(p). Assume that u(z) is a real harmonic function defined on the disc D(zo, p) = {lz- zolp}. Use Cauchy's integral formula to prove the mean value property: -S u(zo) = [ 2π u(zo + reio) do 27T =

5) Determine the slope of the tangent line to the function with the given information: a. f(x, y) = sin(xy) in the direction 2î - 3ĵ, at the point (, -1). b. f(x, y) = 3+x² y² going directly away from the origin, at the point (4,−1). c. f(x, y) = -6x + 2y + 3 in the direction 30° counter-clockwise from the negative y-axis, at the point (-2,2). d. f(x, y) = x³y² + √3x − y in the direction toward (0,5), at the point (3, 1).

8.) Consider the contour C in the complex plane consisting of the portion from z = 1 to z = 2+ i. of the unit circle from z = -i to z = 1 followed by the line segment iR 2+i -2 R 1 Compute the following contour integrals directly by the definition. Then verify your answer by finding an appropriate anti-derivative and eval- uating at the endpoints of the countour. a.) Se b.) C - C2 2 dz dz (Your final answer should involve Arctan(1/2) which is equivalent to the usual real-valued version of arctangent.)

Assume that a rocket is in space and is following the path y = – 6(x − 7)² + 3, from left to right where the origin is set to be a certain point in space. At what point on that path should the engine be turned off in order to coast along the tangent line to another ship that is not moving relative to the origin of the path and is at the point (10, 3)? (Once the engine is turned off, there is no force pushing it in a certain direction, so it continues along the same path that the velocity is pointing at that specific time. This means it continues along a path that is a tangent line to the original curve.) Note: The graph really helps on this one. Go ahead and draw it and you will see that it helps. It's all about the picture on this one! Coordinates: