Questions from General Calculus

Q: (a). Write expressions for the partial derivatives fx (a

(a). Write expressions for the partial derivatives fx (a, b) and fy (a, b) as limits. (b). How do you interpret fx (a, b) and fy (a, b) geometrically? How do you interpret them as rates of change? (c)...

Q: State the Chain Rule for the case where z = f (

State the Chain Rule for the case where z = f (x, y) and x and y are functions of one variable. What if x and y are functions of two variables?

Q: (a). Sketch the curve C with parametric equations x =

(a). Sketch the curve C with parametric equations x = cos t y = sin t z = sin t 0

Q: Let F (x) = (r2 - 2r) x

Let F (x) = (r2 - 2r) x, where x = 〈x, y〉 and r − |x |. Use a CAS to plot this vector field in various domains until you can see what is happening. Describe the appearance of the plot and explain it b...

Q: Match the vector fields F with the plots labeled I–IV

Match the vector fields F with the plots labeled I–IV. Give reasons for your choices. F (x, y) = 〈y, y + 2〉

Q: Match the vector fields F on R3 with the plots labeled I

Match the vector fields F on R3 with the plots labeled I–IV. Give reasons for your choices. F (x, y, z) = x i + y j + 3 k

Q: (a). Set up, but do not evaluate, a

(a). Set up, but do not evaluate, a double integral for the area of the surface with parametric equations (b). Eliminate the parameters to show that the surface is an elliptic paraboloid and set up...

Q: (a). Show that the parametric equations x = a sin

(a). Show that the parametric equations x = a sin u cos v, y = b sin u sin v, z = c cos u, 0 < u

Q: Match the functions f with the plots of their gradient vector fields

Match the functions f with the plots of their gradient vector fields labeled I–IV. Give reasons for your choices. f (x, y) = x (x + y)

Q: Find the area of the part of the sphere x2 + y2

Find the area of the part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2 + y2.