Applied Mathematics 1413 Chapter 12.3: Applied Mathematics 1413 Chapter 12.: Applied Mathematics 1413 Chapter 12.4: Applied Mathematics 1413 Chapter 12.: 12.3 Partial Derivatives

Examine the function for relative extrema and saddle points. g(x, y)x2-y* - 8x - 7y Step 1 Find the first partial derivatives of gx, y)x2 - y2 - 8x Ty with respect to x and y. 2-8 -2y-7 Observe that both the partial derivatives existexist for all x and y. So according to the definition of critical points, the only critical points are where the first partial derivatives are equal to 0. Step 2 To locate the critical points, set gx(x. y)0 and gylx, y) 0. -2y-7 = 0 Solve the equations to find the values of x and y 4,-3.5 This is the one critical point of the function. Step 3 of the function g ecu Findthe val g(x, y)2 - y2 - 8x - 7y relative extrema of the function g occurs at 4, Find the value -11.73x

1. Find the partial derivatives, f,(x, y) and f,(x,y), if f(x,y)-ye" 2. For h(x,y) = ln(x2 + y2), find h(3, 4). 3, Find the second-order partials,å·¦, ,å·¦,and f," if f(x,y)-4x2-3xy2 +y. 4. Find the critical points (if any) of f(x,y)--5r +4xy-y' +16x +10.

Obtain the first and second order partial derivatives of the function f (x,y,z) = xyz - y2 sinx. Define the tangent plane to a surface. By constructing the tangent plane at the point (1.1,1) for the surface x2 - y2 + z3 = 4 obtain an estimate of z at the point x=1/2, y = 1. Use any method (with explanation) to find the minimum distance of the point (0,1,0) from the plane 2x + y +z = 8.