Two-Dimensional Calculus (2011)

Answers to Selected Exercises

Chapter one

Section 1

1.1 a. 2, 1

c. 1, −1

e. 2, 1

1.2 a. |v| = 3, α = 0

c. |v| = 2, α = π

e. |v| = , α = π

1.3 a. −5/ , 5/

c. − , −1

e. 3, 4

1.4 a. 0, 0

c. 0

1.5

c. −10

1.6 a. π

c. π

1.7

1.12 a. |v| = |w|, |v| = |u|, or |u| = |w|

c. u · v = 0, u · w = 0, or v · w = 0

1.13 b. w + u = (v + w)

1.14 a. v − u, v − w

c. u + (v − u) − u = v

e. v

1.15 a. u = v − w ⇔ v = u + w ⇔ w = v − u

c. v = (u + w) ⇔ u = v − u

1.16 ax + by = c, ax₀ + by₀ = c ⇒ a(x − x₀) + b(y − y₀) = 0.

a, b ⊥ x − x₀, y − y₀ .

1.19 Speed is 50 miles per hour; angle with direction of train is cos⁻¹ ~ 37°.

1.21 130 miles per hour

1.23 The wind is from the Northwest.

1.24 The star always appears to be displaced in the direction of the earth’s motion.

1.25 Magnitude 2 , in direction of positive y axis.

1.27 a. Equal magnitude and opposite direction.

1.28 a. Equilibrium point is on line joining the two bodies, at distance d/(1 + ) from body of mass m₁.

1.30 a. All lines perpendicular to v.

Section 2

2.1 a. x = 3t + 1, y = 2t + 1, 0 ≤ t ≤ 1

c. x = − 3t + 2, y = 3, 0 ≤ t ≤ 1

e. x = −t, y = t, 0 ≤ t ≤

2.2 a. y = 2x − 5, 2 ≤ x ≤ 3

c. y = x², 0 ≤ x ≤ 1

e.

2.3 y = 2x² − 1, −1 ≤ x ≤ 1

2.5 x²/a² − y²/b² = 1

2.7 r ≠ 0

2.9 a. 1, 2

c. 2t, 4t³

e. [2/(1 + t²)²] −2t, 1 − t²

2.10 a. 1, 2 /

c. 1, 2t² /(1 + 4t⁴)^1/2

e. −2t, 1 − t² /(1 + t²)

2.11 a. 20

c. 61/27

e. 8

2.12 a. x³ − 3xy + y³ = 0

2.15 a. Yes

b. all t ≠ 0

c. y = |x|

2.16 a. 0, 0 ; −r cos t, sin t = − x − c, y − d ; 0, −g

Section 3

3.7 c. Ex. 3.1b, c; Ex. 3.4a, b; Ex. 3.5a

3.8 a.

c. circles x² + y² = R²

3.9 a. Moved upwards 2 units

c. Moved 2 units in the positive y direction

e. Reflected in the plane x = 0

g. Contracted by a factor of 2 in the x direction

i. Reflected in the origin

3.10 a. They are the same curves, but correspond to different values of the function.

Chapter two

Section 4

4.1 a. f_x = 4x³ + 6xy − 5y³, f_y = 3x² − 15xy²

c. f_x = 3x² + 6xy + 3y², f_y = 3x₂ + 6xy² + 3y²

e. f_x = cos (x + y), f_y = cos (x + y)

g.

i. f_x = − y/(x² + y²), f_y = x/(x² + y²)

k. f_x = y(y² − x²)/(x² + y²)², f_y = x(x² − y²)/(x² + y²)²

m.

4.2 a. 1

c. 2e

4.3 a. 3(x³ + y³)

c.

4.4 a. f_x = e^x cos y = g_y, f_y = −e^x sin y = −g_x

4.9 a. 4(x² + y²)

b. e^2x

c. 0

d. 0

4.12 a. f_x(0, 0) = 1, f_y(0, 0) = 0

c. f_x(0, 0) = 1, f_y(0, 0) = 0

4.13 a. x² + y² ≤ 1, y ≤ x

c. x > 0, y > 0, x + y < 1

4.14 a. The ellipse x² + 4y² = 4

c. The ellipses x² + 4y² = 4 and x² + 4y² = 16

e. The horizontal line segments y = ±2, |x| ≤ 1 and the vertical line segments x = ±1, |y| ≤ 2

4.15 a. Ex. 4.14 a, b

b. For example, in Ex. 4.14c (2, 0) and in Ex. 4.14e (1, 0)

4.17 Ex. 4.14 c, d,f

4.18 a. Ex. 4.14 a, d, e

Section 5

5.1 a. z = 3x − 4y + 5

c. z = 2

e. 25z = 6x + 12y − 20

5.3 a. z = z₀ − (z₀/x₀)^1/2(x − x₀) − (z₀/y₀)^1/2(y − y₀)

5.8 a. f_x(x₀, y₀) = 2x₀ sin 1/x₀ − cos 1/x₀, f_y(x₀, y₀) = 2y₀

b. f_x(0, y₀) = 0, f_y(0, y₀) = 2y₀

5.10 a. f_x = 2y(y² − x²)/(x² + y²)², f_y = 2x(x² − y²)/(x² + y²)²

Section 6

6.1 a. 3, −7

c. 3, −7

6.2 a. 1

c. −1

6.3 a. 2, 0 ; 2, , 0, − , −2, − , 0,

c. 1, 1 ; 1, , 1, 0, −1, − , −1, 0

6.4 a. 1; α = π

c. 1; T_α = ,

6.5 ∇f(x₀, y₀) = 2F'( + ) x₀, y₀

6.9 − , 2

6.12 a. ( , 1); maximum

c. (0,0); minimum

6.14 a. (−2, 1)

c. All points on the line 2x − 3y + 1 = 0

6.17 a = , b = ; r₁ = − , r₂ = , r₃ = −

Section 7

7.1 a. 4 cos 2t

c. 4

e. 50t³

7.2 a. 0

c. 0

e. 0

7.3 x_x(1 + log x)

7.5 ; increasing

7.8 a. Degree 7

c. Not homogeneous

e. Degree 3

g. Degree −

7.10 a. Homogeneous if k = l; then, of degree k

c. Degree k − l

e. Degree 2k

7.16 a. 21.3

c. 0.975

7.17 21.30 < f(2.1, 3.2) < 21.43

7.18 a. (Sin πx)/x

c. (3^x − 2^x)/x

e. (2e⁻² − e^−x)/x

7.19 a. π

7.21 a.

7.25 a. dr/dt = (xdx/dt + y dy/dt)/r

Section 8

8.1 a.

c.

e. −1

8.2 a.

c.

8.4 a.

c.

8.5 a. −b⁴/a²y³

c. (e^y − 1)(e^y + e^x)/(1 − e^x)²

8.8 In Ex. 6.5, level curves are circles about the origin.

8.9 Level curves are:

a. x² + (y − 1)² = 1

c. x² + y² = 2

8.11 a. ∇f = 2 x, −y , ∇g = y, x ; ∇f ⊥ ∇g

8.12 a. f_xg_x + f_yg_y = 0

8.14 a. f_y = 0; cannot solve in the form y = g(x)

c. f_y = 0; can solve as y = g(x), but g(x) not differentiable at x = 0

8.16 a. Cusp at the origin

c. Curve consists of the origin and the circle of radius 1 about the origin

e. “Level curve” has no points on it

8.18 a. If f(t, φ) = φ − kt − sin φ, then f_φ = 1 − φ cos φ > 0 since < 1

c. (x, y) = (a, 0) ⇔ cos φ = 1; (x, y) = (−a, 0) ⇔ cos φ = −1

d.

Section 9

9.1 a. , −

c. 4

e. 1

9.2 (0, −2)

9.4 2 radians

9.6 a. (3, 4)

c. Point on the line 3x + 4y = 25 nearest the origin

9.8 a. Pairs of straight lines parallel to the given one

9.10 a. Pairs of lines, x + y = ±c

c. 4; when x + y = 0

e. No

9.11 a. f(x, y) = π: the segment L; f(x, y) = 0: the y axis except for L; f(x, y) = c, 0 < c < π: a pair of circular arcs through the endpoints of L

b. ( , 0)

9.12 a. 1

c. (0, 0)

Section 10

10.1 a. 3, 1

c. 3, −1

e. −1, −5

g. , −

i. ( − 1)/2, (− − 1)/2

k. (5 + 3 )/2, (5 − 3 )/2

10.3 a. max (1/ , 1/ ), (−1/ , −1/ );

min (1/ , −1/ ), (−1/ , 1/ )

c. max (1/ , −1/ ), (−1/ , 1/ );

min (1/ , 1/ ), (−1/ , −1/ )

10.5 a. III

c. I

e. V

10.6

10.8 a. a² − (λ₁ + λ₂)a + λ₁λ₂ + b² = 0

10.10 a. λ₂ = 0 and λ₁ + λ₂ = a + c ⇒ λ₁ = a + c

10.11 a. λ₂ = a + c

c.

10.14 A = a cos² θ + 2b cos θ sin θ + c sin² θ

B = (c − a) cos θ sin θ + b(cos² θ − sin² θ

C = a sin² θ − 2b sin θ cos θ + c cos² θ

10.18 a. 3X² − Y²

c. X² + Y²

e. X² − Y²

10.19 a. Ellipse; π

c. Hyperbola

e. Parallel straight lines

10.20 a. Ellipse if k > 0; point if k = 0; no locus if k < 0

c. Pair of parallel straight lines if k > 0; single line if k = 0; no locus if k < 0

e. Hyperbola if k > 0 or k < 0; pair of intersecting lines if k = 0

10.22 1,

10.23 a. They are reciprocals.

10.27 2 −

10.29 a.

Section 11

11.1 a. f_xx = 30x⁴ − 60x²y², f_xy = −40x³y + 15y⁴, f_yy = −10x⁴ + 60xy³

c.

e. f_xx = −f_xy = f_yy = 20(x − y)³

11.2 a. f_xx = −fyy = 0

c. f_xx = −f_yy = 2(x³ − 3xy²)/(x² + y²)³

11.8 a. f_xx = 2g'(x² + y²) +4x²g″(x² + y²), f_xy = 4xyg″″(x² + y²),

f_yy = 2g'(x² + y²) + 4y²g″(x² + y²)

11.11 a. f_xxx = e^x sin y, f_xxy = e^x cos y, f_xyy = −e^x sin y, f_yyy = −e^x cos y

c. −e^x sin y

11.12 a. f_xxxx = y⁴e^xy, f_xxyy = (2 + 4xy + x²y²)e^xy, f_xyyy = (3x² + x³y)e^xy

c. (ky^k−1 + xy^k)e^xy

e. 0, if k ≠ ; k! if k =

11.13 a. 2(3 cos² α − 8 cos α sin α + 2 sin² α)

c. α = π

11.15 a. 11/2

11.16 12, 2

11.17 x″f_x + y″f_y + x′²f_xx + 2x′y′ f_xy + y′²f_yy

11.19 a. r_xx = y²/r³, r_xy = −xy/r³, r_yy = x²/y³

c. Positive semidefinite

11.20 c. Upper hemisphere; all curves are concave downward

11.21 b.

11.22

11.26 a. i(i − 1) ... (i − k + 1)cx^i−ky^j, if k ≤ i; 0 if k > i

c. k! ! if k = i, = j; 0 otherwise

Section 12

12.1 a. (0, −1): saddle point; (0,3): local minimum

c. (0, 3), (−4, −3): saddle points; (0, −3): local minimum; (−4, 3): local maximum

12.2 a. See Ex. 6.15

c. If AC − B² > 0, a local maximum if A < 0 and a local minimum if A < 0; if AC − B² < 0, a saddle point

e. A straight line

12.3 a. (0, 0): local minimum

c. (0, 0): neither

e. All points on the lines y = x and y = −x: local minimum

12.6 a, c. Local minimum

e. Neither

12.7 a.

c.

12.8 a. f(x, y) = −11 + 6(x − 1) + 3(y + 2) + 2(x − 1)² −(x − 1)(y + 2)

c. f(x, y) = −6 + 3(x − 1) + 5y + 3(x − 1)² + 3(x − 1)y

c.

e. f(x, y) = 0 + R(x, y)

12.9 a.

c.

where 2m + 1 is the largest odd integer which does not exceed n

e.

12.10 a. z = 51 − 6x − 8y; crosses

12.16 Cube

12.18 10 × 10 × 20 inches

12.20 The absolute maxima and absolute minima are respectively:

a. ; −

c. (3 + )/2; 0

e. 9; 0

g. 4; −3

i. 1; −1

k. 2; 4/e³

12.21 b.

Chapter three

Section 13

13.1 a. Uniform stretching by a factor of 3

c. Reflection in the origin (or rotation through 180°)

e. Horizontal stretching by factor of 2 and vertical stretching by factor of 3, followed by reflection in the horizontal axis

g. “Shearing” motion

t. Double folding over

k. Distortion in vertical direction

m. Map of plane onto square

13.2 a. R = 2r, φ = θ

c. R = r, φ = −θ

e. R = r , φ = θ + π

g. R = r, φ = π − θ

13.4 a. x = u, y = ν

c. x = u − 1, y = u + ν − 1

e. x = (2u + ν)^1/3, y = − (2u − v)^1/5

13.5 a. Bijective

c. Surjective

e. Injective

g. None

i. None

13.6 a. The line y = x

c. The x and y axes

13.7 a. x = 0

c. y = ( − 1)x and y = −( + 1)x

13.8 a. x² + y² = R²/2

c. x²/4R² + y²/9R² = 1

e. (x + 2)² + (y − 3)² = R²

g. x² + y² = R

13.9 a. The x and y axes

c. A rectangular hyperbola whose asymptotes are the x and y axes, lying in the first and third quadrants if d > 0 and in the second and fourth quadrants if d < 0.

e. They start with the lines y = ±x and move outward through a family of hyperbolas.

13.10 a. R = e^x, φ = y

c. It moves counterclockwise around the circle u² + v² = e^2c

e. The positive v axis

g. The positive u axis

i. The image ray rotates about the origin in the counterclockwise direction

13.11 a. (1, 1)

c. The line y = 1, for example

13.13 a. S is upper half-plane: v ≥ 0

c. S is fourth quadrant: u > 0, v < 0; F⁻¹: x = log u, y = log(−v)

e. S is square −π/2 < u < π/2, − π/2 < v π/2; F⁻¹: x = tan u, y = tan v

g. S is right half of parabola: v = u², u ≥ 0

13.14 a. x = u − u² − v² + 2uv, y = v − u

Section 14

14.1 a. Δ = 1

c. Δ = −a² ≠ 0

e. Δ = 0

g. Δ = 1

i. Δ = 0

k. Δ = 0

14.2 a. Δ = −4; x = v, y = u; Δ′ = −

c.

14.3 a. v = − u

c. v = −3u

14.4 a. F⁻¹(e, 2e) is the line x + y = e

14.6 a. φ = −θ

c. φ = θ + π

e. φ = θ − π

14.8 a. c. x² + y² ≤ R²; Δ = 1

e. x² + y² ≤ R²/2; Δ = 2

14.9 c. (0, 0), (a, c), (b, d)

14.10 a. 2

14.11 b. A = a² + c², B = ab + cd, C = b² + d²

c. AC − B² = (ad − bc)²

14.14 a. (—1, 0)

14.17 Δ = ±1

14.18 b. Δ = ad − bc ≠ 0

14.22 a. F( x, y ) = F(x 1, 0 + y 0, 1 ) = xF( 1, 0 ) + yF( 0, 1 )

14.24 b. The equation in part a satisfied by any characteristic value is quadratic and can have at most two roots.

d. If ad − bc < 0, the equation satisfied by λ has positive discriminant. If a ray in the x, y plane rotates in the counterclockwise direction, and the image ray in the clockwise direction, then at some point they will have the same direction.

14.25 a. C is the image of the unit circle under a nonsingular linear transformation.

Section 15

15.1 a.

c.

e.

g.

15.4 The product of the determinants can be zero only if one of the determinants is zero.

15.7 a. u = (x − y)/ , v = (x + y)/

c. u = −y, v = −x

15.8 a. y = x

15.9 The line y = (tan π)x:

15.10 a. − π

c. Reflection in the line y = −x

15.15 a. A dilation of and rotation through − π

c. A dilation of 5 and rotation through α = arc tan ( − ), 0 < α < π

15.17 a. Horizontal stretching by a factor of λι

c. Projection of entire plane onto the y axis along lines parallel to the x axis, followed by reflection in the x axis.

15.26 Use the fact that the determinant of a product of matrices is the product of the determinants.

Section 16

16.1 a.

c.

e.

g.

i.

k.

16.2 a. 1; nowhere

c. 9(x² + y²)²; (x, y) = (0, 0)

e. −2e^2x; nowhere

g. −2x²e^x2+xy; x = 0

i. sin² x cos² y − cos² x sin² y = sin (x + y) sin(x − y); x + y = 2πn or x − y = 2πn, n an integer

k. 0; everywhere

16.5 If F_k is given by u = f_k(x, y), v = g_k(x, y), then f₁ and f₂ have the same gradient, and hence differ by a constant. The same is true for g₁ and g₂.

16.6 a. dF is the linear transformation with matrix .

16.7 a.

16.9 F maps (1, ) onto (—8, 0); is a dilation by factor of 12 composed with rotation through π.

16.11 The image is a horizontal ray to the right of (0, 1).

16.12 a.

c. Orientation-preserving if f′(x₀) and g′(y₀) have the same sign; orientation−reversing if opposite signs.

e. |f′(x₀)| and |g′(y₀)| represent horizontal and vertical stretching at (x₀, y₀).

16.14 a. 1/e

c.

16.16 C regular means that the tangent vector is never zero. F regular means that dF maps nonzero vectors into nonzero vectors, hence vectors tangent to Γ are never zero.

Section 17

17.1 a. z_x(1, 0) = 2, z_y(1, 0) = 0

c. z_x(0, 0) = e, z_y(0, 0) = 0

17.2 a. 2cf_u

17.6 a.

17.7 a. 1

c. e²

e. 4e(1 + π)

17.11 a. F(0, 0) = (1, 1); x_u = , x_v = , y_u = , y_v = −

c. F(0, 0) = (0, 0); x_u = 1, x_v = 0, y_u = 0, y_v = 1

e. F(0, 0) = (1, 0); x_u = 1, x_v = 0, y_u = 0, y_v = 1

g. F(0, 0) = (0, 0); = 1, x_u = 1, y_u = 0, y_v = 1

17.12 a. x = (log u + log v), y = (log u − log v); u > 0, v > 0

c. x = u − v³, y = v; the whole plane

e. x = log (u² + v²), y = arc tan v/u; u > 0, for example

17.14 a. A differentiable function f(x) has a differentiable inverse in some neighborhood of x₀ if and only if f'(x₀) ≠ 0.

Section 18

18.1 a. Dependent; (u − 1)(v − 1) = 1

c.

e.

g. Dependent; u = 2v

i.

18.3 a. If f(x, y) ≡ 0, then Eq. (18.2) holds with λ = 1, μ = 0.

c. Suppose f(x, y), g(x, y) are linearly dependent. If f(x, y) 0, choose c = 0, and f(x, y) ≡ cg(x, y); if g(x, y) ≡ 0, then g(x, y) = 0 · f(x, y); if neither f nor g is ≡ 0, use part b. Conversely, if f(x, y) ≡ cg(x, y) use λ = 1, μ = −cin Eq. (18.2).

18.6 b. The image lies on the curve u = F(t), v = G(t).

18.7 a. π

c. cos⁻¹(2/ ) or tan⁻¹ .

18.8 a. Diffeomorphism; conformai.

c. Diffeomorphism; not conformal

e. Not a diffeomorphism since (x₀, y₀) and (x₀, y₀ + 2π) map into same point

g. Not a diffeomorphism since ∂(u, v)/∂(x, y) = 0 at ( , )

i. A conformai diffeomorphism

Section 19

19.1 a. 0; solenoidal

c. φ'(x) + ψ'(y); solenoidal ⇔ φ(x) = ax ψ(y) = −ay, a constant

e. 2y; not solenoidal

g. 4xy + 6x²; not solenoidal

i. 2 cos x cos y; not solenoidal

k. 2e^x cos y; not solenoidal

19.2 a. Conservative; ax + by

c.

e. Conservative; x²y

g. Not conservative

i. Conservative; −cos x cos y

k. Not conservative

19.3 a, f, l; also part b if a + b = 0; part c if of the form ax, −ay ; and part d if of the form ay, ax

19.9 a. 1, 0

c. 2 x, y

e. (1/x²) −y, x

Section 20

20.1 a. (a² + b²)u_XX + 2(ac + bd)u_XY + (c² + d²)u_YY

c. (X² + Y²)(u_XX + u_YY)

20.6 a.

c. u_θ = −rv_r, v_θ = ru_r

20.9 u = Aθ + B, A and B constants, θ = arc tan (y/x).

20.12

20.14 a. A = a cos² α + 2b cos α sin α + c sin² α

B = −a cos α sin α + b(cos² α − sin² α) + c cos α sin α

C = a sin² α − 2b cos α sin α + c cos² α

20.15 u(x, y) = G(3x − y) + H(3y − x)

20.16 c. u(x, y) = g(y) + xh(y)

20.18 U_Y = U_y cos² α + (v_y − u_x) cos α sin α − v_x sin² α

V_X = −u_y sin² α + (v_y − u_x) cos α sin α + v_x cos² α

20.19 a. (u_x − v_y) cos 2a + (u_y + v_x) sin 2α

c. u_y − v_x

Chapter four

Section 21

21.1 a. −

c.

21.3

21.4 a.

c. 1 − π

e. 0

21.5 1

c. 1

e. −1

21.6 a. π

c. − π

21.7 a. 0

c. 2πab

e. −πab

g. 0

21.9 −2πab

21.13 a. /2

c. π

e. ab(a² + ab + b²)/(a + b)

21.14 ds/dτ = (ds/dt)|dt/dτ|

21.16 a. 3π

c.

e. 1

21.17 a.

c. 13π/3

e. 2πa + π sinh 2a

g. 4πa²

21.18 a.

c. π

21.19 a.

c.

21.20 a.

c.

21.21 c. (2πa)(2πb)

21.22 a. π

c. 1

21.23 a.

c.

Section 22

22.1 a. 0

c. 1

e. 2^e+1 − 1

22.4 a.

c. 11

22.5 a.

c. −11

22.6 c. 2ab

22.8 a. x²y³ + x³y²

c.

22.9 a. 4

c. log − 1

22.11 a. p_y = 3 ≠ q_x = 6; f = 3xy² − y³

22.13 a. v = x + y

c. v = cos x cosh y

e. v = e^x sin y

22.15 a. −1

c. −14

24.7 a. πt

24.9 b. 2π²a²b

d. π

24.10 c.

24.12 a. a²

c. (c⁵ − a⁵ − b⁵)/15, where c² = a² + b²

e. (0, 3πa/16)

g.

24.17 a. h = x³y²/6

c. h = (x² + y²)^5/2/15

24.19 (e − 1)

24.22 a. 0 ≤ y ≤ 2 − 2x, 0 ≤ x ≤ 1

c.

e. e^y ≤ x ≤ 1 + (e − 1)y, 0 ≤ y ≤ 1

24.23 a.

c. 2

Section 25

25.1 a. (c − d)(b − a)

c. 0

25.3

25.4 a. 12

c. 0

25.5 a.

c.

25.6 a. 2π

25.7 a. 2π; the integral over each side is the angle subtended by that side at (X, Y).

25.9 a.

c. y log x + log y − 1/x

25.15 a. ab

c. 0

25.17 a. 2πk

Section 26

26.1 a. 0

c. πab

26.2 a. x = 2 cos t, y = 3 sin t, 0 ≤ t ≤ 2π

c. C₁ : x = 3 cos t, y = 3 sin t, 0 ≤ t ≤ 2π

C₂ : x = 1 + cos t, y = −sin t, 0 ≤ t ≤ 2π

e.

26.3 a. 0

c. 5π

e. 4

26.4 a. πab

c. a²(2 − π)

e. πa²

26.7 a.

c.

Section 27

27.1 a.

c. 2/π

e. a²/12

g. ab/4

27.2 a.

Section 23

23.1 a. h(t) = 1 − t, 0 ≤ t ≤ 1

c. h(t) = t − t², 0 ≤ t ≤ 1

e.

g. h(t) = 2 sinh t, 0 ≤ t ≤ 2

i. h(t) = 2(1 − |t|), −1 ≤ t ≤ 1

k.

m.

o.

23.2 a.

c.

e.

g. 2(cosh 2 − 1)

i. 2

k. 5

m. 5π

o. 12 − π

23.3 a. h(t) = 1 − t, 0 ≤ t ≤ 1

c.

e.

g.

i. h(t) = 2 − t, 0 ≤ t ≤ 2

k.

m. Same answer as Ex. 23.1m

o.

23.5 a.

c.

23.6 c.

23.8

23.9 πa²

23.10 a. Not bounded

c. Not bounded and does not consist of a domain together with its boundary

e. Does not consist of a domain together with boundary (a domain is connected)

23.14 a.

Section 24

24.1 a. 98

c.

24.2 a.

24.3 a. 0

24.4 a.

24.5 a.

c. abc/6

24.6 b. πbc

27.3 a. (e − 1)/2

27.6 Possible choices are:

a. ( , )

c. (1, 2/π)

e. (a/4, a/3)

g. (a/2, b/2)

27.8 4 + 1/4k²

27.10 a.

27.11 a.

27.20 a.

c.

e.

Section 28

28.1 a.

c.

e.

28.2 a. π

c.

e.

28.3 a. ( − 4/3π, 0)

28.4 a.

28.5 a.

c.

28.7 a.

28.8 a.

c.

28.9 (e − 1)²

28.11 a. u = 2x + y, v = x + 3y

28.12 π(1 − e^−R2)/6

28.13 c.

28.16 a. π

c. π

e.