PartII: Problem-SolvingProblems( Showallwork ) PartI: , ,

Calculus Student ID number:

Part I:

,

,

1. (10 pts) Match the surfaces to the functions?

x²+ y²: B , log(x²+ y²): D , xye^−(x2+y2): E , sin(x + y): A , sin(x) sin(y): C .

(A) (B) (C)

(D) (E)

Part II: Problem-Solving Problems (

Show all work)

2. Evaluate the given integrals (a) (5 pts)

∫ _∞

1

x^−4/3dx

∫ _∞

1

x^−4/3dx = lim

R→∞

∫ _R

1

x^−4/3dx = lim

R→∞−3x^−1/3|^R1 = 3 (b) (5 pts)

∫ _∞

0

x^−4/3dx

∫ _∞

0

x^−4/3dx =

∫ ₁

0

x^−4/3dx +

∫ _∞

1

x^−4/3dx

∫ 1

0

x^−4/3dx = lim

R→0⁺

∫ 1

R

x^−4/3dx = lim

R→0⁺−3x^−1/3|¹R DIV

⇒

∫ _∞

0

x^−4/3dx DIV

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Name: Student ID number:

3. (5 pts) Compute the Taylor polynomial of degree 3 about a = 0 for f (x) = e^2x. f (x) = e^2x, f⁽¹⁾(x) = 2e^2x, f⁽²⁾(x) = 4e^2x, f⁽³⁾(x) = 8e^2x. f (0) = 1, f⁽¹⁾(0) = 2, f⁽²⁾(0) = 4, f⁽³⁾(0) = 8.

⇒ P3(x) = 1 + 2x + 4

2!x²+ 8

3!x³ = 1 + 2x + 2x²+4 3x³.

4. (10 pts) Solve dy

dx = x− 1

y , with y(0) =−2, i.e. y = −2 when x = 0.

⇒

∫

y dy =

∫

(x− 1) dx ⇒ 1

2y² = 1

2x²− x + C ⇒ y = ±√

x²− 2x + C

∵ y(0) = −2 ∴ y = −√

x²− 2x + 4

5. Suppose that dy

dx = y(y− 1)(y − 2)

(a) (5 pts) Find the equilibria of the differential equation.

y(y− 1)(y − 2) = 0 ⇒ y = 0, 1, 2 (b) (5 pts) Discuss the stability of the equilibria.

Let g(y) = y(y− 1)(y − 2) = y³− 3y²+ 2y, g⁰(y) = 3y²− 6y + 2.

g⁰(0) = 2 > 0⇒ y = 0 is unstable g⁰(1) =−1 < 0 ⇒ y = 0 is locally stable

g⁰(2) = 2 > 0⇒ y = 0 is unstable

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Name: Student ID number:

6. Given a matrix A =

[ 5 4

−2 −1 ]

(a) (10 pts) Find the eigenvalues and eigenvectors of A

0 = det (A− λI) = λ²− 4λ + 3 = (λ − 1)(λ − 3) ⇒ λ1 = 1, λ₂ = 3.

λ₁ = 1 : Av₁ = λ₁v₁ ⇒

[ 5x + 4y

−2x − y ]

= [ x

y ]

⇒ y = −x ⇒ Take v1 = [ 1

−1 ]

λ₂ = 3 : Av₂ = λ₂v₂ ⇒

[ 5x + 4y

−2x − y ]

= [ 3x

3y ]

⇒ x = −2y ⇒ Take v2 = [ −2

1 ]

(b) (10 pts) Let v1, v2 be two eigenvectors found in part (6a). Find α, β so that

[ −3 2

]

= αv₁+βv₂. Compute A¹⁰ [ −3

2 ]

without using a calculator.

[ −3 2

]

= α [ 1

−1 ]

+ β [ −2

1 ]

⇒ α = −1, β = 1.

⇒ A¹⁰ [ −3

2 ]

= A¹⁰(−v1+ v₂) = −A¹⁰v₁+ A¹⁰v₂ =−λ¹⁰₁ v₁+ λ¹⁰₂ v₂

=− [ 1

−1 ]

+ 3¹⁰ [ −2

1 ]

=

[ −1 − 2 · 3¹⁰ 1 + 3¹⁰

]

7. (5 pts) Let A be a matrix and v₁ and v₂ be two eigenvectors of A, with eigen values λ₁ and λ₂, respectively (i.e. Av₁ = λ₁v₁ and Av₂ = λ₂v₂). Assume that λ₁ 6= λ2, prove that there is no constant c satisfying that v₂ = cv₁

Suppose that v₂ = cv₁ for some c. Then

Av₂ =λ₂v₂ = Acv₁ = cAv₁ = cλ₁v₁ = λ₁v₂.

⇒(λ2− λ1)v₂ = 0⇒ λ1 = λ₂, (∵ v2 6= −→

0 )⇒ contradiction

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Name: Student ID number:

8. (a) (5 pts) Compute lim

(x,y)→(1,2)

2x²y x⁴+ y² lim

(x,y)→(1,2)

2x²y

x⁴+ y² = lim_{(x,y)→(1,2)}2x²y lim_{(x,y)→(1,2)}x⁴ + y² = 4

5. (b) (5 pts) Show that lim

(x,y)→(0,0)

2x²y

x⁴+ y² does not exist.

• Along path x = 0, lim

(x,y)→(0,0)

2x²y

x⁴+ y² = lim

y→0

0

0⁴+ y² = 0

• Along path y = x², lim

(x,y)→(0,0)

2x²y

x⁴+ y² = lim

x→0

2x²x²

x⁴+ x⁴ = lim

x→0

2x⁴ 2x⁴ = 1

⇒ The limit does not exist.

9. (10 pts) Given f (x) = e^xy− x

y, find fx, fy, fxx, fxy and fyy.

f_x = ye^xy −1

y, f_y = xe^xy+ x y², f_xx = y²e^xy, f_xy = (1 + xy)e^xy + 1

y², f_yy = x²e^xy− 2x y³

10. (10 pts) Find the equation of the tangent plane to the surface at the given point.

z = x² − y²+ 1 at (1, 2,−2) Let f (x, y) = x²− y²+ 1

f_x(x, y) = 2x, f_y(x, y) = −2y, f_x(1, 2) = 2, f_y(1, 2) =−4, The equation of the tangent plane:

z = L(x, y) = f (1, 2) + fx(1, 2)(x−1)+fy(1, 2)(y−2) = −2+2(x−1)−4(y −2).

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