Exam Answers Math -, Fall 7. Show, using any metho you like, that log x = x log x x + C. Answer: (x log x x+c) = x x + log x + = log x. Thus log x = x log x x+c.. Compute these. Remember to put boxes aroun your answers. t A. t Answer: t / t = t/ + C. B. sin(θ) θ Answer: sin(θ) θ = cos(θ) + C. C. x log(x ) Answer: Let u = x, so u = x an u = x. Then x log(x ) = (log u) u = log u u = (u log u u + C) = (x log x x + C). Here we have use the result of Problem.. Suppose that y = f(x) is a continuous function an a an b any two numbers. Thoroughly iscuss the istinction between these two objects: f(x) an b a f(x). Answer: f(x) is the antierivative (or inefinite integral) of f(x). It is a function, or rather a family of functions, namely all of the functions whose erivative is f(x). If F (x) is any one of these functions, then all of the others are of the form F (x) + C for some constant C. In contrast, b a f(x) is the (efinite) integral of f(x) over the interval [a, b]. It is a number, representing the (signe) area of the region between the graph of y = f(x) an the x-axis. It is efine as a limit of sums of areas of rectangles that, taken together, approximate that region. 4. Crows like to eat various mollusks. To crack open these har-shelle creatures, they take them up in the air an rop them onto rocks. The longer the rop, the harer the impact an the more likely the mollusk is to crack open. On the other han, flying to high altitues requires a lot of energy. When trying to crack a mollusk, the crow naturally selects its ropping altitue in a way that minimizes the total energy require. Let h be the crow s ropping altitue (in m). Assuming that the crow an mollusk together have mass kg, then each flight to altitue h requires the crow to expen energy h (in kg m /s ). Also, it turns out that the number of rops require (on average) is n(h) = + 6 h.. A. Write a function f(h) that escribes the total energy require to crack a mollusk.
Exam Answers Math -, Fall 7 Answer: The energy per flight is h, an the number of flights require is n(h), so the total energy require is f(h) = h n(h) = h ( + 6 ) h(h.) + 6h = = h + 4.8h h. h. h.. It is worth noting that f(h) is unefine at h =. an negative for < h <.. Therefore the only sensible interval for h is (., ). B. What value of h shoul the crow choose, to minimize its total energy expeniture? Answer: The erivative is [omitting some work] f (h) = = h.4h (.)(4.8) (h.). For h in (., ) this is always efine. It is zero when h.4h (.)(4.8) = ; that is, when h =.4 ± (.4) + 4(.)(4.8) =. ± (.) + (.)(4.8) =. ± 4.. The solution. 4. is outsie (., ), so h =. + 4.. This is the only critical point of f(h) on (., ). There are many ways to see that it is a local minimum. For one, h =. + 4. is the right-han root of the parabola h.4h (.)(4.8) =, which opens up. Thus h.4h (.)(4.8) = passes from negative to positive at h =. + 4., an so must f (h), since its enominator is positive near h =. + 4.. Since f (h) passes from negative to positive, h =. + 4. is a local minimum. Then, because it is the only critical point on (., ), it must also be the absolute minimum on that interval. Thus the best ropping altitue is h =. + 4. m. The average number of rops there is an the total energy spent is n(. + 4.) = + 4., f(. + 4.) = (. + 4.) ( + 4 ). kg m /s. [By the way, these work out to be h 5.6 m, n(h) 4.6, an f(h) 6 kg m /s. These results are corroborate by actual observations of crow behavior.] 5. Recall that the erf function is efine as erf(x) = x e t t. A. What is the omain of erf? That is, which values of x can be plugge into it?
Exam Answers Math -, Fall 7 Answer: Any real number can be plugge in for x. B. It turns out that e t is ifficult to antiifferentiate, so to compute erf we must resort to numerical approximations. In as much etail as possible, explain/show how one coul estimate erf(x) for any given number x using Riemann sums (sums of areas of rectangles). Answer: Let n enote a positive integer, t = x n erf(x) = x e t t = lim n = lim n = x n, an t k = + k t = k x n. Then e t k t e (k x n) x n. To approximate erf(x), we simply pick a large value of n an compute e (k x n) x n. 6. Recall from the previous page that erf(x) = x e t t. A. What is erf(x)? Answer: By the funamental theorem of calculus, erf(x) = x e t t = x e t t = e x. B. I want to solve erf(x) =.5, but this is ifficult, so I resign myself to an approximate solution. In as much etail as possible, explain/show how one coul fin an approximate solution using Newton s metho. Answer: Let f(x) = erf(x).5; then solving erf(x) =.5 is tantamount to fining a zero of f(x). Notice that f (x) = e x iteration formula by Part A of this problem. Newton s metho uses the x n+ = x n f(x n) f (x n ) = x n erf(x n).5. e x n Notice that this formula requires us to compute erf(x n ), which is ifficult in itself; we have to use an approximation, as in Problem 5. Assuming that we have alreay overcome that obstacle, we pick a starting value x an then compute x, x, x 4,... using the iteration formula, until we get answers that agree to the esire number of ecimal places. [By the way, solving erf(x) =.5 (or erf(x) = a for some other a) is useful in the real worl for computing percentiles in ata that is normally istribute. For example, computations like this are carrie out to etermine percentiles for SAT scores.]
Exam Answers Math -, Fall 7 7. My entist keeps asking me about the following function for some reason. To get her off my back, please make a etaile graph of it, incluing intercepts, asymptotes, critical points, inflection points, the correct increasing/ecreasing an concavity behavior, local maxima an minima, an anything else that you think is significant. f(x) = x 8/ x /. Answer: First, it is worth noting that f(x) is an even function, so its graph is symmetric with respect to the y-axis. Now = f(x) = x 8/ x / if an only if x = or x =, so the zeros of f are x =,,. The erivative is f (x) = 8 x5/ x /. This is unefine at x = ; it is zero if an only if 8 x = x = 4 x = ±. So the critical points are x =,,. Now let s analyze the increasing/ecreasing behavior. For x >, f (x) = 8 x5/ x / > 8 x > x >. Hence f(x) is increasing for x > an ecreasing for < x <. Because the graph is symmetric about the y-axis, f(x) is ecreasing for x < an increasing for < x <. Finally, the secon erivative is f (x) = 4 9 x/ + 9 x 4/. This is always positive, so the graph of y = f(x) is always concave up an the critical points at x = ± are local minima. We now know enough to make a goo sketch [see Figure, which I ve generate by computer, for the sake of this answer key]. 8. A space probe is shooting away from Earth. Its velocity at hour t is 7 +. t km/hr. A. Write an integral that represents how far the probe travels in the first 4 hours. Answer: In kilometers, the istance travele is 4 7 +. t t. 4
Exam Answers Math -, Fall 7 4 Figure : The graph of y = x 8/ x / from Problem 7. B. Compute the integral. Answer: 4 7 +.t / t = = [7t +. 4 t4/ ] 4 (7 4 +. 4 44/ ) (7 +. 4 4/ ) = 7 4 +. 4 44/. 9. This problem concerns linear approximation. A. Fin the linear approximation to y = cos x at x = /. Answer: First, y = sin x, so the slope at x = / is sin(/) =. Thus the tangent line is y cos(/) = (x /) y = x + + 6. B. Using your linear approximation from Part A, estimate cos(). Your answer may involve things like an. Answer: cos() + 6 +. C. Using your linear approximation again, estimate cos( ). Is this estimate goo or not? Answer: Accoring to our linear approximation, cos( ) + 6 +. But this is probably not a goo estimate, because is far away from /, where we took the approximation. 5