Let D TCD CI(0.1). Consider the function f(e) Compute for all triangle
Consider the function f(x) = - +1 on the interval (0.1). d. Find the limit by setting up an integral and evaluating that integral.
Problem 1-4. Let AABC be a right triangle with hypotenuse AB. Suppose that D, E, F e AB, BF ZACB. Prove that ZDCE ZECF. FA, ZBDC is a right angle, and CE is the bisector of В E 14 F onu Rnonocitions 1.31 on these-but be sure to label what IT.
2,06 2 b. Consider f (x) = Vx + 1 and let e 0.1. Use graphs and/or algebra to find c and approximate the largest value of & such that f(x) E (2 -e, 2+e) When x E (c-6,c) U (clc+6). Show work and/or graph. CS Vxtl
2,06 2 b. Consider f (x) = Vx + 1 and let e 0.1. Use graphs and/or algebra to find c and approximate the largest value of & such that f(x) E (2...
3. Let f : (a,b) +R be a function such that for all x, y € (a, b) and all t € (0.1) we have (tx + (1 - t)y)<tf(x) + (1 - t)f(y). Prove that f is continuous on (a,b).
Problem (4) Let f(z) denote the function e a f(z) 1 - z Compute f (z) dz where y is any contour that encloses the origin but does not enclose the point z =1
Problem (4) Let f(z) denote the function e a f(z) 1 - z Compute f (z) dz where y is any contour that encloses the origin but does not enclose the point z =1
(2) Consider the function f given by f:R R f(a)1 2 (a) Determine the domain D and range R of the function f. (b) Show that f is not one to one on D. (c) Let ç D be a subset of the domain of f such that for all x ? S, 0 and the function is one to one. Find such a set S. (d) For the set S given in Part (c), find f (x) (e) Determine...
Let a continuously differentiable function f: Rn → R and a point x E Rn be given. For d E Rn we define Prove the following statements: (i) If f is convex and gd has a local minimum at t-0 for every d E R", then x is a minimiser of f. (ii) In general, the statement in (i) does not hold without assuming f to be convex. Hint: For) consider the function f: R2-»R given by
Let a continuously...
4. Consider a function f : X → Y. 4a) (5 pts) Let C, D be subsets of Y. Prove that f (CND)sf1(C)nf-1(D). 4b) (10 pts) Let A, B be subsets of X and assume the function f be one-to- one. Prove that f(A) n f(B)Cf(An B) (Justify each of your steps.) 4c) (4pts) Find an example showing that if the function f is not one-to-on the inequality (1) is violated.
5. Consider a random variable X whose probability mass function is given by 0.1 ifx0.1 0.3 ifx2p p(x)p ifx 3 0 otherwise (a) What is p? (b) Compute P(1.9 S IXI s 3) (c) What is F(p)? What is F(2)? What is F(F(3))? (Here F() denote the cdf for X) d) What is P(2X-3s 41X 2 2.0)? (e) Compute Var(X) ( Compute E(F(p(X))
: R → Rn be a Ci path which solves the 1. Let F : Rn → R be a C1 function, and let differential equation, E'(t)--VF(C(t)), te R. (a) Show that f(t) F((t)) is a non-increasing function of t (ie, f'(t) 30 Vt.) (b) For any t for which F(E(t)) * 0, decreasing in t (ie, f'(t) <0.) show that č is a smooth path, and f(t) is strictly