MATH2023 Multivariable Calculus 2019/20 17 Let u = u i + v j be a unit vector, and let

fu (t) = f (a + tu, b + tv)

From the textbook Calculus - Several Variables (8th) by R. Adams, Addison/Wesley/Longman.
be the single-variable function obtained by restricting the domain of f (x, y) to points of the
straight line through (a, b) parallel to u, If fu (t) is continuous at t = 0 for every unit vector
Homework 3 (Total: 21 questions)
u, does it follow that f is continuous at (a, b)? Conversely, does the continuity of f at (a, b)
Ex. 10.5 guarantee the continuity of fu (t) at t = 0? Justify your answers.
4 Identify the surface represented by the equation and sketch the graph
18 What condition must the nonnegative integers m, n, and p satisfy to guarantee that
x2 + 4y 2 + 9z 2 + 4x − 8y = 8.
xm y n
lim
(x,y)→(0,0) (x2 + y 2 )p
10 Identify the surface represented by the equation and sketch the graph x2 + 4z 2 = 4.
exists? Prove your answer.

Ex. 12.1 Ex. 12.3

xy 2 Find all the first partial derivatives of the function specified and evaluate them at the given
4 Specify the domain of the function f (x, y) = 2 .
x − y2
point
f (x, y) = xy + x2 , (2, 0).
exyz
10 Specify the domain of the function f (x, y, z) = √ .
xyz
8 Find all the first partial derivatives of the function specified and evaluate them at the given
14 Sketch the graph of the function f (x, y) = 4 − x2 − y 2 , (x2 + y 2 6 4, x > 0, y > 0). point
1
y f (x, y) = p , (−3, 4).
24 Sketch some of the level curves of the function f (x, y) = . x2 + y 2
x2 + y2

36 Find f (x, y, z) if for each constant C the level surface f (x, y, z) = C is a plane having intercepts 9 Find all the first partial derivatives of the function specified and evaluate them at the given
C 3 , 2C 3 , and 3C 3 on the x-axis, the y-axis and the z-axis respectively. point
w = x(y ln z) , (e, 2, e).
2 2 2 2
42 Describe the “level hypersurfaces” of the function f (x, y, z, t) = x + y + z + t .

12 Calculate the first partial derivatives of the given function at (0, 0). You will have to use the

Ex. 12.2 definition of first partial derivatives.

2 2
x (y − 1)  2 2
6 Evaluate lim , or explain why it does not exist.  x − 2y
(x,y)→(0,1) x2 + (y − 1)2 if x 6= y,
f (x, y) = x−y

0 if x = y.
x2 y 2
12 Evaluate lim , or explain why it does not exist
(x,y)→(0,0) 2x4 + y 4
28 Show that the given function satisfies the given partial differential equation
x2 + y 2 − x3 y 3
13 How can the function f (x, y) = , (x, y) 6= (0, 0), be defined at the origin so
x2 + y 2 ∂w ∂w ∂w
w = x2 + yz, x +y +z = 2w.
that it becomes continuous at all points of the xy-plane? ∂x ∂y ∂z

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  2xy
 , if (x, y) 6= (0, 0), 21 Assume that f has continuous partial derivatives of all orders and suppose that u(x, y) and
36 Let f (x, y) = x2 + y 2
 v(x, y) have continuous second partial derivatives and satisfy the Cauchy-Riemann equations
0, if (x, y) = (0, 0).
∂u ∂v ∂v ∂u
Note that f is not continuous at (0, 0). Therefore its graph is not smooth there. Show, however, = and =− .
∂x ∂y ∂x ∂y
that fx (0, 0) and fy (0, 0) both exist. Hence the existence of partial derivatives does not imply
that a function of several variables is continuous. This is in contrast to the single-variable case. Suppose also that f (u, v) is a harmonic function of u and v. Show that f (u(x, y), v(x, y)) is
a harmonic function of x and y.
Lecture Note (Exercises for students - (p9) - just after Ex. 1.13), Ex. 2.6 (p16)

Ex. 12.6
6 Use suitable linearization to find approximate value for the given function at the points indi-
cated.
Homework 4 (Total: 18 questions) 2
f (x, y) = xey+x at (2.05, −3.92).
Ex. 12.4
p
4 Find all the second partial derivatives of the function z = 3x2 + y 2 .


2 2
12 By approximately what percentage will the value of w = x2 y 3 /z 4 increase or decrease if x
 2xy(x − y )

if (x, y) 6= (0, 0), increases by 1%, y increases by 2%, and z and increases by 3%?
x 2 + y2
16 Let f (x, y) =


0 if (x, y) = (0, 0).
Calculate fx (x, y), fy (x, y), fxy (x, y) and fyx (x, y) at point (x, y) 6= (0, 0). Also calculate
17 Prove that if f (x, y) is differentiable at (a, b), then f (x, y) is continuous at (a, b).
these derivatives at (0, 0). Observe that fyx (0, 0) = 2 and fxy (0, 0) = −2. Does this results
contradict Theorem 1? Explain why.

2
18 Prove the following version of the Mean-Value Theorem: if f (x, y) has first partial derivatives
+y 2 )/4t
18 Show that the function u(x, y, t) = t−1 e−(x satisfies the two-dimensional heat equation continuous near every point of the straight line segment joining the points (a, b) and (a + h, b +
k), then there exists a number θ satisfying 0 < θ < 1 such that
ut = uxx + uyy .
f (a + h, b + k) = f (a, b) + hfx (a + θh, b + θk) + kfy (a + θh, b + θk).

(Hint: apply the single-variable Mean-Value Theorem to g(t) = f (a + th, b + tk).)

Ex. 12.5
2 Write appropriate versions of the Chain Rule for the indicated derivatives.

∂w/∂t if w = f (x, y, z), where x = g(s), y = h(s, t), and z = k(t).

Ex. 12.7
2xy
6 Let f (x, y) = . Find
x2 + y 2
12 Find the indicated derivative, assuming that the function f (x, y) has continuous first partial
(a) the gradient of the given function at the point (0, 2),
derivatives
∂ (b) an equation of the plane tangent to the graph of the given function at the point (0, 2)
f (yf (x, t), f (y, t))
∂y whose x and y coordinates are given, and
(c) an equation of the straight line tangent, at the point (0, 2), to the level curve of the given
∂3 function passing through that point.
20 Find f (s2 − t, s + t2 ) in terms of partial derivatives of f .
∂t2 ∂s

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10 Find the rate of change of the given function at the given point in the specified direction. Homework 5 (Total: 7 questions)
f (x, y) = 3x − 4y at (0, 2) in the direction of the vector −2 i. Ex. 13.1
4 Find and classify the critical points of the given function f (x, y) = x4 + y 4 − 4xy.

r
14 Let f (x, y) = ln krk where r = x i + y i. Show that ∇f = 2.
krk
1
20 Find the absolute minimum value of f (x, y) = x + 8y + in the first quadrant x > 0, y > 0.
xy
How do you know that an absolute minimum exists?

16 Show that, in terms of polar coordinates (r, θ) (where x = r cos θ, and y = r sin θ), the gradient
of a function f (r, θ) is given by
27 Let f (x, y) = (y − x2 )(y − 3x2 ). Show that the origin is a critical point of f and that the
∂f 1 ∂f b
∇f = r+
b θ, restriction of f to every straight line through the origin has a local minimum value at the
∂r r ∂θ
origin. (That is, show that f (x, kx) has a local minimum value at x = 0 for every k, and that
r is a unit vector in the direction of the position vector r = x i + y j, and b
where b θ is a unit f (x, y) has a local minimum value at y = 0.) Does f (x, y) have a local minimum value at the
vector at right angles to b
r in the direction of increasing θ. origin? What happens to f on the curve y = 2x2 ? What does the second derivative test say
about this situation?

18 In what direction at the point (a, b, c) does the function f (x, y, z) = x2 + y 2 − z 2 increase at
half of its maximal rate at that point?
Ex. 13.3
3 Find the distance from the origin to the plane x + 2y + 2z = 3,
(a) using a geometric argument (no calculus),
21 The temperature T (x, y) at points of the xy-plane is given by T (x, y) = x2 − 2y 2 . (b) by reducing the problem to an unconstrained problem in two variables, and
(a) Draw a contour diagram for T showing some isotherms (curves of constant temperature). (c) using the method of Lagrange multipliers.
(b) In what direction should an ant at position (2, −1) move if it wishes to cool off as quickly
as possible?
(c) If an ant moves in that direction at speed k (units distance per unit time), at what rate
12 Find the maximum and minimum values of f (x, y, z) = x2 + y 2 + z 2 on the ellipse formed by
does it experience the decrease of temperature?
the intersection of the cone z 2 = x2 + y 2 and the plane x − 2z = 3.
(d) At what rate would the ant experience the decrease of temperature if it moves from (2, −1)
at speed k in the direction of the vector − i − 2 j?
(e) Along what curve through (2, −1) should the ant move in order to continue to experience
22 Find the maximum and minimum values of xy + z 2 on the ball x2 + y 2 + z 2 6 1. Use Lagrange
maximum rate of cooling?
multipliers to treat the boundary case.

26 Find a vector tangent to the curve of intersection of the two cylinders x2 +y 2 = 2 and y 2 +z 2 = 2 √
26 What is the shortest distance from the point (0, −1) to the curve y = 1 − x2 ? Can this
at the point (1, −1, 1). problem be solved by the Lagrange multiplier method? Why?

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