PRACTICE SET: MULTIVARIATE CALCULUS

Short Questions

1. What is the domain of z( x,y) = √(x-1).e^y?

2. Let x = (x₁, x₂, x₃). Write down the general form of the Hessian matrix of f(x) at (x*) given that f(x) is twice differentiable in the neighbourhood of (x*).

3. Suppose f(x, y, z) is homogeneous of degree 0. What is the degree of homogeneity of f'_x( x,y, z ) ?

4. Can two level curves of z = f(x, y) intersect? Why?

5. Does this equation define y as an implicit function of x

F ( x, y) = e^y + y² x⁴

Problem 2

Consider the messier function f(x,y) = x³ - 3xy² + y⁴. Find its stationary values and characterize them.

Problem 3

Consider the following function: z = (x₁ + x₂)² + x²₃ + 2x₁x₃ + 2x₂x₃.

1. Find the critical points of the function.

2. Does the function reach a maximum or a minimum? Base your answer on a discussion of the properties of the Hessian matrix of the function.

3. Find the equation of the tangent plane at the critical points identified by the FOC.

Problem 4

Write down all of the second partial derivatives of the function f(x,y) = xln(y). Evaluate these at (x,y) = (-1, 2).

Problem 5

Is the function f(x,y) = (xy^-1+y) homogeneous of degree one? Justify your answer.

Problem 6

Find any maxima or minima of f(x,y) = 3x² - y³ - 3xy and Φ(x,y) = x+ y - 2√xy.

Problem 7

Consider the following function of two variables: f(x,y) = x⁴ - xy + y⁴.

1. Find the gradient vector and the stationary values of the function.

2. Find the Hessian matrix and hence determine the nature of the stationary values you just found.

Problem 8 (I want you to have general understanding what these mean - I would not ask on a test for definitions).

(a) Define a concave function and give an example.
(b) Define a quasiconcave function and give an example.
(c) Prove that a function that is concave is also quasiconcave.
(d) Show by a counterexample that the converse need not be true.

Problem 9

Evaluate the definiteness of the following quadratic forms: f(x,y) = x²+3xy+y² and g(x,y)=x² - y².

Problem 10

Consider the following function: z = x3 + y3 - 6xy.

1. Find the derivative at (1,2,-3) in the direction v^- =(.6,0,.4)
2. Find the equation of the tangent plane at (1,2,-3).

Problem 11

Check whether the following functions have any maxima or minima:
• f(x₁,x₂,x₃)=x₁² +3x₂² -3x₁x₂ + 4x₂x₃ + 6x₃2
• F(x,y) = y²-x².
• F(x,y) = x5y + xy5 + xy
• F(x,y,z) = log(x²+y²+1)

Problem 12

Check the following functions for concavity and quasiconcavity:
• F(x,y)=-x²-y² for x,y>0
• F(x,y)=-(x+1)²-(y+2)² for x,y>0

Problem 13

Let f(x,y,z) = 3x²y + xyz - 4y²z³, where x = 2r-3s, y = 6r + s and z = r - s. Find ∂f/∂r and ∂f/∂s .

Problem 14

Show that U(x,y,z) = (x²+y²+z²)^-1/2 satisfies: ∂²U/∂x² + ∂²U/∂y² + ∂²U/∂z² = 0

Problem 15

Find the planes tangent to the following surfaces at the indicated points.
• x²+2y²+3zx = 10 at (1,2,1/3)
• y² - x² = 3 at (1,2,8)
• xyz = 1 at (1,1,1)
• z = x³+y³-6xy at (1,2,-3)

Problem 16
Consider Q = 2x² + 16xy - 8y².
a) Write Q as a quadratic form.
b) Is Q positive or negative definite (or neither)?
c) Subject to the constraint 2x - 2y = 0, is Q positive or negative definite (or neither)?

Problem 17

For the surface y²xz + 3z² - 2xy² - 8z = 8
a) find the equation of the tangent plane at point (1,1,-1).
b) find the derivative at this point in the direction (-1,-1,1)

Problem 18

Consider the production function Q = 9K^1/3L^1/3T^1/3, where K is capital, L is labour and T is land.

1. What is the output when K = 1000, L = 27 and T = 125?

2. Estimate ∂Q/∂K (the marginal product of labour), ∂Q/∂L (the marginal product of capital) and ∂Q/∂T (the marginal product of land) at (1000, 27, 125).

3. Find all second order partial derivatives. Use Young's theorem to save time.

4. Suppose the number of labour units (workers) increase from 27 to 30, while capital remains at 1000 and land remains at 125. What is the approximate change in output?

Problem 19

Consider f( x,y) = x^3/2y^3/2 + 6x³ and g(x,y) = 3x²y .

1. What is the degree of homogeneity of z(x,y) = f (x,y)/g( x,y)?

2. Show that Euler's theorem holds for z( x,y).

3. Let r = g( x,y) = 3x²y and x, y > 0. Is h( x,y) = ln r homogeneous? Explain.

4. Use the chain rule to find ∂h/∂x and ∂h/∂y.

Problem 21

Find dy/dx for each of the following functions:

a) f(x) = y + y³ = x
b) e^xy = 2x + 3y
c) x² + 2xy+ y² = 8

Problem 22

For the surface x²yz + 3y² = 2xz² -8z:

a) find the equation of the tangent plane at point (1,2,-1).

b) find the equation of the normal line at point (1,2,-1).

c) at what point does that normal line meet the plane x+3y-2z=10?

Problem 23

For the surface y²xz + 3z² = 2xy² + 8z + 4:
a) find the equation of the tangent plane at point (1,2,-1).
b) find the derivative at this point in the direction (.6,0,.4)
c) find the equation of the normal line at point (1,2,-1).
d) at what point does that normal line meet the plane x+3y+2z=10?

Problem 24

Find the directional derivatives for the following functions at the indicated points in the given directions:

a) f(x,y) = x + 2xy -3y², (1,2), v=(3/5,4/5)

b) f(x,y,z) = ex + zy -3y², (1,1,1), v=(1/√3, 1/√3, 1/√3 )

Problem 25

Show that the surface

x² - 2yz + y³ = 4

is ⊥ to any member of the family of surfaces

x² + 1 = (2 - 4a)y² + az²

at the point of intersection (1,-1,2).

Problem 26

a) Find the point- norm equation for the plane passing through the points (3,1,-2), (-1,2,4) and (2,-1,1)

b) Find a parametric representation for the same plane.

c) Find the point- norm equation for the plane passing through the points (2,2,2), (-1,2,4) and (2,-1,1)

d) Find a parametric representation for the normal line to the same plane passing through the origin.

Problem 27 (not a question I would ask on the test)

Consider a point (x₀, y₀) not on the line y = mx + b. Show that the vector of shortest length between the point and the line forms a right angle with the line.

Problem 28

Find the area of the a) triangle, and b) parallelogram with vertices A(2,3,5), B(4,2,-1), C(3,6,4).

Problem 29

Consider the points P(0,0,1), Q(0,1,0) and R=(1,0,0).

a) Find the parametric representation of the plane going through the points.

b) Find the equation of the plane going through the points.

c) Find the equation of the plane orthogonal to the plane in (b) and going through points P and Q.

Problem 30 (not a question I would ask on the test)

Give a non-empty example for each of the following sets in R² (if an example exists) and sketch your answer. If an example does not exist, explain why not (by defining the terms involved and demonstrating their inconsistency).
- A is compact and equal to its own boundary;
- B is open and convex;
- C is neither open nor closed.

Problem 31 (not a question I would ask on the test)
Consider the following simple macro model (IS-LM).

Y = c(Y)+I⁰-i(r) + G

M_s = m(Y,r) + M⁰

Where Y is national income, C is aggregate consumption, G is government spending, I is aggregate investment, r is the interest rate and Ms is the money supply. I0 and M0 are parameters. The partial derivatives have the following signs: cy<0, ir<0, my>0 and mr<0. Linearize the model by total differentiation. Express the result in matrix form and find the government expenditure multiplier by Camer's rule.

Problem 32

Consider the function f (x, y) = 2 log x + log y. Determine whether it is homogeneous. Determine whether it is homothetic.