QUESTION 5 Consider the function f(x,y) – In19 – x? –y?) X+2 a) Find and sketch...
QUESTION 5 a) Find and sketch the domain of the function f(x,y) = \n(x2 - y +1) + VÝ +1. (5 marks) b) Evaluate eży sin(3x +2y) lim (x,y) (-2,3) 3x +2y (6 marks)
2. Find and sketch the domain of the function f(x,y) = V x2 – y + 2.
Question 2 The function r) is defined by 3(x + 1) 1 f12r + 71-4 1+4 () (111) 13 marks Show that f(x)= [4 marks) Find (3) Find the domain off- [2 marks] Given that g(x) = In(x + 1) Find the solution of r if fg(x) = 5, by leaving your answer in terms of e exponential function) [4 marks) (b) Sketch the graph of the function, In)-1-cosx). not by plotting points, but starting with the graph of a...
5. Consider the function f: R -> R given by f (x, y) := e°+v* _ 4. (a) Sketch the level curves of f. (5 marks) (b) Find Vf, the gradient of f, and determine at which points Vf is zero. Remark: These points are called the critical points of f (5 marks) (c) Determine whether the critical points of f are local minima, local maxima, or saddle points by considering the level curves of f. (5 marks) (d) Calculate...
2 + (a) Determine and sketch the domain of the function f(x, y) = (x2 + y2 – 4) 9 – (x2 + y2). [7] x6 – yo (b) Evaluate lim (x,y)+(1,1) - Y [5] (c) What does it mean to say that a function f(x, y) has a relative minimum at (a,b)? [4] (d) Find all second order partial derivatives of the function f(x,y) = 22y.
calc3 #7 sketch by hand please Find the domain of the function f(x,y) = In(x -y + 2) and sketch it in the xy-plane.
Question 1(a&b) Question 3 (a,b,c,d) QUESTION 1 (15 MARKS) Let X and Y be continuous random variables with joint probability density function 6e.de +3,, х, у z 0 otherwise f(x, y 0 Determine whether or not X and Y are independent. (9 marks) a) b) Find P(x> Y). Show how you get the limits for X and Y (6 marks) QUESTION 3 (19 MARKS) Let f(x, x.) = 2x, , o x, sk: O a) Find k xsl and f(x,...
ID Let f(x) = /2x=2 | (a) Find f'(x) as a piecewise function (6) Graph y = f'(x) (c) state the domain of f and the domain of f. Find lin tan 4x cos 3x sin 5x X> 12 Find y if y = (3x+5)*(x+4x) (3 Find y' it ya + 10x tanx 7 Let y= (a) Find (6) Find the equation of the tangent line at (74 y' Elf 8 X3 Prove lim (5-) = 4 (a) write the...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
the function y=f(x)={ 0-4), 14x+16, x20 x<0 Consider 1. (a) Sketch the graph off. (3 pts.) (b) Verify that the function is continuous everywhere using the properties of the definition and possibly calculating the limit at a particular point. (2 pts.) (c) Show f'(x) is not continuous at x-0. (5 pts.) the function y=f(x)={ 0-4), 14x+16, x20 x