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School of Mathematics & Physics

EXAMINATION

Semester One Final Examinations, 2020

MATH1052 Multivariate Calculus and Ordinary Differential Equations

1. (10 marks total)

A plane is given by P = (0, 1, −1) + λ(1, 0, 2) + µ(0, 2, −4), where λ, µ ∈ R.

(i) (4 marks) Determine a normal vector to P.

(ii) (4 marks) Let X be a point on the line

Write down an expression for the distance from the point X to the plane P.

(iii) (2 marks) Using this, determine the point of intersection between the line from (ii) and the plane P.

2. (10 marks total)

Let a conic section be given by

(i) (5 marks) Determine the type of conic section.

(ii) (5 marks) Sketch the conic section. Be sure to identify the center, foci, or asymptotes.

3. (10 marks total)

Let

(i) (5 marks) Find all critical points of h(x, y).

(ii) (5 marks) Classify each critical point as a local minimum, local maximum, or saddle point.

4. (10 marks total)

Consider a particle moving through space with velocity

(i) (3 marks) Determine its acceleration vector.

(ii) (7 marks) Determine the position vector, supposing the particle starts at position (3, −2, 3) at time t = 0.

5. (10 marks)

Consider the ODE

and show that it becomes separable using the substitution y(x) = xu(x). (You do not need to solve the ODE.)

6. (10 marks total)

Consider a particle moving in the plane along the curve

for some constants R, ω > 0.

(i) (5 marks) Determine the distance the particle travels for t ∈ [0, 2π].

(ii) (5 marks) Suppose the plane has a voltage given by

Determine the change in voltage the particle experiences at time t.

7. (10 marks)

Find the general solution y(x) to the ODE

8. (10 marks)

Find the maximum and minimum values of the function subject to the constraint

9. (10 marks total)

Let a vector field in be defined by

Evaluate where C is the curve

10. (10 marks total)

(i) (5 marks) Determine all equilibrium solutions to the ODE

(ii) (5 marks) Determine if each equilibrium solution is stable or not. Justify your answer.

Formula sheet

● Linear and quadratic approximations

● Directional derivative

● Chain rule

● Implicit differentiation

● Hessian

● Lagrange Multipliers

● Work