(iii)h(x,y,z)=

√

x^2 +y^2 +z^2 at the points (−1, 2, 2) and (3, 2, 4)

(iv)l(x,y,z)=ex

2

y^4 coszat

(

0 ,− 2 ,

π

3

)

2.Sketch the surfaces represented byz=f(x,y)where

(i) z= 1 − 3 y (ii) x^2 +y^2 +z^2 = 9

3.Determine∂z/∂x,∂z/∂yin each case

(i) z=x^2 +y^2 (ii) z=

x

y

(iii) z=x^3 +x^2 y+y^4

(iv) z=

1

√

x^2 +y^2

(v) z=exycos( 3 y^2 ) (vi) z=ln( 1 +xy)

(vii) z=e−xy( 2 + 3 xy) (viii) z=

x^2 +y^2

√

1 +y

(ix) z=x^3 tan−^1

(

x

y

)

4.For each of the functions in Q3 evaluate

∂z

∂x

( 0 , 0 ),

∂z

∂y

( 1 , 2 )whenever possible.

5.Determine all first order partial derivatives

(i) w=x^2 + 2 y^2 + 3 z^2 (ii) w=

1

√

1 −x^2 −y^2 −z^2

(iii) w=xyz (iv) w=xcos(x+yz)

(v) w=exyln(x+y+z)

6.Determine all second order partial derivatives for the functions in Q3.

7.Determine all second order partial derivatives for Q5(i), (iii), (iv).

8.Show thatT(x,t)=ae−b

(^2) t
cosbx,whereaandbare arbitrary constants, satisfies the
equation
∂T
∂t

∂^2 T
∂x^2
9.Determinedzfor the functions
(i) z=x^2 − 3 y^2 (ii) z= 3 x^2 y^3
(iii) z=ln(x^2 +y^2 ) (iv) z=cos(x+y)
(v) z=x^2 e−xy
10.Ifz= 3 x^2 + 2 xy−y^2 andxandyvary with timetaccording tox= 1 +sintand
y=3cost−1evaluate
dz
dt
directly and by using the total derivative (chain) rule.

16.7 Applications

1.Partial differential equations are equations containing partial derivatives – analogous
to ordinary differential equations of Chapter 15. Such equations occur frequently in