Tutorial 10 Multivariable Calculus
1. If 𝑓(𝑤, 𝑥, 𝑦, 𝑧) = 13𝑥 2 𝑦 + 5𝑤 3 − 4𝑥𝑦𝑧 , find the following.
𝜕𝑓
𝜕𝑓
(a) 𝜕𝑤
2. Evaluate z =
𝜕𝑓
(b) 𝜕𝑥
𝑥 2 +𝑦 2
2 2
𝑒 𝑥 +𝑦
at
𝜕𝑓
(c) 𝜕𝑦
∂z
|
∂x 𝑥=0
𝑦=0
(d) 𝜕𝑧
∂z
and ∂y|𝑥=1
𝑦=1
3. Suppose that a company’s sales are related to its television advertising by
𝑠 = 20,000 + 10𝑛𝑡 + 20𝑛2
where n is the number of commercials per day and t is the length of the commercials in
seconds. Find the partial derivative of s with respect to n, and use the result to find the rate
of change of sales with respect to the number of commercials per day, if the company is
currently running ten 30-second commercials per day.
4. Test z = y2 – x2 for relative maxima and minima.
5. A manufacturer’s joint-cost function for producing qA units of product A and qB units of
product B is given by
𝑞𝐴2 (𝑞𝐵3 + 𝑞𝐴 )1/2
1/3
c=
+ 𝑞𝐴 𝑞𝐵 + 600
17
where c is in RM.
(a) Find the marginal cost functions with respect to qA and qB.
(b) Evaluate the marginal cost functions with respect to qA when qA = 17 and qB = 8. Round
your answer to two decimal places.
(c) Use your answer to part (b) to estimate the change in cost if production of product A is
decreased from 17 to 16 units, while production of product B is held constant at 8 units.
6. Suppose the cost of producing x units of product X and y unit of product Y is given by
∂c
c = √2𝑥 + 3𝑦 where 𝑥 = 3𝑡 + 5 and y = 𝑡 2 + 2𝑡 + 1, evaluate ∂t when t = 1.
7. Use Lagrange to find the critical points of
constraints x + y + z = 4 and x − y + z = 4.
f(x, y, z) = 𝑥 2 + 𝑦 2 + 𝑧 2 subject to the
Self-Practice Questions
1. If 𝑧 = 𝑥 3 𝑦 − 3𝑥𝑦 2 + 4, find each of the second partial derivatives of the function.
2. Examine f(x, y) = x4 + (x − y)4 for relative extrema.
3. Find the minimum value of the function 𝑧 = 𝑥 3 + 𝑦 3 + 𝑥𝑦 subject to the constraint
x + y – 4 = 0.