos + sin cos =1 2 1 + cos cos + sin − sin =1 2 1 + cos2 − sin2 =1 2 2 cos2 = cos2 Hence 1 2 ( + sin cos ) is an anti-derivative of cos2 b) Because = 1 2 ( + sin cos ) is an anti-derivative of cos2 , applying the Fundamental Theorem of Calculus we get ∫04 cos2 = 4 − 0 = 1 + sin cos 24 44 − 1 2 0 + sin 0 cos 0 =1 2 + 1 1 4 22 − 0 = 8 − 1 4 = −2 8 70 Example Problem A race car can accelerate from a standing start to a speed of = −0.062 + 6 (m per sec) after seconds (for 0 ≤ ≤ 45) a) Find the distance it will travel from start in the first sec b) Use your result in (a) to find the distance the car will travel in the first 20 sec. Solution (a) If () denotes the distance the car travels in the first sec, then its speed is = = −0.062 + 6. Hence = ∫ −0.062 + 6 = −0.023 + 32 + At = 0, 0 = 0 = 0 + . Hence = 0. This gives = − 0.023 + 32. (b) The answer is 20 = … (find it yourself) 71 Finding Integrals with Wolfram Alpha • Finding anti-derivatives (indefinite integrals): integrate [function in x] dx or int [function in x] dx • Finding definite integrals: Integrate_a^b [function in x] dx or int_a^b [function in x] dx 72 Another Motivating Problem Problem Suppose a tree grows in such a way that its height is increasing at a rate of 2(1+0.8 cos(2)) +2 (m/yr) in a continuous manner. Here is the number of years since the tree has been planted from a seed. What is the height of the tree at 10 years after it was planted? Solution Let ℎ() denote the height of the tree at year after the tree was planted. It is given that = 2(1+0.8+co2s(2)). Hence the solution is ∫010 2(1+0.8 cos(2)) +2 . However, it is not easy to find the anti-derivative of 2(1+0.8 cos(2)) +2 by hand. We shall use Wolfram Alpha instead, by inputting int_0^10 2(1+0.8cos(2 pi t))/(t+2) dt 73 Another Motivating Problem Solution (cont.) The following is the output of Wolfram Alpha • This gives the height of the tree at = 10 years after planted as ℎ 10 = 3.593 m. 74 Exercises Problem 1 A flu epidemic is spreading in a town, and the number of people newly infected on day is modelled by a function = 122 − 3 (where 0 ≤ ≤ 12). Find the instantaneous rate of change of this number on (a) day 5 and interpret your answer; (b) day 10 and interpret your answer. Problem 2 The population (in millions) of a city can be approximated by () = 3.86(1.02) where is the number of years since 1990. What is the instantaneous rate of change at the year (a) 2000 and (b) 2010? Problem 3 Find the derivatives of the following functions (i) by hand computation and (ii) by Wolfram Alpha. (a) () = cot (= cos sin ) (b) = 2− 2+1 (c) = 2 + 1 75 Exercises Problem 4 Water is running into a cylindrical tank at a rate of 1000 litre per minute. If the cross section area of the tank is 500 m2, what is the rate of change of the water level? Problem 5 Concentration of a drug in the blood stream at time (hours) is given by = 5 9+2 (where 0 ≤ ≤ 30). When is the concentration increasing? When is it dropping? When is the concentration being highest? Problem 6 When a muscle lifts a load, it does so according to the Hill’s equation, + + = , where is the load that the muscle is lifting, is the velocity of contraction of the muscle, and , , and are constants. Use implicit differentiation to find . 76 Exercises Problem 7 A race car accelerate from a standing start to a speed of = −0.052 + 5 (m per sec) after seconds (for 0 ≤ ≤ 50) a) Find the distance it will travel from start in the first sec (0 ≤ ≤ 50) b) Use your result in (a) to find the distance the car will travel in the first half minute Problem 8 The number of bottles of water of a certain brand that a super market will sell per day and their price (dollars) per bottle are related through the equation 2 = 3800 − 602 − 80( + 5). If the current price is $5 and the price is falling at a rate of $0.2 per day, find how the sales will change. (Hint: assume that both and are functions of time (in days), and apply implicit differentiation.) 77 References • G.C. Berresford, A.M. Rockett, Applied Calculus (5th edition), 2010, Brooks/Cole, Cengage Learning 78

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