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📝 Functions of Two or More Variables (14 MCQs)

📖 From Calculus • 14. Partial Derivatives Calculus • 14 questions available

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What is Functions of Two or More Variables?

Definition:
A function of two or more variables assigns a unique output value $z = f(x, y)$ to each ordered pair or tuple in its domain within $\mathbb{R}^n$ .

Example:
The temperature distribution on a metal plate is modeled by $T(x, y) = 100 - x^2 - y^2$ , where input coordinates determine the scalar temperature output.

Reason:
This concept extends single-variable calculus to model real-world phenomena dependent on multiple independent factors like space and time.

Easy

Medium

Hard

📝 All Functions of Two or More Variables MCQs

Q1. A temperature distribution on a metal plate is modeled by $T(x,y) = 100 - x^2 - 2y^2$ . An ant at point $(2,3)$ wishes to cool down as rapidly as possible. In which direction should it move?

A.The direction of

\langle 4, 12 \rangle

B.The direction of

\langle -4, -12 \rangle

✅

C.The direction of

\langle 2, 6 \rangle

D.The direction of

\langle -2, -3 \rangle

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: To decrease temperature most rapidly, the ant must move opposite to the gradient vector. Computing $\nabla T = \langle -2x, -4y \rangle$ at $(2,3)$ yields $\langle -4, -12 \rangle$ . The direction of steepest descent is exactly this vector, not its negation or a scaled version, making option B correct for fastest cooling.

Q2. Consider $f(x,y) = x^3 + y^3 - 3xy$ . A student claims that since $f_x(1,1) = 0$ and $f_y(1,1) = 0$ , the point $(1,1)$ must be a local minimum. What is the fundamental flaw in this reasoning?

A.The partial derivatives were computed incorrectly.

B.Vanishing first derivatives only identify critical points; second-order analysis is required to classify them. ✅

C.The function is not differentiable at (1,1).

D.Local minima cannot occur when both variables are positive.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Setting first partials to zero identifies critical points but does not determine their nature. The student skipped the Second Derivative Test or other classification methods. At (1,1), the discriminant $D = f_{xx}f_{yy} - (f_{xy})^2$ is negative, indicating a saddle point, not a minimum, revealing the error in assuming extremum from first derivatives alone.

Q3. Given contour lines of $z = f(x,y)$ that become increasingly spaced apart as one moves rightward along the x-axis while holding y constant, what can be definitively concluded about $f_x$ in that region?

f_x

is increasing.

f_x

is decreasing in magnitude. ✅

f_x

is constant.

f_{xy} > 0

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Contour spacing reflects gradient magnitude: wider spacing indicates smaller rate of change. Since contours spread apart moving rightward with fixed y, the slope in the x-direction diminishes. Thus, $|f_x|$ decreases. This interpretation links visual topographic information directly to partial derivative behavior without requiring explicit functional form, testing graphical literacy in multivariable calculus.

Q4. If $w = f(u,v)$ where $u = x+y$ and $v = x-y$ , and it is known that $\frac{\partial w}{\partial x} = 5$ and $\frac{\partial w}{\partial y} = 3$ at a specific point, what is the value of $\frac{\partial f}{\partial u}$ at the corresponding (u,v)?

A.4 ✅

B.2

C.8

D.1

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Applying chain rule: $w_x = f_u u_x + f_v v_x = f_u + f_v$ and $w_y = f_u u_y + f_v v_y = f_u - f_v$ . Adding equations gives $w_x + w_y = 2f_u$ , so $f_u = (5+3)/2 = 4$ . This requires synthesizing transformation rules with system solving, integrating multiple concepts beyond rote differentiation.

Q5. Which statement best explains why $f_{xy} = f_{yx}$ holds for most functions encountered in applied mathematics, despite being logically independent conditions?

A.It is always true by definition of partial derivatives.

B.Continuity of second mixed partials guarantees equality via Clairaut’s Theorem, which applies to smooth physical models. ✅

C.Mixed partials represent the same geometric curvature regardless of order.

D.The equality follows from the commutativity of addition in limits.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: While mixed partials aren't inherently equal, real-world phenomena are typically modeled by sufficiently smooth functions where second derivatives are continuous. Clairaut’s Theorem then ensures symmetry. This conceptual distinction prevents overgeneralization while acknowledging practical applicability, emphasizing mathematical rigor versus empirical convenience in modeling contexts involving heat flow or fluid dynamics.

Q6. A company's profit is $P(x,y) = -x^2 - 2y^2 + 4x + 6y$ , where x and y are production levels constrained by $x + y \leq 5$ . Without using Lagrange multipliers, how would you verify whether the unconstrained maximum satisfies the constraint?

A.Compute gradient and check if sum of components equals 5.

B.Find critical point of P; if x+y<5, it's valid; else optimum lies on boundary. ✅

C.Substitute y=5-x into P and maximize single-variable function.

D.Check second derivatives to confirm concavity before evaluating constraint.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Unconstrained critical point occurs at $(2, 1.5)$ with sum 3.5 < 5, satisfying constraint. Since P is strictly concave, this interior point is global max under constraint. This approach prioritizes checking feasibility before invoking advanced methods, demonstrating efficient problem-solving strategy in optimization scenarios with inequality constraints common in economics and operations research.

Q7. Student computes directional derivative of $f(x,y)=x^2y$ at (1,2) in direction $\vec{v}=\langle 3,4 \rangle$ as $\nabla f \cdot \vec{v} = 28$ . Why is this result incorrect despite correct gradient calculation?

A.Directional derivative requires unit vector;

\vec{v}

wasn't normalized. ✅

B.Gradient was miscalculated at the point.

C.Dot product formula doesn't apply to non-linear functions.

D.Directional derivative is undefined for polynomial functions.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Directional derivative definition mandates unit direction vector. Here $|\vec{v}|=5$ , so correct computation divides dot product by 5, yielding 5.6. Student omitted normalization, a pervasive misconception. Recognizing this error reinforces precise understanding of directional rates versus raw projections, crucial for applications like gradient descent algorithms where step size depends on proper scaling.

Q8. For $f(x,y) = \sqrt{x^2 + y^2}$ , why do partial derivatives fail to exist at the origin despite the function being continuous everywhere?

A.The function has a cusp-like singularity where tangent plane is undefined. ✅

B.Partial derivatives require differentiability, which fails at origin.

C.Limits defining partials yield infinite values.

D.Continuity implies existence of partials only in open domains.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Although continuous, $f$ represents a cone with sharp vertex at origin. Approaching along axes gives finite one-sided derivatives, but no unique tangent plane exists. Partial derivatives measure linear approximation quality; conical singularity violates this. This distinguishes continuity from differentiability, highlighting geometric intuition behind analytical definitions in multivariable contexts beyond mere limit computations.

Q9. Given $z = e^{xy} \sin(x+y)$ , which computational strategy minimizes algebraic errors when finding $z_{xy}$ ?

A.Differentiate w.r.t. x first, treating y constant, then differentiate result w.r.t. y.

B.Use logarithmic differentiation before applying product rule.

C.Apply symmetry of mixed partials to compute easier order first. ✅

D.Expand sin(x+y) using trig identity before differentiating.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Since $z_{xy}=z_{yx}$ for smooth functions, choosing computationally simpler order reduces mistakes. Here $z_y$ involves fewer terms than $z_x$ due to exponential structure. Strategic ordering leverages theoretical properties for practical efficiency, exemplifying meta-cognitive problem-solving in complex differentiation tasks where brute force invites errors despite mechanical correctness.

Q10. In thermodynamics, internal energy U(S,V) has partials $(\partial U/\partial S)_V = T$ and $(\partial U/\partial V)_S = -P$ . If experimental data shows $(\partial T/\partial V)_S \neq -(\partial P/\partial S)_V$ , what does this imply?

A.Measurement errors violate Maxwell relations.

B.U is not a state function or system isn't in equilibrium. ✅

C.Temperature and pressure definitions are inconsistent.

D.Second law of thermodynamics is violated.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Maxwell relations derive from exactness of dU, requiring U to be well-defined state function. Observed inequality suggests either non-equilibrium conditions, path-dependent measurements, or invalid state function assumption. This connects abstract mathematics to physical reality, testing ability to diagnose model breakdown through mathematical consistency checks rather than accepting formulas uncritically in scientific contexts.

Q11. When approximating $f(1.02, 1.97)$ for $f(x,y)=x^3+y^2$ using linearization at (1,2), which factor contributes MOST to approximation error?

A.Magnitude of second-order partial derivatives near expansion point.

B.Distance between evaluation point and center of linearization. ✅

C.Nonlinearity of cubic term versus quadratic term.

D.Choice of base point (1,2) versus nearby alternatives.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Linear approximation error scales quadratically with displacement $\Delta x, \Delta y$ . Here $\Delta x=0.02, \Delta y=-0.03$ , small but nonzero. While higher derivatives matter theoretically, actual error dominance comes from geometric distance in domain. Understanding this guides adaptive refinement strategies in numerical methods, emphasizing spatial proximity over functional complexity in error budgeting.

Q12. A student argues that because $f_x(a,b)>0$ and $f_y(a,b)>0$ , function f must increase along every path through (a,b). What counterexample disproves this claim?

A.Any function with negative mixed partial derivative.

B.Path approaching diagonally where directional derivative becomes negative.

C.Function decreasing along curve tangent to negative gradient direction. ✅

D.No counterexample exists; positivity of partials implies omnidirectional increase.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Positive partials guarantee increase only along coordinate axes. Along direction $\vec{v}=\langle -1,-1 \rangle$ , directional derivative could be negative if cross-effects dominate. For instance, $f=x+y-10xy$ at (0,0) has positive partials but decreases along y=x. This exposes limitation of axis-aligned reasoning, reinforcing need for full gradient analysis in multidimensional behavior prediction.

Q13. For production function $Q(K,L)=K^{0.3}L^{0.7}$ , marginal rate of technical substitution (MRTS) equals $-\frac{Q_L}{Q_K}$ . If capital K increases while maintaining output Q constant, how does MRTS change?

A.Increases due to diminishing returns to labor.

B.Decreases as isoquants become flatter. ✅

C.Remains constant along Cobb-Douglas isoquants.

D.Changes sign depending on elasticity parameters.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Along isoquant, MRTS = $-(0.7/0.3)(K/L)$ . As K rises with Q fixed, L must fall, increasing K/L ratio. But MRTS magnitude decreases because isoquants convexify. This reflects economic principle of diminishing MRTS, linking calculus to microeconomic theory through dynamic interpretation of partial ratios under constraints, beyond static computation.

Q14. Suppose $f(x,y)$ satisfies $f(x,y) = g(x)h(y)$ and $f_{xy}(a,b) = 0$ at some point. Which conclusion is necessarily valid?

A.Either

g&#039;(a)=0

h&#039;(b)=0

. ✅

B.Both

g(a)=0

and

h(b)=0

C.Function f has a critical point at (a,b).

D.Product structure implies separability of variables globally.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For separable functions, $f_{xy} = g'(x)h'(y)$ . Zero mixed partial requires at least one factor's derivative vanish at respective coordinate. This doesn't imply function value zero nor critical point status. Recognizing structural implications of vanishing mixed partials tests deep understanding of functional forms versus pointwise properties, distinguishing necessary from sufficient conditions in advanced analysis.

🔗 Related Topics (MCQs)

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