4. Consider the set S = {(x, y) | x ∈ [0, 1], x2 ≤ y ≤ x}. Prove that S is a Jordan region and integrate the function 2x 2 + 3y 2 on S.
4. Consider the set S = {(x, y) | x ∈ [0, 1], x2 ≤ y ≤ x}. Prove that S is a Jordan region and integrate the function 2x 2 + 3y 2 on S.
3. Consider the function f(x,y) = 4 + 2x - 3y - x2 + 2y2 - 3xy. a) (5 pts.) Calculate the partial derivative functions, and use them to calculate the gradient vector evaluated at c = b) (5 pts.) Write down the affine approximation to at the e given in a) /(x) = f(c)+ Vf(e)'(x - c) . Use it to calculate (1.1, 1.1). (Hint: it should be close to f(1.1, 1.1))
5) Given the function fix.y) - x2 and region R bounded by x 0, y x and 2x+y 6 (a) Sketch the region R (label lines, intercepts, axes and shade region) (b) SET UP the integral over this region (c) Assuming f(x.y)- xa is the density function for the lamina R given above, determine the mass for R
5) Given the function fix.y) - x2 and region R bounded by x 0, y x and 2x+y 6 (a) Sketch the...
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)
10) Integrate f(x, y) = sin (Vx2 + y2) over the region 0 < x2 + y2 = 16
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(1) y"+2y'+y=x2-1-3, y(0)=-2, y'(0)=1 (2) + y'= -8cos2x+6sin 2 x (3) y*- 3y + 2 y =e" (x2 + 2x - 1)
8. Consider the region bounded by the y = x2 - 2x + 1 and y = 1 + 2x - x? Find the area of the region. a. b. Find the volume of the solid when the region is rotated about the x-axis. c. Find the volume of the solid when the region is rotated about the y-axis. d. Find the volume of the solid when the region is rotated about the line x = 5. e. If the...
Consider the following LP z= 2x1-x2 st s r 22 0 1. Prove the feasible region of the above LP is convex set. (Note: You could not prove using graphical representation) (2 points) 2. Find extreme directions of the feasible region. (2 points)
Consider the following LP z= 2x1-x2 st s r 22 0 1. Prove the feasible region of the above LP is convex set. (Note: You could not prove using graphical representation) (2 points) 2. Find extreme directions...
The area of the region enclosed by the curves y = 2x, 3y = X and 2y = -x+5 is none of these o o 3,5 0 2.5 0 o o
Evaluate the integral. 4) S -2x cos 7x dx Integrate the function. dx (x2+36) 3/2 5) S; 5) Express the integrand as a sum of partial fractions and evaluate the integral. 7x - 10 6) S -dx x² . 44 - 12 6)
7.2.18) Consider the predatory-prey mode 2x 2x 2 1 x where 0 and x, y 2 0. Prove this system has no closed orbits by invoking Dulac's for suitable choice of a (Hofbauer and criterion with the function g(x.y Sigmund 1998).
7.2.18) Consider the predatory-prey mode 2x 2x 2 1 x where 0 and x, y 2 0. Prove this system has no closed orbits by invoking Dulac's for suitable choice of a (Hofbauer and criterion with the function g(x.y...