Contentsazhou/teaching/17F/hw-problems.pdf1 3 (u+ v + w). The point Pis the centroid of the...

MATH 32A-1 (17F) (L) R. Killip Calculus of Several Variables

Contents

1 Homework 1 3

2 Homework 2 7

3 Homework 3 11

4 Homework 4 15

5 Homework 5 19

6 Homework 6 23

7 Homework 7 25

8 Homework 8 29

1


HOMEWORK 1

Problem 1 (13.1.44). Find the vector of length 3 in the direction of v = 4i + 3j.

Problem 2 (13.1.46). Find the unit vector in the direction opposite to v =

(−2

4

).

Problem 3 (13.1.60). Sketch the parallelogram spanned by v =

(1

4

)and w =

(5

2

). Add the

vector u =

(2

3

)to the sketch and express u as a linear combination of v and w.

Problem 4 (13.1.64). Determine the magnitude of the forces F1 and F2 in Figure 1a, assumingthat there is no net force on the object.

(a) Forces on a block (b) “Midpoint connectors” bisect each other

Figure 1: (a) 13.1.64; (b) 13.1.70

Problem 5 (13.1.70). Use vectors to prove that the segments joining the midpoints of oppositesides of a quadrilateral bisect each other (Figure 1b).

Hint: show that the midpoints of these segments are the terminal points of

1

4(2u + v + z) and

1

4(2v + w + u).

Problem 6 (13.2.44). Find the parametric equation for the line through (1,−1, 0) and (0,−1, 2).

Problem 7 (13.2.50). Find the point of intersection of the lines

r(t) =

1

0

0

+ t

−3

1

0

and s(t) =

0

1

1

+ t

2

0

1

.

3

Homework Problems MATH 32A-1 (17F)

Problem 8 (13.2.67). A median of a triangle is a segment joining a vertex to the midpoint ofthe opposite side. Referring to Figure 2, prove that the medians of triangle ABC intersect at the

terminal point P of the vector1

3(u + v + w). The point P is the centroid of the triangle.

Hint: show, by parametrizing the segment AA′, that P lies two-thirds of the way from A to A′. Itwill follow similarly that P lies on the other two medians.

Figure 2: Centroid of a triangle (13.2.67)

Problem 9 (13.3.26). Find the angle between the vectors

3

1

1

and

2

−4

2

.

Problem 10 (13.3.30). Find a vector that is orthogonal to

−1

2

2

.

Problem 11 (13.3.36). Simplify the expression

(v + w) · (v + w)− 2v ·w.

Problem 12 (13.3.62). Compute the component of u =

3

0

9

along v =

1

2

2

.

Problem 13 (13.3.68). Find the decomposition

a = a‖b + a⊥b

with respect to b for

a =

4

−1

5

and b =

2

1

1

.

Problem 14 (13.3.84). Let P and Q be antipodal points on a sphere of radius r centered at theorigin and let R be a third point on the sphere (Figure 3). Prove that PR and QR are orthogonal.

4


Figure 3: Right angle formed by antipodal points (13.3.84)

5


HOMEWORK 2

Problem 1. Let a and v be non-zero vectors. Show that the point on the line a + tv nearest theorigin is given by a⊥v. (Use your knowledge of one-variable calculus.)

Problem 2. Find all unit vectors v that are perpendicular to the vector

(5

12

).

Problem 3 (13.3.77). The methane molecule CH4 consists of a carbon molecule bonded to fourhydrogen molecules that are spaced as far apart from each other as possible. The hydrogen atomsthen sit at the vertices of a tetrahedron, with the carbon atom at its center, as in Figure 1a. We canmodel this with the carbon atom at the point (1/2, 1/2, 1/2) and the hydrogen atoms at (0, 0, 0),(1, 1, 0), (1, 0, 1), and (0, 1, 1). Use the dot product to find the bond angle α formed between anytwo of the line segments from the carbon atom to the hydrogen atom.

(a) A methane molecule (b) An iron crystal

Figure 1: (a) 13.3.77; (b) 13.3.78

Problem 4 (13.3.78). Iron forms a crystal lattice where each central atom appears at the center ofa cube, the corners of which correspond to additional atoms, as in Figure 1b. Use the dot productto find the angle β between the line segments from the central atom to two adjacent outer atoms.

Hint: take the central atom to be at the origin and the corner atoms to be at (±1,±1,±1).

Problem 5. Let P and Q be points in R3. Show that if R is the midpoint of PQ, then a point X

is equidistant from P and Q if and only if−−→XR is orthogonal to

−−→PQ.

Problem 6 (13.4.10). Calculate

2

0

0

×−1

0

1

.

7


Problem 7 (13.4.12). Calculate

1

1

0

×0

1

1

.

Problem 8 (13.4.26). In Figure 2, which of the following form a right-handed system?

(a) {v,w,u} (b) {w,v,u} (c) {v,u,w}(d) {u,v,w} (e) {w,v,−u} (f) {v,−u,w}

Figure 2: Right and left handedness (13.4.26)

Problem 9 (13.4.30). Find the two unit vectors orthogonal to both a =

3

1

1

and b =

−1

2

1

.

Problem 10 (13.4.36). Find the volume of the parallelepiped spanned by u,v,w in Figure 3a.

(a) Volume of a parallelepiped (b) Area of a triangle

Figure 3: (a) 13.4.36; (b) 13.4.44

8


Problem 11 (13.4.44). Use the cross product to find the area of the triangle whose vertices areP = (1, 1, 5), Q = (3, 4, 3), and R = (1, 5, 7) (Figure 3b).

Problem 12 (13.5.4). Write the equation of the plane with normal vector n =

2

−4

1

passing

through the point (1/3, 2/3, 1) in the scalar form ax+ by + cz = d.

Problem 13 (13.5.18). Find an equation of the plane passing through the points P = (5, 1, 1),Q = (1, 1, 2), and R = (2, 1, 1).

Problem 14 (13.5.22). Find the equation of the plane which passes through (4, 1, 9) and is parallelto x+ y + z = 3.

Problem 15 (13.5.26). Find the equation of the plane which contains the lines

r1(t) =

t

2t

3t

and r2(t) =

3t

t

8t

.

Problem 16 (13.5.30). Are the planes 2x− 4y − z = 3 and −6x+ 12y + 3z = 1 parallel?

9


HOMEWORK 3

Problem 1. We draw a curve by attaching a pen to the circumference of a circle (say of radiusone) and rolling it around the circumference of another (fixed) circle of equal radius.

(a) Parametrize the resulting curve.

(b) Determine the total length of the curve.

Problem 2 (12.1.66). A 10-ft ladder slides down a wall as its bottom B is pulled away from thewall (Figure 1a). Using the angle θ as a parameter, find the parametric equations for the pathfollowed by (a) the top of the ladder A, (b) the bottom of the ladder B, and (c) the point P located4 ft from the top of the ladder. Show that P describes an ellipse.

(a) Ladder sliding down wall (b) Property of the cycloid

Figure 1: (a) 12.1.66; (b) 12.1.76

Problem 3 (12.1.76; property of the cycloid). Prove that the tangent line at a point P on thecycloid always passes through the top point of the rolling circle as indicated in Figure 1b. Assumethe generating circle of the cycloid has radius 1.

Problem 4 (14.1.12). Match the space curves (Figure 2) with the following vector-valued functions:

(a) r1(t) =

cos(2t)

cos t

sin t

(b) r2(t) =

t

cos(2t)

sin(2t)

(c) r3(t) =

1

t

t

Figure 2: Space curves

11


Problem 5 (14.2.30). Find a parameterization of the tangent line to r(t) =

(cos(2t)

sin(3t)

)at t = π/4.

Problem 6 (14.2.56). Find the location and velocity at t = 4 of a particle whose path satisfies

dr

dt=

2t−1/2

6

8t

, r(1) =

4

9

2

.

Problem 7 (14.2.63). Prove that the Bernoulli spiral (Figure 3) given by the parameterization

r(t) =

(et cos 4t

et sin 4t

)has the property that the angle ψ between the position vector and the tangent

vector is constant. Find the angle ψ.

Figure 3: Bernoulli spiral

Problem 8 (14.3.18). What is the velocity vector of a particle traveling to the right along thehyperbola y = x−1 with constant speed 5 cm/s when the particle’s location is (2, 1/2)?

Problem 9 (14.3.35). Find the arc length parameterization of the line y = mx for an arbitraryslope m.

Problem 10 (14.3.39). The unit circle with the point (−1, 0) removed has parameterization

r(t) =

(1−t21+t2

2t1+t2

), −∞ < t <∞.

Use this parameterization to compute the length of the unit circle as an improper integral.

Hint: the expression for ‖r′(t)‖ simplifies.

Problem 11 (14.3.40). The involute of a circle (Figure 4), traced by a point at the end of a threadunwinding from a circular spool of radius R, has parameterization

r(θ) =

(R(cos θ + θ sin θ)

R(sin θ − θ cos θ)

).

Find an arc length parameterization of the involute.

12


Figure 4: The involute of a circle

Problem 12 (14.4.4). Calculate r′(t) and T(t), and evaluate T(1), for r(t) =

1 + 2t

t2

3− t2

.

Problem 13 (14.4.8). Calculate the curvature function κ(t) for r(t) =

4 cos t

t

4 sin t

.

Problem 14 (14.4.22). Show that the curvature at an inflection point of a plane curve y = f(x)is zero.

13


HOMEWORK 4

Problem 1. Fix 0 < b ≤ a and consider the parametric curve

r(t) =

(a cos t

b sin t

),

which is an ellipse.

(a) Find the unit tangents T(t) and normals N(t) to the ellipse.

(b) Find the curvature of the ellipse as a function of t.

(c) Find the points of minimal and maximal curvature (if a > b) and mark them on a sketch.

(d) Find the osculating circles at the points of maximal and minimal curvature and mark themon a sketch. Include explicitly the locations of the centers.

(e) Defining c =√a2 − b2, show that the perimeter of the triangle with vertices

r(t), F1 =

(c

0

), F2 =

(−c0

)is independent of t and determine its value. The points F1 and F2 are the foci of the ellipse.

(f) Show that the two vectors point to r(t) from each of the foci make equal angles with N(t).

Problem 2 (14.4.34). The Cornu spiral is the plane curve r(t) =

(x(t)

y(t)

), where

x(t) =

∫ t

0

sin

(u2

2

)du, y(t) =

∫ t

0

cos

(u2

2

)du.

Verify that κ(t) = |t|. Since the curvature increases linearly, the Cornu spiral is used in highwaydesign to create transitions between straight and curved road segments (Figure 1a).

(a) Cornu spiral (b) The curvature at P is the quantity |dθ/ds|.

Figure 1: (a) 14.4.34; (b) 14.4.69

15


Problem 3 (14.4.69). The angle of inclination at a point P on a plane curve is the angle θbetween the unit tangent vector T and the x-axis (Figure 1b). Assume that r(s) is an arc lengthparameterization, and let θ = θ(s) be the angle of inclination at r(s). Prove that

κ(s) =

∣∣∣∣dθds∣∣∣∣ .

Hint: observe that T(s) =

(cos θ(s)

sin θ(s)

).

Problem 4 (14.4.82). Let r(s) be an arc length parameterization of a closed curve C of length L.We call C an oval if dθ/ds > 0 (see Exercise 14.4.69 / Problem 3). Observe that −N points to theoutside of C. For k > 0, the curve C1 defined by r1(s) = r(s)− kN is called the expansion of C inthe normal direction.

(a) Show that ‖r′1(s)‖ = ‖r′(s)‖+ kκ(s).

(b) As P moves around the oval counterclockwise, θ increases by 2π (Figure 2A). Use this and a

change of variables to prove that

∫ L

0

κ(s) ds = 2π.

(c) Show that C1 has length L+ 2πk.

Figure 2: As P moves around the oval, θ increases by 2π.

Problem 5 (14.5.4). For r(t) = etj− cos(2t)k, calculate the velocity and acceleration vectors andthe speed at time t = 0.

Problem 6 (14.5.49). A space shuttle orbits the earth at an altitude 400 km above the earth’ssurface, with constant speed v = 28 000 km · h−1. Find the magnitude of the shuttle’s acceleration(in kilometers per square hour), assuming that the radius of the earth is 6378 km (Figure 3).

Figure 3: Space shuttle orbit

16


Problem 7 (14.6.17). The total mechanical energy (kinetic energy plus potential energy) of aplanet of mass m orbiting a sun of mass M with position r and speed v = ‖r′‖ is

E =1

2mv2 − GMm

‖r‖.

(a) Prove the equations

d

dt

(1

2mv2

)= v · (ma),

d

dt

(GMm

‖r‖

)= v ·

(−GMm

‖r‖3r

).

(b) Then use Newton’s law F = ma to prove that energy is conserved, i.e. dE/dt = 0.

Problem 8 (14.R.26). A specially trained mouse runs counterclockwise in a circle of radius 0.6 mon the floor of an elevator with speed 0.3 m/s, while the elevator ascends from ground level (alongthe z-axis) at a speed of 12 m/s. Find the mouse’s acceleration vector as a function of time. Assumethat the circle is centered at the origin of the xy-plane and the mouse is at (0.6, 0, 0) at t = 0.

Problem 9 (14.R.40). If a planet has zero mass (m = 0), then Newton’s laws of motion reduce tor′′(t) = 0 and the orbit is a straight line r(t) = r0 + tv0, where r0 = r(0) and v0 = r′(0) (Figure4). Show that the area swept out by the radial vector at time t is A(t) = 1

2‖r0 × v0‖t, and thusKepler’s second law continues to hold (the rate is constant).

Figure 4: Kepler’s second law for m = 0

17


HOMEWORK 5

Problem 1 (15.1.18). Match each of graphs (A) and (B) with one of the following functions:

(i) f(x, y) = (cosx)(cos y) (ii) g(x, y) = cos(x2 + y2)

Figure 1: Graphs of functions of two variables

Problem 2 (15.1.20). Match the functions (a)-(d) with their contour maps (A)-(D) in Figure 2.

(a) f(x, y) = 3x+ 4y

(b) g(x, y) = x3 − y

(c) h(x, y) = 4x− 3y

(d) k(x, y) = x2 − y

Figure 2: Contour maps of functions of two variables

19


Problem 3 (15.1.26). Sketch the graph of f(x, y) =1

x2 + y2 + 1and draw several horizontal and

vertical traces.

Problem 4 (15.1.56). The function f(x, t) = t−1/2e−x2/t, whose graph is shown in Figure 3, models

the temperature along a metal bar after an intense burst of heat is applied at its center point.

(a) Sketch the vertical traces at times t = 1, 2, 3. What do these traces tell us about the way heatdiffuses through the bar?

(b) Sketch the vertical traces x = c for c = ±0.2,±0.4. Describe how temperature varies in timeat points near the center.

Figure 3: Graph of f(x, t) = t−1/2e−x2/t beginning shortly after t = 0

Problem 5. By repeatedly completing the square, find changes of variables of the form

X = x+ ay + b Y = y + c

that bring the paraboloids

z = x2 + 6xy + 10y2 + 4x+ 10y + 5

z = x2 − 2xy + 4y − 4

into standard position. Identify them as either elliptic or hyperbolic.

Problem 6. By repeatedly completing the square, determine for each of the following equationswhether they describe a cone, ellipsoid, cylinder, or hyperboloid (of how many sheets):

2x2 + 2xy + 2xz + 3y2 − 4yz + 3z2 = 1 2x2 − 2xz + y2 + 2yz + z2 = 0

2xy + y2 + 2z2 − 4x− 12z = 14 2xy + y2 + 2z2 − 4x− 12z = −25.

Problem 7 (13.6.32). Sketch the surface y2 + z2 = 1.

Problem 8 (13.6.36). Sketch the surface x2 − 4y2 = z.

20


Problem 9 (13.6.40). Find the equation of the elliptic cylinder passing through the points markedin Figure 4a.

(a) Points on an elliptic cylinder (b) Quadric surface

Figure 4: (a) 13.6.40; (b) 13.6.42

Problem 10 (13.6.42). Find the equation of the quadric surface shown in Figure 4b.

Problem 11 (13.6.47). Let S be the hyperboloid x2 + y2 = z2 + 1 and let P = (α, β, 0) be a pointon S in the xy-plane. Show that there are precisely two lines through P entirely contained in S(Figure 5).

Hint: consider the line r(t) = (α + at)i + (β + bt)j + tk through P . Show that r(t) is contained inS if (a, b) is one of the two points on the unit circle obtained by rotating (α, β) through ±π/2.

Remark: this proves that a hyperboloid of one sheet is a “doubly ruled surface”, which means thatit can be swept out by moving a line in space in two different ways.

Figure 5: Hyperboloids of one sheet are doubly ruled surfaces

21


HOMEWORK 6

Problem 1. Use the ε-δ definition to show that the following are continuous at (x, y) = (0, 0):

f(x, y) = 7, f(x, y) = x2 − y2.

Problem 2 (15.2.14). Let f(x, y) = xy/(x2 + y2). Show that f(x, y) approaches zero along the xand y axes. Then prove that lim

(x,y)→(0,0)f(x, y) does not exist by showing that the limit along the

line y = x is non-zero.

Problem 3 (15.2.20). Evaluate the limit lim(x,y)→(0,0)

x2 − y2

x2 + y2or show that it does not exist.

Problem 4 (15.2.28). Evaluate the limit lim(z,w)→(−1,2)

(z2w − 9z) or show that it does not exist.

Problem 5 (15.2.40). Evaluate the limit lim(x,y)→(1,1)

x2 + y2 − 2

|x− 1|+ |y − 1|or show that it does not exist.

Hint: Rewrite the limit in terms of u = x− 1 and v = y − 1.

Problem 6 (15.3.16). Compute the first-order partial derivatives of V = πr2h.

Problem 7 (15.3.32). Compute the first-order partial derivatives of P = e√y2+z2 .

Problem 8 (15.3.34). Compute the first-order partial derivatives of z = yx.

Problem 9 (15.3.52). Calculate ∂P/∂T and ∂P/∂V , where pressure P , volume V , and temperatureT are related by the ideal gas law PV = nRT (here R and n are constants).

Problem 10 (15.3.68). Compute the derivative uxx for u(x, t) = t−1/2e−x2/4t.

Problem 11 (15.3.76). Show that u(x, t) = sin(nx)e−n2t satisfies the heat equation for any n:

∂u

∂t=∂2u

∂x2.

Problem 12 (15.4.8). Find an equation of the tangent plane to g(x, y) = ex/y at (2, 1).

Problem 13 (15.4.14). Write the linear approximation to f(x, y) = x(1 + y)−1 at (a, b) = (8, 1) inthe form

f(a+ h, b+ k) ≈ f(a, b) + fx(a, b)h+ fy(a, b)k.

Use it to estimate 7.98/2.02 and compare with the value obtained using a calculator.

Problem 14 (15.4.39). The volume V of a right circular cylinder is computed using the values 3.5m for diameter and 6.2 m for height. Use the linear approximation to estimate the maximum errorin V if each of these values has a possible error of at most 5%. Recall that V = πr2h.

Problem 15 (15.5.12). Use the chain rule to calculate ddtf(r(t)) for

f(x, y) = x2 − 3xy, r(t) = cos ti + sin tj, t = π/2.

23


Problem 16 (15.5.30). Calculate the directional derivative of g(x, y, z) = x ln(y+z) in the directionof v = 2i− j + k at P = (2, e, e). Remember to normalize the direction vector.

Problem 17. Consider the surface z =√x2 + y2, similar to that in Exercise 15.5.48.

(a) Show that the plane tangent to this surface at any point (x, y, z) 6= (0, 0, 0) passes throughthe origin.

(b) Explain why the point (0, 0, 0) was excluded.

Problem 18. Consider a gas whose pressure P , volume V , and temperature T are related by(P +

1

V 2

)V = T,

as well as the following path, defined for t > 0,(V (t)

T (t)

)=

(t

t−3/2

).

(a) Writing P as a function of V and T , determine

∂P

∂V(V (t), T (t)).

(b) Now computed

dtP (V (t), T (t)).

24


HOMEWORK 7

Problem 1 (15.5.32). Find the directional derivative of f(x, y) = x2 + 4y2 at the point P = (3, 2)in the direction pointing to the origin.

Problem 2 (15.5.34). Suppose you are hiking on a terrain modeled by z = x2 + y2 − y. You areat the point (1, 2, 3).

(a) Determine the slope you would encounter if you headed due east (in the positive x-direction)from your position. What angle of inclination does that correspond to?

(b) Determine the slope you would encounter if you headed due north (in the positive y-direction)from your position. What angle of inclination does that correspond to?

(c) Determine the slope you would encounter if you headed due northeast from your position.What angle of inclination does that correspond to?

(d) Determine the steepest slope you could encounter from your position.

Problem 3 (15.5.58). Find a unit vector n that is normal to the surface z2 − 2x4 − y4 = 16 atP = (2, 2, 8) that points in the direction of the xy-plane. (In other words, if you travel in thedirection of n, you will eventually cross the xy-plane.)

Problem 4. Consider the surface z2 = 1 + x2 + y2.

(a) Find all points on the surface where the tangent plane is perpendicular to

1

1

2

.

(b) Find all points on the surface whose tangent plane passes through (0, 0, 1/2).

Problem 5 (15.6.4). Calculate ∂f/∂r and ∂f/∂t for

f(x, y, z) = xy + z2; x = r + s− 2t, y = 3rt, z = s2.

Problem 6 (15.6.13). Evaluate ∂g/∂θ at (r, θ) =(2√

2, π/4)

for

g(x, y) =1

x2 + y2; x = r cos θ, y = r sin θ.

Problem 7 (15.6.20). The law of cosines states that c2 = a2 + b2 − 2ab cos θ, where a, b, c are thesides of a triangle and θ is the angle opposite the side of length c.

(a) Compute ∂θ/∂a, ∂θ/∂b, and ∂θ/∂c using implicit differentiation.

(b) Suppose that a = 10, b = 16, and c = 22. Estimate the change in θ if a and b are increasedby 1 and c is increased by 2.

Problem 8 (15.6.21). Let u = u(x, y) and let (r, θ) be polar coordinates. Verify the relation

‖∇u‖2 = u2r +1

r2u2θ.

Hint: Compute the right-hand side by expressing ur and uθ in terms of ux and uy.

25


Problem 9 (15.6.30). Compute ∂r/∂t and ∂t/∂r using implicit differentiation if r2 = tes/r.

Problem 10 (15.6.44). A function f(x, y, z) is homogeneous of degree n if

f(λx, λy, λz) = λnf(x, y, z)

for all λ ∈ R.

Prove that if f(x, y, z) is homogeneous of degree n, then

x∂f

∂x+ y

∂f

∂y+ z

∂f

∂z= nf.

Hint: Let F (t) = f(tx, ty, tz) and calculate F ′(1) using the chain rule.

Problem 11 (15.6.46). Suppose that f is a function of x and y, where x = g(t, s) and y = h(t, s).Show that

ftt = fxx

(∂x

∂t

)2

+ 2fxy

(∂x

∂t

)(∂y

∂t

)+ fyy

(∂y

∂t

)2

+ fx∂2x

∂t2+ fy

∂2y

∂t2.

Problem 12. Compute all partial derivatives of

F (x, y, z) =

∫ y

x

ezs2 − 1

sds,

and so determined

dt

∫ 2t

t

ets2 − 1

sds.

Problem 13 (15.7.4). Use the contour map in Figure 1 to determine whether the critical pointsA,B,C,D are local minima, local maxima, or saddle points.

Figure 1: Contour map

Problem 14 (15.7.12). Find the critical points of the function f(x, y) = x3 + y4− 6x− 2y2. Thenuse the second derivative test to determine whether they are local minima, local maxima, or saddlepoints (or state that the test fails).

26


Problem 15 (15.7.24). Show that f(x, y) = x2 has infinitely many critical points (as a functionof two variables) and that the second derivative test for all of them. What is the minimum valueof f? Does f(x, y) have any local maxima?

Problem 16 (15.7.26). Let f(x, y) = (x2 + y2)e−x2−y2 .

(a) Where does f take on its minimum value? Do not use calculus to answer this question.

(b) Verify that the set of critical points of f consists of the origin (0, 0) and the unit circlex2 + y2 = 1.

(c) The second derivative test fails for points on the unit circle (this can be checked by somelengthy algebra). Prove, however, that f takes on its maximum value on the unit circle byanalyzing the function g(t) = te−t for t > 0.

Problem 17. In class, we introduced the paraboloid of best approximation to f(x, y) at (0, 0) as

p(x, y) = f(0, 0) +∇f(0, 0) ·

(x

y

)+

1

2

∂2f

∂x2(0, 0)x2 +

∂2f

∂x∂y(0, 0)xy +

1

2

∂2f

∂y2(0, 0)y2.

Show that for any vector u =

(a

b

), we have

d

dt

∣∣∣∣t=0

f(tu) =d

dt

∣∣∣∣t=0

p(tu)

andd2

dt2

∣∣∣∣t=0

f(tu) =d2

dt2

∣∣∣∣t=0

p(tu).

Hint: Problem 15.6.46 helps with the latter.

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HOMEWORK 8

Problem 1. (a) Find all critical points of the function

f(x, y, z) = (x+ 4y + z) exp(1/2− x2 − y2 − z2).

(b) Pick one and determine the approximating paraboloid at that point.

(c) Complete squares to determine if the critical point is a maximum, a minimum, or a saddle.

Problem 2 (15.7.35). Find the maximum of

f(x, y) = x+ y − x2 − y2 − xy

on the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 (Figure 1).

Figure 1: The function f(x, y) = x + y − x2 − y2 − xy on the boundary segments of the square0 ≤ x ≤ 2, 0 ≤ y ≤ 2.

Problem 3 (15.7.44). Determine the global extreme values of the function

f(x, y) = (4y2 − x2)e−x2−y2

on the domain x2 + y2 ≤ 2.

Problem 4 (15.7.54). A box with a volume of 8 m3 is to be constructed with a gold-plated top,silver-plated bottom, and copper-plated sides. If gold plate costs $120 per square meter, silver platecosts $40 per square meter, and copper plate costs $10 per square meter, find the dimensions thatwill minimize the cost of the materials for the box.

Problem 5. Consider three fixed points x1, x2, and x3 in space. Find the point x that minimizesthe sum of distances squared, that is, which minimizes

f(x) = ‖x− x1‖2 + ‖x− x2‖2 + ‖x− x3‖2.

Incidentally, if x is tied to the three points via (idealized) elastic bands, this minimum would bethe equilibrium position (i.e. the position of least energy).

Hint: Use vectors as much as possible - writing out all the components gets very messy.

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Problem 6 (15.8.2). Find the extreme values of f(x, y) = x2 + 2y2 subject to the constraintg(x, y) = 4x− 6y = 25.

(a) Show that the Lagrange equations yield 2x = 4λ and 4y = −6λ.

(b) Show that if x = 0 or y = 0, then the Lagrange equations give x = y = 0. Since (0, 0) doesnot satisfy the constraint, you may assume that x and y are non-zero.

(c) Use the Lagrange equations to show that y = (−3/4)x.

(d) Substitute in the constraint equation to show that there is a unique critical point P .

(e) Does P correspond to a minimum or a maximum value of f? Refer to Figure 2 to justify youranswer.

Hint: Do the values of f(x, y) increase or decrease as (x, y) moves away from P along theline g(x, y) = 0?

Figure 2: Level curves of f(x, y) = x2 + 2y2 and graph of the constraint g(x, y) = 4x− 6y− 25 = 0.

Problem 7 (15.8.10). Find the minimum and maximum values of f(x, y) = x2y4 subject tox2 + 2y2 = 6.

Problem 8 (15.8.22). Use Lagrange multipliers to find the maximum area of a rectangle inscribedin the ellipse (Figure 3)

x2

a2+y2

b2= 1.

Figure 3: Rectangle inscribed in the ellipse x2/a2 + y2/b2 = 1.

30


Problem 9 (15.8.52). This exercise shows that the multiplier may be interpreted as a rate ofchange in general. Assume that the maximum of f(x, y) subject to g(x, y) = c occurs at a point P .Then P depends on the value of c, so we may write P = (x(c), y(c)) and we have g(x(c), y(c)) = c.

(a) Show that

∇g(x(c), y(c)) ·

(x′(c)

y′(c)

)= 1.

Hint: Differentiate the equation g(x(c), y(c)) = c with respect to c using the chain rule.

(b) Use the chain rule and the Lagrange condition ∇fP = λ∇gP to show that

d

dcf(x(c), y(c)) = λ.

(c) Conclude that λ is the rate of increase in f per unit increase in the “budget level” c.

Problem 10. Fix a unit vector n as well as two points x1 and x2 in the plane.

(a) Show that any point x on the linen · x = 0

that minimizesf(x) = ‖x− x1‖+ ‖x− x2‖

must have the property (x− x1

‖x− x1‖

)⊥n

+

(x− x2

‖x− x2‖

)⊥n

= 0.

We assume here that neither x1 nor x2 lie on the line n · x = 0.

(b) Observe that the previous relation can be expressed as follows: “the angle of incidence isequal to the angle of reflection,” at least if both points are on the same side of the ‘mirror’occupying the line n · x = 0.

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