13. a) Show that the equation 2 10 must have a solution 'c' somewhere in the interval (0,co) b) Suppose the domain of some function g is (0,0) and g(c)-2 where Is the following true or fa...
4. The function f is continuous on the closed interval (-2, 1). Some values of f are shown in the table below. --2 f(x) -3 -1 0 1 7 k3 The equation f(x) = 3 must have at least two solutions in the interval [-1,1) if k = a. 1 b. C. 2 CONN NICO d. 5. If k(r) is a continuous function over the interval (-2, 4) such that k(-2) = 3 and k(4) = 1, then k(2) 0...
Suppose f : B(0.1) C is holomorphic, with irg:) 1 for every z є B(0,1). Suppose also that f(0)-0, so f(z)g(2) for some holomorphic function g: B(0,1)C. (a) By applying the Maximum Principle to g on B(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...
Consider the following function: where x1 2 0,0,A 0, 0<1. a) Suppose that the preferences of a consumer regarding the consumption of goods x1 and x2 are represented by function = f(x1,x2) above. Suppose also that the consumer is endowed with some disposable income Y > 0 and faces prices pı and p2 respectively for goods x1 and x2. i) Derive and describe the demand of the consumer for goods x1 and x2. [20 marks] i) Are demand functions affected...
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly...
Please help and show steps. Thank you.
= 2, and you We consider some function () whose domain of g is all real numbers except are given and g" as follows: ' ) = e't-3) (1-2)2 (r - 2) "(t)er? - 6r + 10) a) Find all points where ) = 0 or g'(c) does not exist. Give a sign diagram for g'(x). Indicate where g is increasing or decreasing. Find all local minima and maxima of g(x). b) Find...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
A student calculated a solution for a differential equation g'(x) with domain x > 0. The results are shown below. b g'(x) = cos(az) + x +1 sin(ax) b g(x) = (x + 1)2 There may have been some mistakes in the calculation. Indicate to the student which terms of the differential equation were integrated correctly. The terms are separated by a + or - sign. to a o a) First term is integrated incorrectly, second term is integrated correctly...
1 Suppose V is the plane in R3 that contains the ponts a, b and g, where a (1,2,3), b-(1,-2,1) and c-(0,1,2) Suppose L is the lne in R3 that contains the pont a and is perpendicular to the plane V (a) Fınd an equation for V (b) Find an equation for L 2 Consıder the R2 - R function f defined by f (,y) (0,0) 2 (a) Write down the largest possıble doman Df of f (b) Prove from...
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...
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true or false questions
17. The smallest positive real number is c, where c = card(0,1). 18. To show that two sets A and B are equal, show that x A and x B. 19. If (vx)P(e) is false, then P(x) is never true for that domain. 20. If R is a relation on A and if (a, a) is true for some a in A, then R is reflexive. 21. If f:A → B is a function, then...