📝 Local linear approximation multivariable (14 MCQs)

📖 Explanation: Linear approximations depend on the function being nearly planar locally. While first derivatives define the tangent plane, the accuracy of $L(x,y)$ is governed by the remainder term involving second derivatives. Without knowing bounds on $f_{xx}, f_{xy}, f_{yy}$, one cannot guarantee precision solely from gradient information, making option B the most rigorous assessment.

📖 Explanation: Contour density reflects gradient magnitude, not approximation error. Error in $L(x,y)$ relates to how quickly the surface bends away from the tangent plane, determined by second derivatives. Even if movement is orthogonal to steepest ascent, high curvature in that direction could cause significant deviation. Students often confuse rate of change with linearity quality.

📖 Explanation: The tangent plane depends only on first derivatives, but its predictive accuracy hinges on second-order behavior. A large mixed partial $f_{xy}$ indicates the surface twists rapidly, meaning the plane deviates quickly from the true surface even for small displacements. This invalidates assumptions of local flatness regardless of correct gradient computation.

📖 Explanation: Since $f(Q)=g(Q)$ and $\nabla f(Q)=\nabla g(Q)$, their linearizations $L_f$ and $L_g$ are mathematically identical. However, error terms involve Hessians: positive definiteness means consistent over/under estimation, while indefiniteness causes sign-varying errors. Recognizing that same linear model can have divergent reliability tests deep conceptual understanding beyond formula application.

📖 Explanation: Although directional derivatives exist in all directions at origin, $f$ is not differentiable there because no single plane approximates the cone's vertex. The nonexistence of a total derivative means $L(x,y)$ cannot be constructed via standard formulas. This distinguishes mere existence of partials from true differentiability, a subtle but critical HOTS distinction.

📖 Explanation: While $L$ correctly identifies local sensitivity ranking, real-world decisions require considering parameter constraints, costs, and domain of validity. Large $\partial P/\partial r$ might correspond to biologically impossible rates or expensive interventions. Additionally, linearity breaks down for substantial changes. Effective modeling integrates mathematical insight with practical limitations beyond pure calculus.

📖 Explanation: Taylor expansion gives $f-L \approx \frac{1}{2}(x+y)^2 e^\xi$ for some $\xi$ between 0 and $x+y$. For tiny displacements, $e^\xi \approx 1$, validating the claim. But as displacement grows, $e^\xi$ amplifies error beyond pure quadratic prediction. The statement lacks crucial qualification about neighborhood size, testing nuanced grasp of asymptotic vs global behavior.

📖 Explanation: Given $\nabla f(0,0)=(0,0)$, we have $L(x,y)=0$. The bound $|f| \leq r^2$ implies $|f-L|/r \leq r \to 0$ as $r\to 0$, satisfying differentiability definition. Thus $L$ not only exists but provides superlinear accuracy. This connects inequality constraints to differentiability without requiring smoothness assumptions.

📖 Explanation: While $L=x+y$ is valid near origin, approaching $x+y=-1$ makes $f$ and its derivatives singular. The linear model cannot capture asymptotic behavior, and differentiability itself fails near boundary. Using $L$ outside convergence radius isn't merely inaccurate—it's fundamentally unjustified. This tests recognition that formal expressions lack universal applicability.

📖 Explanation: At $(0,0)$, $\nabla f = (0,0)$ so $L \equiv 0$, losing all geometric information about the saddle. At $(1,1)$, $L = 2(x-1)-2(y-1)$ captures tilt but still misses curvature. The qualitative difference lies in whether $L$ reflects nondegenerate geometry. Critical points produce trivial linearizations that fail to distinguish surface types, unlike regular points.

📖 Explanation: Standard linear propagation assumes independence and local linearity. Correlated errors require covariance terms absent in simple sum. Simultaneously, if $V$ curves sharply over $[\pm\Delta I, \pm\Delta T]$, quadratic contributions add systematic bias. Real experiments often face both issues, making naive linear estimates optimistic. This integrates multivariable calculus with statistical reasoning.

📖 Explanation: At $(1,1)$, $f=-1$, $f_x=f_y=0$, so $L\equiv -1$. Along $y=x$, $f(t,t)=2t^3-3t^2$. Let $t=1+h$: $f=-1+3h^2+2h^3$. For small $h>0$, $f>-1$; for $h<0$, $2h^3$ dominates making $f<-1$. Wait—rechecking shows $f(-1+h)$ actually exceeds -1 for both signs initially? No: expansion gives $-1 + 3h^2 + 2h^3$. For negative $h$, $2h^3<0$ but $3h^2>0$; near zero, quadratic dominates so $f>-1$ always. Correction: actually $f \geq -1$ locally, so $L$ underestimates. Option B captures directional nuance missed by naive inspection.

📖 Explanation: Gradient descent assumes $f(\mathbf{x}-\eta\nabla f) \approx f(\mathbf{x}) - \eta\|\nabla f\|^2$, valid only if $\eta$ is small relative to inverse Lipschitz constant of $\nabla f$. Large Hessian eigenvalues mean steep curvature, requiring extremely small $\eta$ for linear approximation to hold. Even tiny steps can escape basin of attraction if curvature varies rapidly. This links optimization practice to theoretical foundations of linearization validity.

📖 Explanation: Tangency guarantees matching value and gradient at a point but doesn't preclude further intersections. For $f(x,y)=x^3$, the tangent plane at origin is $z=0$, which intersects the surface along the entire y-axis. Similarly, developable surfaces contain straight lines lying in tangent planes. Unique intersection occurs only for strictly convex/concave functions. This misconception reveals shallow understanding of geometric versus analytic definitions.