1) use the total differential to approximate the change (round to two digits after decimal point)
Δz from f(3,4) to f(3.04,3.98) where f(x,y) = √( x^2 + y^2)
2) for z = (cosx)(sinY) and x = u-v and y = u+v find (partial derivative symbol) ϑz/ϑu and ϑz/ϑv.
3) at a certain instant, the height h of a right circular cone is 30 inches and is increasing at a rate dh/dt of 2inches/sec. At the same instant, the radius r of the base is 20 inches and is increasing at the rate dr/dt of 1 inch/sec. at what rate dV/dt is the volume increasing at that instant?
(note that volume of a right circular cone is V(r,h) = 1/3pi r^2h)
4)find the first partial derivative of z with respect to x for x^3y + y^3z + z^3x = 11
5)calculate the directional derivative of the following f(x,y,z) at the point (-1,0,3) in the direction of the vector v = 3i + j -5k
f(x,y,z) = z^2 e^(xy)
6) find the gradient of the following f(x,y,z) at the point (-1,0,3) f(x,y,z) = z^2 e^(xy)
7) find the maximum value of the directional derivative of the following f(x,y) at the point (-2,0)
f(x,y) = x^2 e^y
8)find an equation for the tangent plane to the surface given by the following f(x,y) at the point (-2,-1,2)
z= f(x,y) = 1/7(4x^2 – 2y^2)
9) find a set of symmetric equations for the normal line to the surface given by the following f(x,y) at the point (-2,-1,2)
z= f(x,y) = 1/7(4x^2 – 2y^2)
10) find a unit tangent vector to both the following F(x,y,z) and G(x,y,z) at the point (2,1,-1)
F(x,y,z) = 3y^2 +3z^2 -15 G(x,y,z) = x + y^2 +z^2 -4
11)find the angle in
radians
of inclination of the tangent plane to the following F(x,y,z) at the point (0,2,-2)
F(x,y,z) = (x^2/4)+(y^2/4)+(z^2/2)-3
12) find the critical points of the following f(x,y), then determine if it is a relative minimum or relative maximum or saddle point.
f(x,y) = x^2 +x -3xy+ y^3 -5
13) the sum of length L, width W and height H of a rectangular box is 12 inches. Find the length L, width W and height H when the sum of the squares of length L, width W and height H is a minimum.
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