Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 5
Each set of homework questions is worth 100 marks
Due: at the beginning of the tutorial session Thursday/Friday, 25/26 February 2016
Name:
Use Mathematica to check your answers.
1. Use appropriate forms of the chain rule to find
x
sin 2y ;
2
z = cos
∂z
∂v
where
x = 2u + 3v , y = u3 − 2v 2 .
2. The Taylor series is given by
∞
X
f (~x) =
k1 ,...,kn
∂1k1 · · · ∂nkn f (x~o ) k1
∆x1 · · · ∆xknn ,
k1 ! · · · kn !
=0
(1)
where we denote
f (xo1 , . . . , xon ) ≡ f (x~o ) ,
f (x1 , . . . , xn ) ≡ f (~x) ,
xi − xoi ≡ ∆xi
(2)
k
and ∂i0 f ≡ f ; ∂ik f ≡ ∂∂xfk is the k-th partial derivative of f with respect to xi . Thus, the
i
Taylor series is an expansion in powers of ∆xi . One can write
f (~x) = f (x~o + ∆~x) .
(3)
F (t) = f (x~o + t∆~x) ,
(4)
Then, introducing the function
one sees that the Taylor expansion of F (t) in powers of t evaluated at t = 1 produces
the Taylor series of f (~x) in powers of ∆xi . Use this observation to show that the Taylor
series can be equivalently written as
f (~x) =
n
∞
X
∂ q f (x~o )
1 X
∆xi1 · · · ∆xiq .
q!
∂x
·
·
·
∂x
i
i
q
1
q=0
i ,...,i =1
1
3. Let r =
Pn
i=1
(5)
q
xi ei and r = |r|, where ei form an orthonormal basis of vectors in Rn . Find
∂f (r)
,
∂xi
i = 1, 2, · · · , n ,
and ∇f (r) ,
|∇f (r)|2 ,
where f is a smooth function of a single variable.
4. Show that if z = f (x, y), x = r cos θ, y = r sin θ, then
∂ 2z ∂ 2z
∂ 2 z 1 ∂z
1 ∂ 2z
+
= 2+
+
.
∂x2 ∂y 2
∂r
r ∂r r2 ∂θ2
1
5. Consider the surface
z = f (x, y) = ln
1 2/3 p
e 3 8x2 − 6xy 2 − y 3 + 32 − 12 sin(2x − y)
2
.
(a) Find an equation for the tangent plane to the surface at the point P = (1, 2, z0 )
where z0 = f (1, 2).
(b) Find points of intersection of the tangent plane with the x-, y- and z-axes.
(c) Sketch the tangent plane, and show the point P = (1, 2, z0 ) on it.
(d) Find parametric equations for the normal line to the surface at the point P =
(1, 2, z0 ).
(e) Sketch the normal line to the surface at the point P = (1, 2, z0 ).
6. Show that the equation of the plane that is tangent to the cone
x2 y 2 z 2
+ 2 − 2 =0
a2
b
c
at (x0 , y0 , z0 ) can be written in the form
y0
z0
x0
x + 2y − 2z = 0.
2
a
b
c
7. Prove: If the surfaces z = f (x1 , . . . , xn ) and z = g(x1 , . . . , xn ) intersect at P = (xo1 , . . . , xon , z o ),
and if f and g are differentiable at (xo1 , . . . , xon ), then the normal lines at P are perpendicular if and only if
n
X
∂f (xo , . . . , xo ) ∂g(xo , . . . , xo )
1
i=1
n
∂xi
1
∂xi
2
n
= −1 .