AP Calculus (Mr. Surowski)

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1 AP Calculus Mr. Surowski) Lesson 14 Indeterminate forms Homework from Chapter 4 and some of Chapter 8) 8.2) l Hôpital s rule and more on its; see notes below.) 5 30, 35 47, 50, ): 1 12, 13 28, 35 38, 42, 51. Lesson 15 Relative rates of growth Explain the following growth table: Real Slow Slow Moderate Fast Real Fast Rediculuously Fast ln ln x ln x x n a x x x a ax A) Let fx) = a x, gx) = x x, x > 0) and show that f = og). Read the footnotes on polynomial-time algorithms. 2,3 Quiz over Lessons ): 1 4, 11 18, 19 26, For you should: a) state the natural domain of the given function; b) graph the function over its domain may use graphing calculator); c) determine all critical points in the interior of the domain; Lesson 16 Relative and absolute extrema d) determine all relative and absolute extrema of the function over its domain. A) Find the extreme values of the function fx) = 3x + cos x on the interval [0, 2π]. B) The entropy function of a binary source is defined to be Hx) = x ln x 1 x) ln1 x), where 0 x 1. Using l Hôpital s rule show that this definition makes sense at x = 0 by computing x ln x. Graph this function and compute the maximum entropy on the interval 0 x 1. 1 I wish that I had $1 for every time I saw a student fall into this trap! 2 As you know, mathematics is full of algorithms. One more familiar algorithm is that of solving a system of n linear equations in n unknowns. Notice that there are n 2 + n inputs: the coefficients of the unknowns and the n constants in the problem. The efficiency of an algorithm to solve such a problem would be measured as follows. Assume that fn) is the number of computations required to by a particular algorithm to solve the problem. How many computations are required to solve two equations in two unknows?) We would call an algorithm a polynomial-time algorithm if there is a polynomial function pn) such that fn) = Opn)). You should be able to convince yourselves that the standard methods based on eination and/or substitution) are polynomial-time algorithms. In fact, it can be shown that for such methods, fn) = On 3 ). 3 A much more complicated problem is the following. Let k be a very large number, say one with n digits. Is there a polynomial-time algorithm for determining the primality of k? This is a problem that stood for several decades. A polynomialtime primality testing algorithm was discovered in 2002 by Manindra Agarwal and two of his Ph.D. students Nitin Saxena, and Neeraj Kayal.

2 Homework from Chapter 4 and some of Chapter 8), cont d 4.2): 1 7, 11, 12, 15 22, 26 28, 37, 38, 43. Lesson 17 The Mean Value Theorem A) Refer to the function f in exercise A) of Lesson 16. Determine the intervals where f is increasing and the intervals where f is decreasing. Can you conclude that the function is 1 1 over its domain? Doesn t this prove that f 1 exists? Compare with footnote 5 of the Chapters 2 3 syllabus. B) The surge function looks like ft) = ate bt a, b > 0 are constants). a) Graph y = ft), t 0. Use use a = 2 and b = 0.05.) b) Find ft). t + c) Find the relative extrema of f d) Determine where f is increasing and where f is decreasing.

3 Lesson 18 Applications graphing Homework from Chapter 4 and some of Chapter 8), cont d 4.3): 3, 4, 5, 7 10, 15 18, 20, 21 24, 29, 30, 35 37, 48, 51. L A) The logistic function looks like fx) = 1 + Ce kt L, C, k > 0 are constants). to a) Compute ft) and ft). t + t b) The number L is called the carrying capacity of the logistic function. Why do you think this is? c) Show that if y = fx), then y = k yl y). L d) Show that f x) = 0 where fx) = L/2. Use part c) ) e) The point of inflection is called the point of diminishing returns. Why do you think this is? B) Miniproject: Consider the graph y = fx) where fx) is the cubic polynomial fx) = x + 5)x 1)x 4). i) For each pair of zeros of fx), compute the average a, and then draw the line tangent to the graph at the point a, fa)). Do you notice anything? ii) Prove that your observation is true for this cubic polynomial. iii) Try to generalize this to arbitrary cubic polynomials. y 50 y=x+5)x-1)x-4) x Equation 1: y=x+5)x 1)x 4)

4 Homework from Chapter 4 and some of Chapter 8), cont d Miniproject C) Miniproject: Consider the graph of y = gx), where gx) is the quartic polynomial gx) = x 4 4x x + 1. A line is drawn through its two points Q and R of inflection and meets the graph in two other points P and S. 10 y 4 3 y=x -4x +10x+1 S 5 R Q x P 5 Equation 1: y=x^4-4x^3+10x+1 Equation 2: y=2x+1 i) Show that the distances P Q = RS and that QR P Q = 1 + 5, 2 the so-called Golden Ratio. ii) Do you think that this is just an accident? 4.4): 1 3, 5, 8, 9, 12a), 13 15, 17, 19a), 20 4, 42. Lesson 19 Optimization A) A rumor spreads among a group of 1,000 people. The number of people, Nt), who have heard the rumor by time t hours) is often modeled by a logistic function of the form 1000 Nt) = e.69t a) Approximately how long will it take for half of all the people to have heard the rumor? b) Approximately when is the rumor spreading the fastest? B) If t measures years since 1990, one popular world-population 40 model is P t) = P is in billions) e 0.08t a) What does this model predict for the maximum sustainable population of the world? b) According to this model, when will the population be growing the fastest? C) The concentration C of a drug measured in ng/ml) as a function of t hours) is often modeled by a surge function. Assume that the model is Ct) = 12.4te 0.2t. How many hours after administering this drug will the concentration reach its greatest? 4 This problem is formally identical with the Dogs Know Calculus problem!

5 Homework from Chapter 4 and some of Chapter 8), cont d Lesson ): 1 6, 7 note that this is just the first two terms of the binomial expansion), 8, 13, 14, 15, 20, 23 26, 27, 29, 36 39, Linearization 46, 48, 55. Lesson 21 Related Rates 4.6): 1 5, 8, 11, 13, 14, 18, 19, 22, 33, 35. Review for Chapter 4 test Chapter 4 test The 0/0 Indeterminate Form and l Hôpital s Rule Some weeks back, we already encountered a fundamental 0/0 indeterminate form, namely the it sin x x 0 x. What makes this it interesting is that both numerator and denominator tend to zero in the it, making impossible a naive computation of the iting ratio. Indeed, most interesting its such as those defining the derivative are indeterminate in the sense fx) that they are of the form where the numerator and denominator both tend to 0 x a gx) or to ). Students learn to compute the derivatives of trigonometric functions only after they have been shown that the it sin x x 0 x = 1. At the same time, you ll no doubt remember that the computation of this it was geometrical in nature and involved an analysis of the diagram to the right. The above it is called a 0/0 indeterminate form because the its of both the numerator and denominator are 0. You ve seen many others; here are two more: 2x 2 7x + 3 x 3 x 3 and x 5 1 x 1 x 1. Note that in both cases the its of the numerator and denominator are both 0. Thus, these its, too, are 0/0 indeterminate forms.

6 While the above its can be computed using purely algebraic methods, there is an alternative and often quicker method that can be used when algebra is combined with a little differential calculus. fx) Formally, a 0/0 indeterminate form is a it of the form where both x a gx) fx) = 0 and gx) = 0. Assume, in addition, that f and g are both differentiable x a x a and that f and g are both continuous at x = a a very reasonable assumption, indeed!). Then we have ) fx) fx) x a gx) This result we summarize as x a = ) x a gx) x a ) fx) x a x a = ) gx) x a x a = f a) g a) f x) x a = x a g x) by continuity of the derivatives) l Hôpital s Rule 0/0). Let f and g be functions differentiable on some interval containing x = a, that fx) = 0 = gx), and assume that f and g are continuous at x = a. x a x a Then As a simple illustration, watch this: fx) f x) x a gx) = x a x a g x). 2x 2 7x + 3 x 3 x 3 which agrees with the answer obtained algebraically. = 4x 7 x 3 1 = 5, x 3 In a similar manner, one defines / indeterminate forms; these are treated as above, namely by differentiating numerator and denominator: l Hôpital s Rule / ). Let f and g be functions differentiable on some interval containing x = a, that fx) = ± = gx), and assume that f and g are continuous at x a x a x = a. Then

7 fx) f x) x a gx) = x a x a g x). There are other indeterminate forms as well: 0, 1, and 0. These can be treated as indicated in the examples below. Example 1. Compute x2 ln x. Note that this is a 0 indeterminate form. It can easily be converted algebraically to an indeterminate form and handled as above: x2 ln x = ln x 1/x 2 ) l H 1/x = 2/x = x = 0. Other indeterminate forms can be treated as in the following examples. Example 2. Compute 1 4 x. Here, if we set L equal to this it if it exists!), ) then we have, by continuity of the logarithm, that ) x ln L = ln 1 4 = ln 1 4 ) x = x ln 1 4 ) ) ln 1 4 x = x 1/x l H 4/ x 2 1 = x)) 4 x 1/x 2 = x x This says that ln L = 4 which implies that L = e 4. ) = 4 Example 3. This time, try θ π/2) cos θ)cos θ. The same trick applied above works here as well. Setting L to be this it, we have

8 It follows that Exercises θ ln L = ln θ)cos θ π/2) cos θ = lncos θ)cos θ π/2) = cos θ ln cos θ θ π/2) = θ π/2) l H = θ π/2) cos θ)cos θ = 1. θ π/2) ln cos θ 1/ cos θ tan θ sec θ tan θ = Using l Hôpital s rule if necessary, compute the its indicated below: x 3 1 a) x 1 4x 3 x 3 cosπx/2) b) x 1 3 x 1) 2 c) x d) θ 0 sin 3θ sin 4θ sin θ 2 e) θ 0 θ f) θ π/2 2x 2 5x x 3 x sin θ 1 + cos 2θ lnx + 1) g) x log 2 x h) ln x ln sin x) Hint: you need to convert this indeterminate form to one of the forms discussed above!) i) ln 2x lnx + 1)). x j) ) x k) 1 + a l) x 1/x 1) x 1 m) x x 3 e x ) x n) xa e x, a > 0 o) ln x ln1 x) p) ln x ln1 x) Are o) and p) really x 1 different?)