Questions: Total Differential and Linear Approximation

Question 1

You want to estimate how f(x, y) = xy² changes when x goes from 3 to 3.02 and y goes from 2 to 1.99. You know ∂f/∂x = y² = 4 and ∂f/∂y = 2xy = 12 at (3, 2). Which expression gives the correct total differential approximation?

• A. Δf ≈ (∂f/∂x)(∂f/∂y) = 4 · 12 = 48
• B. Δf ≈ (∂f/∂x)Δx + (∂f/∂y)Δy = 4(0.02) + 12(−0.01) = −0.04
• C. Δf ≈ (∂f/∂x)Δx = 4(0.02) = 0.08, since only x changes enough to matter
• D. Δf ≈ (∂f/∂y)Δy = 12(−0.01) = −0.12, since y's percentage change is larger

Answer: (B)

The total differential sums all variable contributions: df = (∂f/∂x)dx + (∂f/∂y)dy. With Δx = 0.02 and Δy = −0.01: Δf ≈ 4(0.02) + 12(−0.01) = 0.08 − 0.12 = −0.04. You cannot drop one term based on which change is 'larger' — both partial derivatives contribute independently. Options A, C, and D all represent common errors: multiplying partials, or using only one term.

Question 2

The total differential df of a function f(x, y) at a point is best described as:

• A. The exact change in f when x and y each change by specific amounts
• B. The rate at which f increases in the direction of steepest ascent
• C. The best linear approximation to the change in f for simultaneous small changes in x and y
• D. The product of the two partial derivatives ∂f/∂x and ∂f/∂y

Answer: (C)

The total differential is an approximation, not an exact change — the true change Δf and the differential df differ by higher-order terms that shrink faster than the displacement. 'Best linear approximation' is precise: df is exact to first order, meaning the error |Δf − df|/|(Δx, Δy)| → 0 as the displacement goes to zero. Option B describes the gradient direction; option D is not a standard concept.

Question 3

In the total differential df = (∂f/∂x)dx + (∂f/∂y)dy, the quantities dx and dy are independent variables that can represent arbitrary (small) changes in x and y.

• T. True
• F. False

Answer: True

This is subtle but important: dx and dy in the total differential are not fixed increments — they are free variables representing an arbitrary direction of displacement. This is what makes df a linear map on displacements (a 1-form), not a specific number. Only when you substitute specific values Δx and Δy do you get a numerical approximation to Δf.

Question 4

If a function f(x, y) has partial derivatives ∂f/∂x and ∂f/∂y at a point, then the total differential df = (∂f/∂x)dx + (∂f/∂y)dy is very likely to be a valid linear approximation to Δf near that point.

• T. True
• F. False

Answer: False

Existence of partial derivatives is not sufficient for differentiability. A function can have both partial derivatives at a point yet fail to be differentiable there — meaning the total differential does not accurately approximate the change in all directions. Differentiability (the stronger condition) requires that Δf equals df plus an error term that is o(|(Δx, Δy)|). This is why multivariable differentiability is defined separately from the existence of partial derivatives.

Question 5

Explain why the total differential sums the partial derivative contributions rather than, say, taking the larger one or multiplying them.

Model answer: Each partial derivative measures the rate of change due to one variable alone, holding the other fixed. When both variables change simultaneously, their first-order contributions to Δf are additive and independent: the change due to Δx is approximately (∂f/∂x)Δx, and the change due to Δy is approximately (∂f/∂y)Δy. The cross-term ΔxΔy is second-order and negligible for small displacements. Summation correctly captures all first-order contributions; taking just one term would ignore part of the change, and multiplying them has no geometric meaning.

This additivity is exactly the linearity of the best linear approximation. Near any differentiable point, f behaves locally like a plane, and the plane's height change is a linear (additive) function of the displacements in x and y. The total differential is the equation of that tangent plane, expressed in terms of the displacement variables dx and dy.