8. Calculate Lj,P) and UJ, P) (as defined in lecture) for the following. Make sure to...
7. Calculate L(f, P) and U(f, P) (as defined in lecture) for the following. Make sure to justify your work, especially your m, and M, computations. (a) f(x)-ln(x), x E [1,2]; an arbitrary partition P {xi);-o. ππ 4' 2' -1 if EQ-1,1]; any parti ifEQ (c) f(x)= 7. Calculate L(f, P) and U(f, P) (as defined in lecture) for the following. Make sure to justify your work, especially your m, and M, computations. (a) f(x)-ln(x), x E [1,2]; an arbitrary...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
We consider an even and periodic function of period p = 6 defined by: Calculate f (17.75). Justify your answer. f(x) = 2 + e-*, pour 0 < x < 3.
Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all partitions P of [0, 1]. b) Let m E N. Show that the cardinality of the set A :-{х є [0, 1] : f(x) > 1/m} is bounded by m(m...
5. A return to the circular disc problem examined in class (Lecture 2): (Despite all of the text below you are required to do very little. Please read on.) A thin, circular plate assumed to lie on the ry-plane is rotating about its center O, located at (0, 0), with constant angular speed w. (w > 0 means that the plate is rotating in the counterclockwise direction.) Using the results obtained in class, show that the velocity field of of...
Question l: Consider the function f(x) = sin(parcsinx),-1 < x < 1 and p E R (a) Calculate f(0) in terms of p. Simplify your answer completely fX) sin(p arcsinx) f(o) P The function fand its derivatives satisfy the equation where f(x) denotes the rth derivative of f(x) and f (b) Show thatf0(n2p2)f(m)(o) (x) is f(x). (nt2) (nti) (I-x) (nt 2 e 0 (c) For p E R-仕1, ±3), find the MacLaurin Series for f(x), up to and including the...
7 points Question 3. An Unusual Integrable Function (Show Working) Consider the function f : 10, 11 → R defined by 1 if r-for some nEN; f(x) = 0 for all other x E [0,1 (1 subpts) (a) Draw a rough diagram of the graph of f. When we study the formal definition of the continuity of a function later in the course, we will be able to prove that this function is discontinuous at those domain values r such...
Exercise 15. Are the following functions norms on the vector spaces they are defined? Prove your answer. (i 21 - 3|r2 for x - (71 12)Т € R?. (1x2)f(x)|d for f(x) e C[0, 1] (i) _ (iii) pl dо + 2la| + 3/az| for p(z) — аz2? + ајя + ao є Pз. Exercise 15. Are the following functions norms on the vector spaces they are defined? Prove your answer. (i 21 - 3|r2 for x - (71 12)Т €...
1. (2 pts each) The graph of some unknown function f is given below. 10 6/ 8-64-2 624 10 12 Use the graph to estimate the following quantities: (0 f (9) (g) f(4) b) lim (a) lim (e) (d) lim ( 6) (e) lim f(x) (c) lim f(x) if g(x)f(x) 6) a value of r where f is continuous but not differentiable (k) a value of r where f"(x) 0 and f"(x)>0 (1) the location of a relative maximum value...
For questions 5-8, consider the following experiment: Suppose the location of a particle in the plane is restricted to be within the region of the first quadrant enclosed by y = 0, y = 2, and 1-1 and that the z and y coorlinates of the point are d(scribed by the jointly continuous random variables X and Y, respectively, with joint pdf Uz, y) = cryl(0,1) (z)10.ェ2)(y) 5. Given this joint pdf, (a) Find the value of c that makes...