Make sure to give lots of significant digits for (floating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2 3 instead of 8, sin(3* pi/2) instead of -1, e A (In(2)) instead of 2, (2+ tan(3)) * (4- sin(5)) A6-7/8 instead of 27620.3413, etc. Problem 1. (1 point) Problem 4. (1 point) Let f(x, y) = 9xsin(y +5) + ery. The second derivative a of is axdy Evaluate the integral fo fo (ycosx - 3) dydx. equal to which of the following? . A. 45xy* cos(y5 + 5) + 45y cos(ys + 5) + 5x ery . B. 45xy cos(ys + 5) + 25x8 ery Problem 5. (1 point) . C. 45xy* cos(ys + 5) + 45y cos(y) + 5) + 5xetsy + 5yx ery Calculate the double integral of f(x, y) over the triangle indicated in the following figure: . D. 45y# cos(y + 5) + 5xtet y + 5yx etsy . E. 45xy* cos(ys + 5) + 5xter y + 5xx etsy Problem 2. (1 point) Given z = f(x,y), x= x(u, v), y= y(u, v), with x(1,2) = 2 and y(1,2) = 4, calculate Zu (1, 2) in terms of some of the values given in the table below. fx (1,2) = p fy(1,2) = -5 Xu(1,2) = r yu(1,2) = 3 fx (2, 4) = c fv(2, 4) = q x(1,2) = a yv(1,2) = d Zu (1,2) = Problem 3. (1 point) Suppose z = x2 siny, x= -s2 + 512, y = -4st. X A. Use the chain rule to find & and & as functions of x, y, s and 2 34 t. (That is leave ; and ; in terms of x and y.) f(x, y) = -20yet Answer : oz Problem 6. (1 point) B. Find the numerical values of g, and $7 when (s, t) = (-2, -3). Evaluate the double integral /24 2dA, where R is the region that lies between the circles x2 + y? = 16 and x2 + y2 = 49, by 95 (-2, -3) = changing to polar coordinates. &F (-2, -3) = Answer:Problem 7. (1 point) Problem 11. (1 point) Suppose R is the shaded region in the figure. As an iterated integral in polar coordinates, Use cylindrical coordinates to calculate : (f f(x,y)da = [ S'f(rcos(@), rsin(@) rar de () >trav Wixty