Quiz: Multivariable Functions and Optimization — 10 questions

1. What does a function of two variables assign to each ordered pair in its domain?

A. A set of possible input pairs

B. A collection of related equations

C. A unique real-number output

D. A variable-length ordered tuple

Answer: (C)

Explanation

A function of two variables assigns exactly one real number to every ordered pair in its domain. A relation can associate multiple outputs with one input pair, so it does not necessarily satisfy the function condition.

2. What is a function of two variables?

A. A function that only depends on one variable.

B. A set of ordered pairs without any specific rule.

C. A rule that assigns a unique real number to each ordered pair (x,y) in its domain.

D. A rule that assigns multiple values to each pair (x,y).

Answer: (C)

Explanation

A function of two variables assigns a unique real number to each ordered pair (x,y) within its domain. The other options describe relations that are not functions or are incomplete.

3. Which expression represents an ordered set of n input elements for a function of several variables?

A. $[x_1+x_2+\cdots+x_n]$

B. $x_1x_2\cdots x_n$

C. $\{x_1,x_2,\ldots,x_n\}$

D. $(x_1,x_2,\ldots,x_n)$

Answer: (D)

Explanation

An n-tuple is written with parentheses because the order of its n elements matters. A set written with braces does not preserve element order in the same way.

4. What is the primary purpose of defining the domain of a multivariable function?

A. To specify the set of all possible input pairs where the function is defined

B. To find the limit of the function as variables approach infinity

C. To determine the maximum and minimum values of the function

D. To identify the points where the function is continuous

Answer: (A)

Explanation

The domain of a multivariable function specifies the subset of \\mathbb{R}^2 where the function is defined, which is essential for understanding where the function can be evaluated. The other options relate to properties or values of the function, not its domain.

5. Which set describes the domain of a multivariable function?

A. The collection of equations used to define the function

B. The subset of $\mathbb{R}^2$ containing all admissible ordered pairs

C. The set of real numbers produced as function values

D. The subset of $\mathbb{R}^2$ containing pairs with equal coordinates

Answer: (B)

Explanation

The domain consists of every ordered pair in $\mathbb{R}^2$ for which the function is defined. The resulting output values form the range, not the domain.

6. When was the concept of limits and continuity in multivariable functions formally established in mathematical analysis?

A. In the early 20th century, during the development of rigorous calculus.

B. In the 17th century, with the advent of calculus by Newton and Leibniz.

C. In the mid-20th century, with the rise of modern mathematical analysis and topology.

D. In the late 19th century, as part of the formalization of analysis by Cauchy and Weierstrass.

Answer: (D)

Explanation

The formal study of limits and continuity for multivariable functions was developed in the late 19th century, notably through the work of Cauchy and Weierstrass, who contributed to the rigorous foundations of analysis. The other options refer to earlier or later periods that are less associated with the formalization of multivariable limits and continuity.

7. What is the domain of $f_1(x,y)=\frac{-x^2+y^3-5}{x-1}$?

A. $\{(x,y)\in\mathbb{R}^2\mid x>1\}$

B. $\{(x,y)\in\mathbb{R}^2\mid x\ne1,\ y\ne0\}$

C. $\{(x,y)\in\mathbb{R}^2\mid y\ne1\}$

D. $\{(x,y)\in\mathbb{R}^2\mid x\ne1\}$

Answer: (D)

Explanation

The denominator is zero when $x=1$, so those input pairs are excluded from the domain, while every real value of $y$ remains allowed. The numerator imposes no additional restriction on either variable.

8. How do second partial derivatives relate to the symmetry of mixed derivatives in multivariable calculus?

A. Mixed derivatives are unrelated to each other, so $f_{xy}(x,y)$ and $f_{yx}(x,y)$ can differ arbitrarily.

B. Mixed derivatives are always equal regardless of continuity, so $f_{xy}(x,y)=f_{yx}(x,y)$ holds universally.

C. If the mixed derivatives are continuous, then $f_{xy}(x,y)=f_{yx}(x,y)$ at every point.

D. The equality $f_{xy}(x,y)=f_{yx}(x,y)$ only holds at points where the derivatives are zero.

Answer: (C)

Explanation

When the mixed partial derivatives are continuous, Clairaut's theorem guarantees that $f_{xy}(x,y)=f_{yx}(x,y)$ at every point. If the derivatives are not continuous, the equality may not hold, making continuity a key condition.

9. Who is credited with establishing the fundamental relationship that connects second partial derivatives in multivariable calculus, specifically the equality of mixed derivatives under certain conditions?

A. Cauchy, who introduced the concept of second derivatives and their symmetry in the context of complex analysis.

B. Newton, who developed the calculus and the fundamental theorem relating derivatives and integrals.

C. Lagrange, known for formulating the method of Lagrange multipliers and related second derivative tests.

D. Clairaut, who proved that if the mixed derivatives are continuous, then they are equal at every point.

Answer: (D)

Explanation

Clairaut is credited with the theorem stating that if the mixed partial derivatives are continuous, then they are equal, which is fundamental in the symmetry of second derivatives. Cauchy contributed to the development of analysis but is not specifically credited with this particular result.

10. What is the primary cause for the classification of returns to scale in a Cobb-Douglas production function?

A. The sum of the exponents $\alpha + \beta$ \\) determines the degree of returns to scale.

B. The value of the constant A in the function influences the returns to scale.

C. The individual values of labor L and capital K directly affect the returns to scale.

D. The specific form of the function, whether linear or nonlinear, causes the classification.

Answer: (A)

Explanation

The sum of the exponents $\alpha + \beta$ \\) in the Cobb-Douglas function directly determines whether the returns to scale are increasing, decreasing, or constant. A sum equal to 1 indicates constant returns, greater than 1 indicates increasing, and less than 1 indicates decreasing returns.