Mock Examination for MATH 1310 (1) In analogy with the definition of ln(x), define S(x) = ∫ x 0 sin(v2)dv. (a) Prove that S is increasing on the interval [0, √ pi]. (b) Use the Intermediate Value Theorem to prove that S is not increasing on the interval [0, √ 2pi + ] for any positive . (c) Evaluate the derivative S ′(x) explaining your calculation and stating any theorem you use. (d) If T is the inverse of S on the interval [0, √ pi], evaluate T ′. (2) Suppose that limn→∞ an = L and 0 < L < 1. (a) Does the series ∑∞ n=1 an converge? Explain your reasoning. (b) Does the series ∑∞ n=1 a n n converge? Explain your reasoning. (3) Let F (x) = ∫ x e √ e4 ln(t)2 t2 − 1dt for x in the interval [e, e2]. What is the length of the graph of F between e and e2? Explain your reasoning. (4) Explain why the following statement is true or false? If limn→∞ an = 0 then ∑∞ n=1(−1)nan converges. (Read this carefully!) (5) If f(x) is continuous and g(x) and h(x) are differentiable functions, find a formula for d dx h(x)∫ g(x) f(t)dt. (6) If fave[a, b] denotes the average value of f(x) on [a, b] and a < c < b, show that fave[a, b] = c− a b− afave[a, c] + b− c b− afave[c, b]. (7) Find the area of the region between y = 2x2 and y = 4− 2x2. (8) Suppose that S(x) = ∫ x 0 sin(pix2)dx. (a) Compute S(0). (b) Compute the derivative S ′(x). (c) Evaluate lim x→0 S(x) x3 explaining your reasoning. (9) If f is a continuous, positive function defined on the interval (0, 1] such that limx→0+ = ∞ we have seen how to make sense of the area of the infinite region bounded by the graph of f , the x-axis and the vertical lines x = 0 and x = 1 with the definition of the improper integral∫ 1 0 f(x)dx Consider the function f(x) = x sin(1/x) defined on (0, 1] and note that f is not defined at 0. • Would it make sense to argue similarly and define the length of the infinite curve y = f(x) on the interval (0, 1]? • If yes, what would be a reasonable suggestion for defining such a notion? • For bonus marks find a differentiable (or, at least, continuous) function f such that limx→0+ f(x) = 0 and the graph of f on [0, 1] has infinite length. (10) Find a power series representation for the function f(x) = x ln(1+x2) and determine the interval of convergence. (Taylor series will work, but there is an easier approach.) 1 (11) Show that the function f(x) = ∞∑ n=0 (−1)n x 2n (2n)! is a solution of the differential equation f ′′(x) + f(x) = 0 (12) A metal washer is made by revolving the function 1 (x+ 2)(x+ 1) restricted to the interval [1.1, 1.2] around the y-axis. What is the volume of the washer? (13) What is the volume of the solid obtained by revolving the function H(x) = x√ 2− x2 around the x-axis between x = 0 and x = 1? How would you evaluate the volume of the solid obtained by revolving the function around the x-axis between x = 0 and x = √ 2? (14) News report of the corona virus epidemic often mention the doubling rate, the number of days it takes for the infected population to double. The infected rate is often described as exponential and this is one of the rare times the term exponential is actually used accurately in the media. • Assuming a doubling rate of 3.8, provide a function that models the infected rate of a population that start with 5 infected individuals. • How long would it take for the number of infect individuals to reach 10,000? • Explain why the graphs of infected rates are often displayed with the vertical axis scaled logarithmically. If the graph of the function you calculated in the first part were plotted on such a graph, why would it be a straight line and what slope would that line have?
