1. Obtain the Jacobian of the system of three following functions: f(xyz)-5x-2y +6zyx-2
What is the gradient of the function f(x,y,z) = 5x^2 + 2y^3 + 7z^4 + xyz ?
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Let F =< 5x + 2, 2y + z, 10x + 10y > be a vector field in R3. Evaluate the following surface integral directly: XFX)DA Il vaš Š = [] E. (ru xro D S Where S is the part of the plane 5x + 2y + z = 10 in the first octant (with upward orientation).
Q20 (5 pts). Solve the system u x 2y and vx + y for x and y and find the Jacobian( 2. Find the volume of the region R using this transformation (u,v)
Q20 (5 pts). Solve the system u x 2y and vx + y for x and y and find the Jacobian( 2. Find the volume of the region R using this transformation (u,v)
Let Ě =< 5x + 2, 2y +z, 10x + 10y> be a vector field in R3. Evaluate the following surface integral directly: Si F.25 = [] #(x 7)AA S D Where S is the part of the plane 5x + 2y+z= 10 in the first octant (with upward orientation). SHOW ALL OF YOUR WORK!
2. Solve the following equations: (4 pts ea.) a. 5x – 2y = 15 & 3x + 2y = -7 b. 2x + 3y = 3 & - 10x + 2y = -32
Suppose that the functions f and g are defined as follows. 5x +2 Find f+g and fg. Then, give their domains using interval notation. Domain of f+g Domain of fg
5. Find the derivatives of the following functions: (a) f(x) = 3" sin?(5x)
9. The linear function f is defined by f(x,y) = (x + 2y, 5x - y). (a) (5 pts) If ū=(-1,2) and 7 =(3, 1), check that fü+v) = f(ū) + f(ū). (b) (10 pts) Find the standard matrix for f.
1. Let F(x,y,z) =< 32, 5x, – 2y >. Use Stokes's Theorem to evaluate the integral Scurl F.ds, where S is the part of the paraboloid z = x² + y2 that lies below the plane z = 4 with upward- pointing normal vector.
Obtain the Laplace transform of the functions plotted in Figures
1 and 2.
Obtain the Laplace transform of the functions plotted in Figures 1 and 2. ft) f(t) 14 0 10 20 30 40 Figure 1 Figure 2