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The sales S (in thousands of units) of a seasonal product are given by the model below, where t is the time in months, with t-1 corresponding to January. Find the average sales for each time period. (Round your answers to two decimal places.) s-74.5543.94 sin It 6 sin (a) The first quarter (O sts 3) thousand units (b) The second quarter (3 s ts 6) thousand units (c) The entire year (0štš12) thousand units

The sales S (in S 74.92+43.01 sin t thousand units (b) The second quarter (3 6) thousand units (c) The entire year (o sts 12) thousand units Need Help?

3.) For f(xy)- xe. Find a.) the rate of change of f at the point P(2,0) in the direction from P to@(5.4) b.) the direction of the maximum rate of change of f c.) the maximum rate of change? 4)ノThe temperature on the surface of a long, flat plate is given by T(x,y)-xsiny degrees Celsius. A bug is located at 1 a.) In what direction should the bug move for the most rapid decrease in temperature? If the bug moves in the direction(-2.1), will the temperature increase or decrease and at what rate? b.) c.) In what direction (there are two) can the bug move so that the temperature remains the same asīt is at 11 ?

16. The temperature at a point in the plane is T(x,v) = 100-3x2-2y3. A bug is at the point (1,-1). (a) Compute ▽T(1,-1). (b) Find the rate of change of temperature in the direction of v (3, -4). (c) Find the direction in which the bug should move to increase its temperature the fastest. (d) Find a direction in which the bug should move to NOT change its temperature.

3. The temperature at (x, u) on a surface is T(r, y)2 +ry2y (a) (1 credit) With initial location (2, 1, 8), in which direction should a heat seeking bug move to increase its temperature fastest? (b) (1 credit) If the bug wants to avoid a change in temperature, in which direction should it head?

Help Entering Answers (1 point) Verify that Stokes' Theorem is true for the vector field F-yi-zj + xk and the surface S the hemisphere x2 + y2 +22-25, y the direction of the positive y-axis 0 oriented in To verify Stokes' Theorem we will compute the expression on each side. First compute curl F - dS The surface S can be parametrized by S(s, 1) - (5 cos(t) sin(s), S2 curl F dS- dt ds where curl F dS - Now compute /F dr The boundary curve C of the surface S can be parametrized by: r(t) - (5 cos(t), (Use the most natural parametrization) 2π dt 0 F -dr-

Help Entering Answers (1 point) Use Stokes' Theorem to evaluate F dr where F(x, y, z) 5yzi 2xzj2ek and C is the ciry4,z 3 oriented counterclockwise as viewed from above. Since the circle is oriented counterclockwise as viewed from above the surface we attach to the circle is oriented upwards curl F = The easiest surface to attach to this curve is the disk x2 + y2 〈 4, z-3. Using this surface in Stokes' Theorem evaluate the following. F dr- J C where yi X2ー Evaluate P-dr

1. Evaluate the following line integrals. 1to (b) 2y+100) )y where C'is the circle with radius 2 centeresd at the origin oriented clockwise (e) Use Stokes' Theorem to evaluate F dr where F(r,y,) (ry, 3,2y) and C is the intersection of the planeェ+2 = 4 and the cylinder 2 +y2-9 oriented counterclockwise when viewed from above.

(6) Use Stokes' Theorem to compute CF-dr where F = (z2y,T3/3, xy) and C is the intersection of the hyperbolic paraboloid z = y2 and the cylinder 2 y1. C is oriented counter-clockwise when viewed from above

(c)L F. di, C is the intersection of + ะ-4 and 2,2 + y2-9 oriented counter- clockwise when viewed from the above and F(z, y,z)-(zy,3:, 2y), (Hint: Use Stoke's Theorem.) (2m + y2) dr + (2 ry) dy + (r' + 3~2) d: where C s a path froln (1,-1,2) to (2,2,3) (d)

(c) /-F. d7, C is the intersection of z + z ?? and + y. 9 oriented counter- clockwise when viewed from the above and Fy,a)-(ry, 3.2u) Hint: Use Stoke's Theorem.) (2x: + y2) dr + (2zy) dy + ( 2 + 3z2) dz where C s a path from(1,1,2) to (2,2,3) (d)