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📝 Differentiability Differentials and Local Linearity (14 MCQs)

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

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What is Differentiability Differentials and Local Linearity?

Definition:
Multivariable differentiability means $f$ can be approximated by linear map $L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$ with error vanishing faster than distance.

Example:
Near $(1,2)$ , $f(x,y)=x^2+y$ is approximated by $L(x,y) = 3 + 2(x-1) + 1(y-2)$ .

Reason:
Differentiability justifies linear approximation, enabling error analysis, numerical methods, and defining tangent planes rigorously beyond mere partial existence.

Easy

Medium

Hard

📝 All Differentiability Differentials and Local Linearity MCQs

Q1. A function $f(x,y)$ has partial derivatives $f_x(0,0) = 2$ and $f_y(0,0) = -1$ , but is not differentiable at the origin. Which scenario best explains this failure of local linearity despite existing partials?

A.The function is discontinuous along every path approaching the origin except the coordinate axes.

B.The directional derivative exists in all directions but does not vary linearly with the direction vector.

C.The tangent plane approximation error decays slower than the distance from the origin along some non-axial path. ✅

D.The mixed partial derivatives are unequal, violating Clairaut’s theorem at the critical point.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Differentiability requires that the linear approximation error vanishes faster than the norm of the displacement vector. Existing partial derivatives only guarantee behavior along axes; if the remainder term in the Taylor expansion does not satisfy $o(\|\mathbf{h}\|)$ along some curve, the function lacks a true tangent plane, making local linearity fail despite defined partials.

Q2. An engineer uses the differential $df = f_x dx + f_y dy$ to estimate measurement error in $f(x,y)=x^2y$ at $(3,2)$ . If actual errors are $\Delta x=0.1, \Delta y=-0.05$ , but the linear estimate significantly overpredicts the true change, what is the most likely mathematical reason?

A.The partial derivatives were computed incorrectly due to product rule misapplication.

B.The second-order terms in the Taylor expansion have the same sign as the first-order terms, reinforcing rather than canceling error. ✅

C.The function is not continuous at the evaluation point, invalidating differential use.

D.The differential formula requires absolute values of increments, which were omitted.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The total differential provides only a first-order approximation. When second-order partial derivatives are large and aligned in sign with the first-order contribution, the neglected quadratic terms accumulate constructively. This causes systematic overestimation, highlighting that differentials assume local linearity holds sufficiently well, which breaks down when curvature effects dominate near the operating point.

Q3. Given a contour plot of $f(x,y)$ where level curves near point $P$ appear as parallel straight lines, but become hyperbolic just outside this region, what can be definitively concluded about differentiability at $P$ ?

A.The function is differentiable at

P

because contours are locally linear.

B.The function may still be non-differentiable at

P

if the gradient magnitude changes discontinuously across infinitesimal neighborhoods. ✅

C.Differentiability cannot be assessed from contour plots alone without explicit formulas.

D.The hyperbolic outer pattern proves non-differentiability at

P

due to global inconsistency.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Parallel straight contours suggest constant gradient direction and magnitude locally, supporting differentiability. However, visual regularity on discrete plots can mask pathological behavior like oscillatory gradients or undefined limits in derivative definitions. True differentiability demands analytical verification of the limit definition; graphical evidence is suggestive but insufficient for rigorous confirmation of local linearity.

Q4. In constrained optimization using Lagrange multipliers, why is the condition $d(f - \lambda g) = 0$ equivalent to finding stationary points, and how does this relate to local linearity?

A.It enforces that the differential of the objective equals zero independently of constraints.

B.It ensures the combined function’s linear approximation vanishes in all feasible tangent directions, reflecting alignment of gradients. ✅

C.It guarantees global minima by eliminating second-order effects through linearization.

D.It replaces nonlinear constraints with linear ones via differential substitution.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The equation $d(f - \lambda g) = 0$ means the total differential of the Lagrangian vanishes along admissible variations. This occurs precisely when $\nabla f$ and $\nabla g$ are parallel, so their linear approximations align on the constraint surface. Thus, stationarity reflects cancellation of first-order changes in feasible directions, embodying local linearity within the constrained manifold.

Q5. A student computes $\Delta z \approx dz$ for $z = \sqrt{x^2 + y^2}$ at $(0,0)$ using $dx=0.01, dy=0.02$ , obtaining $dz=0$ . The actual $\Delta z = 0.0224$ . What fundamental misconception caused this error?

A.Assuming the square root function is linear everywhere.

B.Forgetting that partial derivatives do not exist at the origin for this radial function. ✅

C.Using Euclidean norm instead of Manhattan distance in differential calculation.

D.Misapplying chain rule when computing partial derivatives.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The function $\sqrt{x^2+y^2}$ has undefined partial derivatives at the origin because the limit defining $f_x$ or $f_y$ depends on the approach path. Since differentials require existing partials, applying $dz = f_x dx + f_y dy$ at a non-differentiable point is invalid. The nonzero actual change confirms absence of a tangent plane, exposing misuse of linear approximation where it mathematically cannot apply.

Q6. In a thermodynamic model, pressure $P(V,T)$ satisfies $dP = -\frac{nRT}{V^2}dV + \frac{nR}{V}dT$ . If volume uncertainty dominates temperature uncertainty by factor 10, but $|\partial P/\partial V|$ is 100 times larger than $|\partial P/\partial T|$ , which variable contributes more to output variance and why?

A.Temperature, because its relative uncertainty is smaller but sensitivity compensates.

B.Volume, because sensitivity amplifies its larger input uncertainty multiplicatively. ✅

C.Both contribute equally due to inverse proportionality in partial magnitudes.

D.Neither, since differentials assume independence violated in real gases.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Output variance in linear error propagation scales as $(\partial P/\partial x)^2 \sigma_x^2$ . Even though temperature uncertainty is smaller, volume’s partial derivative is two orders of magnitude larger while its uncertainty is only one order larger. Squaring the sensitivity makes volume’s contribution dominate by factor $100^2 / 10^2 = 100$ , demonstrating how differential-based sensitivity analysis prioritizes variables with high leverage in multivariable systems.

Q7. Why does the existence of all directional derivatives at a point not guarantee differentiability, whereas existence of partial derivatives plus continuity of those partials does?

A.Directional derivatives ignore coordinate system alignment needed for linear maps.

B.Continuity of partials ensures the Jacobian matrix varies smoothly, making the linear approximation uniformly valid in all directions. ✅

C.Directional derivatives are scalar quantities lacking vector structure required for differentials.

D.Partial derivatives inherently encode more information than directional derivatives.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Differentiability requires a single linear map approximating the function in all directions simultaneously. Directional derivatives may exist individually yet fail to assemble into a consistent linear transformation if they don’t vary linearly with direction. Continuous partial derivatives imply the Jacobian is well-behaved nearby, ensuring the remainder term in the differentiability definition vanishes uniformly, thus bridging directional data into coherent local linearity.

Q8. On a 3D surface plot, two candidate planes intersect the surface at point $Q$ . Plane A matches surface height and slope along x and y axes but diverges rapidly along diagonal paths. Plane B shows slight height mismatch but maintains close proximity in all visible directions. Which plane represents the true tangent plane if the function is differentiable?

A.Plane A, because it exactly matches partial derivatives defining the tangent plane.

B.Plane B, because visual closeness in multiple directions outweighs exact axial matching.

C.Neither, since differentiable functions must have unique tangent planes matching both criteria perfectly. ✅

D.Plane A is correct mathematically, but Plane B suggests numerical instability in plotting.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: If a function is differentiable at $Q$ , the tangent plane is uniquely defined by partial derivatives and must approximate the surface with error $o(\|\mathbf{h}\|)$ in every direction. Plane A failing diagonally violates this condition, implying either non-differentiability or incorrect partial computation. Plane B’s height mismatch contradicts tangency. Thus, neither qualifies unless both axial and omnidirectional agreement hold, confirming uniqueness of the true tangent plane under differentiability.

Q9. Construct a function $f(x,y)$ that is differentiable at $(0,0)$ with $df=0$ , yet whose partial derivatives are discontinuous there. What property must $f$ satisfy to reconcile these seemingly contradictory features?

A.Partial derivatives must oscillate with amplitude decaying faster than linearly toward origin. ✅

B.The function must be identically zero in a neighborhood of the origin.

C.Mixed partials must exist and be equal despite individual partial discontinuity.

D.The gradient must vanish along all rational-slope lines through origin.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Differentiability at a point only requires the linear approximation error to be $o(\|\mathbf{h}\|)$ ; it does not demand continuous partials. Functions like $f(x,y)=(x^2+y^2)\sin(1/(x^2+y^2))$ for $(x,y)\neq(0,0)$ and 0 at origin have vanishing differential at origin but oscillating partials. The key is that partial discontinuities occur with amplitude diminishing sufficiently fast so that the difference quotient still converges to zero, preserving differentiability despite lack of $C^1$ smoothness.

Q10. In the expression $dw = \frac{\partial w}{\partial x}dx + \frac{\partial w}{\partial y}dy + \frac{\partial w}{\partial z}dz$ , what do the symbols $dx, dy, dz$ fundamentally represent in modern analysis?

A.Infinitesimal changes in independent variables as conceived by Leibniz.

B.Basis covectors dual to the standard coordinate basis vectors in the cotangent space. ✅

C.Small finite differences used in numerical differentiation schemes.

D.Arbitrary constants scaling the partial derivative contributions.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Modern differential geometry interprets $dx, dy, dz$ not as infinitesimals but as linear functionals forming the dual basis to $\partial/\partial x, \partial/\partial y, \partial/\partial z$ . The total differential $dw$ is then a covector field acting on tangent vectors to produce directional derivatives. This abstraction resolves historical ambiguities about infinitesimals while preserving computational utility in multivariable calculus and physics applications.

Q11. For $f(x,y) = e^{xy}$ , the linear approximation at $(0,0)$ is $L(x,y)=1$ . Within what region is $|f-L| < 0.01$ guaranteed, and why does this region shrink asymmetrically?

A.A circular disk of radius 0.1, because exponential growth is isotropic near origin.

B.A rectangular region where

|xy| < 0.01

, since error depends on product not individual coordinates. ✅

C.An elliptical region aligned with eigenvectors of Hessian, reflecting anisotropic curvature.

D.No bounded region exists because exponential exceeds any linear bound eventually.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The error $|e^{xy}-1|$ is dominated by $|xy|$ near origin via Taylor expansion. Thus, accuracy depends on the product $xy$ , not Euclidean distance. The valid region is hyperbolic-shaped $|xy|<\epsilon$ , elongated along axes where one variable is small. This asymmetry arises because cross-term curvature governs deviation, illustrating that linear approximation domains reflect functional structure, not just metric proximity.

Q12. When transforming $df = f_x dx + f_y dy$ to polar coordinates via $x=r\cos\theta, y=r\sin\theta$ , why must we substitute $dx = \cos\theta dr - r\sin\theta d\theta$ rather than treating $dr, d\theta$ as independent increments?

A.Because

r

and

\theta

are curvilinear coordinates with non-constant basis vectors. ✅

B.Because the Jacobian determinant accounts for area distortion in integration.

C.Because partial derivatives transform contravariantly under coordinate changes.

D.Because

d\theta

has units of radians while

dr

has length units.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Differentials transform via the chain rule as covectors, requiring substitution of $dx, dy$ in terms of $dr, d\theta$ using partial derivatives of the coordinate map. Treating $dr, d\theta$ as naive increments ignores that the basis vectors $\partial/\partial r, \partial/\partial \theta$ vary spatially in Cartesian embedding. Correct substitution preserves the invariant meaning of $df$ as a linear functional, ensuring consistency across coordinate systems in physical modeling.

Q13. A researcher models population growth as $P(t,x)$ and assumes $\Delta P \approx P_t \Delta t + P_x \Delta x$ holds globally because partials are constant. Field data shows systematic prediction bias increasing with $\Delta x$ . What flaw in reasoning explains this?

A.Constant partials imply global linearity, contradicting observed nonlinear dynamics.

B.Population models inherently violate conservation laws assumed in differential formulation.

C.The assumption confuses local linearity with global affinity; constant partials only ensure linearity if domain is convex and function is affine. ✅

D.Measurement noise accumulates quadratically, overwhelming linear signal.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Even with constant partial derivatives, a function is affine (linear plus constant) only if defined on a convex domain and satisfying integrability conditions. In ecological models, state spaces often have boundaries or topological constraints preventing global extension of local linearity. Observed bias indicates the model extrapolates beyond the neighborhood where the differential approximation is valid, mistaking local tangent behavior for universal law.

Q14. Given $F(x,y,z)=0$ defines $z$ implicitly as $z=g(x,y)$ , and $dF = F_x dx + F_y dy + F_z dz = 0$ , why is solving $dz = -(F_x/F_z)dx - (F_y/F_z)dy$ valid only when $F_z \neq 0$ , and what geometric interpretation accompanies this condition?

A.Division by zero corresponds to vertical tangent plane where implicit function theorem fails.

B.Nonzero

F_z

ensures level set is transverse to z-axis, enabling projection onto xy-plane.

C.Both statements are correct and equivalent descriptions of regularity condition. ✅

D.The condition prevents division by zero algebraically but has no geometric significance.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Algebraically, $F_z \neq 0$ avoids undefined coefficients in the solved differential. Geometrically, it means the gradient $\nabla F$ has nonzero z-component, so the level surface isn’t vertical relative to xy-plane, allowing unique local representation as graph $z=g(x,y)$ . This transversality is precisely the implicit function theorem’s hypothesis, linking analytic solvability of differentials to geometric regularity of level sets in multivariable calculus.

🔗 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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