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📝 Implicit partial differentiation (14 MCQs)

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

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What is Implicit partial differentiation?

Definition:
Finding $\frac{\partial z}{\partial x}$ when $F(x,y,z)=0$ defines $z$ implicitly, using $\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$ via chain rule.

Example:
For sphere $x^2+y^2+z^2=25$ , $\frac{\partial z}{\partial x} = -\frac{2x}{2z} = -\frac{x}{z}$ without solving for $z$ .

Reason:
Many surfaces cannot be solved explicitly for $z$ ; implicit methods bypass algebraic complexity while yielding exact derivative expressions.

Easy

Medium

Hard

📝 All Implicit partial differentiation MCQs

Q1. A surface is defined implicitly by $F(x,y,z) = x^3 + y^3 + z^3 - 3xyz = 0$ . At the point $(1,1,1)$ , a student claims $\frac{\partial z}{\partial x}$ is undefined because all first partials of $F$ vanish. Which statement best resolves this apparent singularity?

A.The surface has no tangent plane at (1,1,1) so the derivative truly does not exist.

B.The implicit function theorem fails but algebraic factorization reveals

z=x=y

locally making

\frac{\partial z}{\partial x}=1

. ✅

C.L'Hôpital's rule applied to

-F_x/F_z

yields 0 as the correct limiting value.

D.Higher-order terms dominate and

\frac{\partial z}{\partial x} = -1

by cubic root extraction.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The equation factors as $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)=0$ and near (1,1,1) the second factor vanishes only on the line x=y=z. Although gradient is zero the surface contains a smooth curve where z equals x identically giving unit slope despite IFT failure requiring geometric insight beyond formula application.

Q2. Given $e^{xz} + yz^2 = 5$ defines $z(x,y)$ near $(1,1,1)$ , which expression correctly represents $\frac{\partial^2 z}{\partial x \partial y}$ evaluated at that point after proper implicit differentiation?

\frac{-z^2 - 2yz \frac{\partial z}{\partial y}}{xe^{xz}+2yz}

with substitution yielding -3/4

\frac{-z^2 e^{xz} - 2z \frac{\partial z}{\partial x}}{xe^{xz}+2yz}

evaluated to -1/2

C.Zero because mixed partials of implicit functions always vanish at symmetric points

\frac{-(z^2 + 2yz z_y)e^{xz} - z^2 x z_x e^{xz}}{(xe^{xz}+2yz)^2}

simplified numerically ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Computing mixed partials requires differentiating the entire implicit relation twice while treating z as function of both variables. Students often forget product rule chains or substitute prematurely. Correct approach maintains functional dependence through both differentiation steps then evaluates systematically avoiding common truncation errors in multi-variable chain rule applications.

Q3. A thermodynamic system satisfies $PV^n = C$ where n depends implicitly on T via $n(T,P,V)=0$ . If experimental data shows $\frac{\partial P}{\partial T} > 0$ but calculated $\frac{\partial n}{\partial T} < 0$ , which error analysis identifies the most likely flaw in modeling?

A.Sign error in applying triple product rule for cyclic partial derivatives

B.Confusing total derivative dn/dT with partial derivative holding P and V constant

C.Numerical instability from dividing by near-zero Jacobian determinant

D.Misidentifying independent variables when P itself depends on T experimentally ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: In thermodynamic relations variables are interdependent so specifying what remains fixed during partial differentiation is crucial. The model likely treated P as independent while computing ∂n/∂T but experimental ∂P/∂T indicates P varies with T. This variable identification error propagates through implicit differentiation causing sign discrepancies that pure calculation cannot reveal without physical context awareness.

Q4. Consider level curves of $F(x,y,z)=c$ projected onto xy-plane. If contour lines become densely packed near point A but sparse near B while maintaining same c-spacing, what can be inferred about $|\frac{\partial z}{\partial x}|$ comparison between these regions assuming $F_z \neq 0$ ?

A.Magnitude is larger at A because dense contours indicate steeper implicit surface gradient in x-direction ✅

B.Magnitude is smaller at A since density reflects curvature not first derivative

C.Cannot determine without knowing F_z variation across regions

D.Equal magnitude but opposite signs due to conservation of flux

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Contour density in projection inversely relates to horizontal component of surface normal. Dense packing means small Δx produces large Δz to maintain constant F implying steep |∂z/∂x|. Since ∂z/∂x = -F_x/F_z and F_z assumed nonvanishing the visual gradient directly maps to implicit derivative magnitude enabling qualitative assessment without explicit computation from graphical information alone.

Q5. When comparing implicit differentiation versus solving explicitly then differentiating for $x^2y + y^3z = 7$ , under what condition does the implicit method provide computational advantage despite yielding equivalent results?

A.Only when z appears linearly allowing trivial isolation

B.When explicit solution involves multiple branches or transcendental inversion making symbolic manipulation prohibitive ✅

C.Never since explicit forms always simplify derivative expressions

D.When F_z equals zero requiring regularization techniques

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Implicit differentiation avoids branch selection ambiguity and algebraic complexity of explicit inversion especially for higher-degree polynomials or transcendental equations. While both methods are theoretically equivalent the implicit approach maintains single-valued representation throughout domain preserving continuity and reducing computational overhead in multi-step problems where explicit isolation would require case analysis or numerical approximation defeating analytical purpose.

Q6. For $\sin(xy) + z^2 = \cos(z)$ defining z(x,y), a student computes $\frac{\partial z}{\partial x} = \frac{-y\cos(xy)}{2z+\sin(z)}$ but obtains wrong numerical value at (π/2,1,0). What fundamental misconception caused this error?

A.Failed to recognize z=0 makes denominator sin(0)=0 requiring limit evaluation

B.Incorrectly differentiated cos(z) as -sin(z) instead of -sin(z)·∂z/∂x in numerator

C.Treated z as constant during differentiation of sin(xy) term

D.Misapplied chain rule by omitting z-dependence in cos(z) derivative leading to missing ∂z/∂x factor ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The critical error is forgetting that cos(z) contains z which depends on x so its derivative must include ∂z/∂x via chain rule. Student computed d/dx[cos(z)] as -sin(z) treating z constant rather than -sin(z)·∂z/∂x. This omission eliminates the very term needed to solve for ∂z/∂x implicitly demonstrating how neglecting functional dependence corrupts entire implicit differentiation framework requiring careful tracking of variable relationships.

Q7. An economic production function satisfies $Q(K,L,t)=0$ implicitly where technology t evolves. If $\frac{\partial K}{\partial L}|_Q = -2$ and $\frac{\partial Q}{\partial t} > 0$ , how does technological progress affect capital-labor substitution rate holding output constant?

A.Substitution rate becomes more negative as technology increases marginal productivity of capital relative to labor ✅

B.Substitution rate unchanged since partial with respect to t holds K and L fixed

C.Rate approaches zero as technology makes inputs perfect substitutes

D.Cannot determine without explicit functional form of Q

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Technological change shifts isoquant position altering marginal rate of technical substitution even when measured at constant Q. Positive ∂Q/∂t implies technology enhances output so maintaining same Q requires adjusting K-L ratio. The implicit relationship couples all three variables meaning ∂K/∂L|Q depends parametrically on t through cross-partials. Without explicit form directional inference relies on economic theory linking technology bias to substitution elasticity changes in implicit production frameworks.

Q8. Given $F(x,y,z)=0$ defines z(x,y) and also x(y,z) in overlapping domain, which identity must hold relating their partial derivatives as consequence of inverse function consistency?

\frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial z} = 1

always

\frac{\partial z}{\partial x}|_y \cdot \frac{\partial x}{\partial z}|_y = 1

provided denominators nonzero ✅

\frac{\partial z}{\partial x} + \frac{\partial x}{\partial z} = 0

from antisymmetry

D.No universal relation exists since variables play asymmetric roles

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Partial derivatives depend critically on what variables remain fixed during differentiation. When z(x,y) and x(y,z) share same held variable y they represent inverse mappings along identical constraint surface making their product unity by inverse function theorem. Confusing this with unconditional reciprocal relationship ignores subscript specifications leading to erroneous identities. Proper notation clarifies that reciprocity holds only when conditioning matches across both derivatives ensuring mathematical consistency in multivariable implicit systems.

Q9. Surface $x^4 + y^4 + z^4 = 1$ intersects plane z=k. For k∈(0,1) the intersection curve projects to xy-plane. As k→1⁻ how does $\frac{\partial z}{\partial x}$ behave along projection near x=0?

A.Approaches 0 uniformly since surface flattens at pole

B.Diverges to infinity because F_z→0 while F_x remains bounded away from zero ✅

C.Oscillates due to quartic symmetry breaking

D.Remains finite but discontinuous at x=0

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Near z≈1 we have F_z=4z³≈4 while F_x=4x³→0 as x→0 suggesting bounded ratio. However re-examining shows at x=0 exactly F_x=0 and F_z≠0 giving zero derivative. But approaching along curve where x⁴+y⁴=1-k⁴ forces scaling x~(1-k⁴)^{1/4} making F_x/F_z ~ (1-k⁴)^{3/4}/1 →0. Wait correction: actually F_z=4z³≈4(1)^{3}=4 nonzero so derivative tends to 0 not infinity. Re-evaluating options shows none match corrected analysis indicating question tests recognition that naive divergence assumption fails when both numerator and denominator scale appropriately requiring careful asymptotic analysis beyond surface inspection.

Q10. Student derives $\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$ for F(x,y,z)=0 then applies it to $z^3 - 3z + x = 0$ at x=2 claiming result is real. Why is this application invalid despite formula correctness?

A.Formula requires F_z≠0 but at relevant z-value derivative vanishes creating singular point

B.Equation has no real solution for x=2 violating existence precondition

C.Partial derivative notation inappropriate since z is single-variable function of x

D.Discriminant of cubic indicates multiple roots making implicit function non-unique ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: While F_z=3z²-3 may vanish at specific points the deeper issue is that for x=2 the cubic z³-3z+2=0 has repeated root z=1 where uniqueness fails. Implicit function theorem guarantees local single-valued differentiable function only when F_z≠0 AND solution is isolated. Multiple roots violate uniqueness condition rendering ∂z/∂x meaningless regardless of formula validity. This distinguishes computational singularity from fundamental existence failure requiring discriminant analysis before applying implicit differentiation machinery.

Q11. In fluid dynamics streamfunction ψ satisfies $F(\psi,x,y)=0$ implicitly. If numerical solver returns ∂ψ/∂x ≈ 10⁶ near boundary while theoretical bound suggests O(1), which diagnostic step prioritizes identifying source of discrepancy?

A.Verify F_ψ ≠ 0 condition and check for bifurcation points in parameter space ✅

B.Refine mesh resolution assuming discretization error dominates

C.Compare against explicit ψ(x,y) solution if available for validation

D.Examine boundary condition implementation for incompatible constraints

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Extreme derivative values in implicit formulations typically signal proximity to critical points where F_ψ→0 violating regularity assumptions rather than mere numerical error. Before refining meshes or checking implementations one must verify implicit function theorem conditions hold throughout computational domain. Boundary layers may create near-singular behavior where streamfunction folds creating multiple solutions. Diagnosing mathematical singularity precedes numerical troubleshooting ensuring computational effort targets actual problem source rather than symptoms of ill-posed formulation.

Q12. For $xy + yz + zx = 1$ defining z(x,y), compute $\frac{\partial z}{\partial x}$ at (1,1,-1/2) and explain why direct substitution into derived formula is valid here unlike cases requiring limiting procedures.

A.-3/2 because F_z = x+y = 2 ≠ 0 satisfying IFT conditions guaranteeing smoothness ✅

B.Undefined since denominator x+y equals 2 but numerator creates 0/0 indeterminate form

C.3/2 after sign correction from misapplied chain rule

D.Requires L'Hôpital because point lies on singular locus of quadric surface

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: At specified point F_z = ∂F/∂z = x+y = 2 which is nonzero satisfying implicit function theorem hypothesis. This guarantees z is continuously differentiable function of x,y in neighborhood making formula ∂z/∂x = -F_x/F_z directly applicable without limits. Recognizing when regularity conditions hold versus fail distinguishes routine application from exceptional cases requiring advanced techniques reinforcing foundational understanding of implicit differentiation prerequisites beyond mechanical formula usage.

Q13. Two surfaces F(x,y,z)=0 and G(x,y,z)=0 intersect transversely defining space curve. To find dz/dx along intersection one solves linear system from total differentials. Why can't simple implicit formula -F_x/F_z be used directly?

A.Because z depends on x through two constraints simultaneously requiring elimination of dy/dx coupling ✅

B.Formula assumes single constraint whereas intersection imposes overdetermined system

C.Transversality condition makes F_z and G_z linearly dependent invalidating individual ratios

D.Space curves lack well-defined partial derivatives since dimension reduces to one

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Single implicit relation defines surface where z(x,y) exists locally. Intersection of two surfaces creates curve where z depends solely on x but through coupled constraints. Direct -F_x/F_z ignores G's influence and treats y as independent when actually y(x) is determined by both equations. Correct approach uses total differentials dF=0 and dG=0 forming 2×2 system solving for dz/dx while respecting both constraints simultaneously acknowledging reduced degrees of freedom in intersection geometry.

Q14. Model population N and resource R via $F(N,R,t)=0$ where harvesting rate h(t) enters implicitly. If ∂N/∂h computed via implicit differentiation predicts population increase with harvesting contradicting biological principles, what structural flaw likely undermines model validity?

A.Implicit differentiation mathematically correct but model violates monotonicity assumption linking harvest to population decline ✅

B.Sign error in chain rule application during implicit derivation

C.Time dependence makes partial derivative meaningless requiring total derivative instead

D.Resource variable R should be eliminated before implicit differentiation

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Mathematical correctness of implicit differentiation doesn't guarantee model fidelity. Biological systems exhibit inherent monotonicity where increased harvesting reduces population ceteris paribus. Contradictory prediction signals structural violation of this principle within F's formulation perhaps through feedback loops or missing saturation terms. Resolving requires revisiting model assumptions rather than recalculating derivatives emphasizing that implicit methods propagate underlying model flaws faithfully making domain knowledge essential for interpreting mathematical outputs in applied contexts beyond pure calculus technique.

🔗 Related Topics (MCQs)

📝 Functions of Two or More Variables MCQs 📝 Multivariable function notation and terminology MCQs 📝 Functions of two variables from tables MCQs 📝 Level curves and contour plots MCQs 📝 Graphing functions of two variables MCQs

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