Download Partial Derivatives

Tutorial – 4
Review of Module -2
1
Given w = x.Cos(yz2). If x = Cos t, y = t2 and z = et, find the rate of change of w with respect
to t at t = π, (a) using chain rule and (b) using substitutuion, expressing w as a function of t.
2
Let w = f(x, y, z) be differentiable at (1,0,2) with fx = 1, fy = 2 and fz = 3 at that point. If x =t,
y = sin (πt) and z = t2 + 1, find dw/dt when t =1.
3
Use total differential to approximate the changes in the function f(x,y,z) = 2xy2z3 as a point
moves from P (-1,-2,4) to Q (-1.04, -1.98, 3.97). Compare this with the actual change and see
the error in approximating the change in f(x,y,z) using its total differential.
4
Find the linear approximation of the function
(4 x )
at the point (1,1,1)
( y +z )
5
Suppose that a function f(x,y,z) has a local linear approximation L(x, y, z) = x+2y+3z+4 at
(0, -1,-2). Then evaluate f, fx, fy and fz at that point.
6
Suppose that the percentage error in measuring two quantities x and y are at the most r% and
s% respectively. Use differentials to approximate the maximum possible percetage error in
calculating (a)
x
y
and (b)
x
3
√y
7
The legs of a right triangle are measured to be 3cm and 4 cm, with a max error of 0.04cm in
each measurement. Use differentials to approximate the maximum possible error in
calculating (a) hypotenuse and (b) area of the triangle.
8
Find all the critical points and classify them for the function,
2
3
2
2
f (x , y)=3 x y + y −3 x −3 y +2
9
Find the absolute maximum and absolute minimum for the function
2
2
f (x , y )= x −x y+ y −3 x+3 y in the domain, D = {(x, y)/ 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 }