R X R → R X R where F(x, y) = (1 – x, 3y – 1) ∀ (x, y)
in R → R.
Prove it is onto, and one-to-one
5. Let A =R x R and f: A+ A be given by the rule f(x, y) = (x – y, x + y). (a) Prove f is one-to-one. (b) Prove f is onto A. (Comment: don't forget that if given b E A, you construct a such that f(a) = b, you must also show a E A.) (c) What is the inverse function? (d) Is f a permutation? Explain.
HwK 'E THAT HwK PROVE THAT f(x,y)= (x + y ² , x = 3y) is DIFFERENTIABLE AT ANY Point a in IR? GIVE THE TOTAL DERIVATIVE TO
QUESTION 4 Evaluate the double integral. 6x2 - 3y) da, where R = [(x, y)/05 x 54 and 1sys 3) -304 304 208 -208 QUESTION 5 T F(x, ) dx dy 1. Change the order of integration of S F(x, y) dy dx Click Save and Submit to save and submit. Click Save All Answers to save all ans esc
11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
3) Find Fodr where F(x, y) = (cos x, sec? y) and r(t) =(4,4). 4) Determine whether F(x, y) = (x+ - 3y?)i + (4y3 - 6xy)jis conservative. If it is, find its potential function.
Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all x, y ∈ R. Prove that for some constant a ∈ R, f(x) = ax. Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all X, Y E R. Prove that for some constant a ER, f(x) = ax.
4. Co ider dĀ, where R is the parallelogram enclosed by the lines x-3y=0, x-3y=4, 2x-y=2, Å 2x - y and 2x-y=7. Fill in the boxes: Let u=x-3y, and v= 2x - y. Then in terms of u and v, we can set up the PX - 3 ingen i 19 = 3/d2=SHH dvdu. (You do not actually evaluate the integral.) dvdu van de integral as: JJ 2 actually salane te imeni)
13. A linear transformation T takes Nº into f". T[ +y - y 2.1 + 3y y (a) Is T one to one? Justify your answer. If not, then give two vectors with the same image. (b) Is T onto? Justify your answer. If not then give a vector in R? that is not an image.
1 Use Stokes' theorem to evaluate the integrals: F(x, y, z) dr a) where F(r, y,z)(3yz,e, 22) and C is the boundary of the triangle i the plane y2 with vertices b) where F(x, y,z (-2,2,5xz) and C is in the plane 12- y and is the boundary of the region that lies above the square with vertices (3,5, 0), (3,7,0),(4,5,0), (4,7,0) c) where F(x, y,z(7ry, -z, 3ryz) and C is in the plane y d) where intersected with z...
Suppose X and Y have the joint pdf f (x, y) = 3y, 0 < y < 1, y − 1 < x < 1 − y 0 otherwise a) Give an expression for P (X > Y ). b) Find the marginal pdfs for Y . c) Find the conditional pdf of X given Y = y, where 0 < y < 1. d) Give an expression for E[XY ]. e) Are X and Y independent?