Questions: Total Differential and Linear Approximation

Question 1

You are estimating the volume V = πr²h of a cylinder with r = 3 cm and h = 10 cm, with measurement errors dr = 0.1 cm and dh = 0.2 cm. What is the approximate error in V using the total differential?

• A. π(0.1)(0.2) ≈ 0.063 cm³ — multiply the errors together
• B. π(0.1 + 0.2) ≈ 0.94 cm³ — add the errors
• C. 2πrh · dr + πr² · dh = 2π(3)(10)(0.1) + π(9)(0.2) = 6π + 1.8π = 7.8π ≈ 24.5 cm³
• D. π(dr)²(dh) — use second-order terms since the errors are small

Answer: (C)

The total differential gives dV = (∂V/∂r)dr + (∂V/∂h)dh = 2πrh·dr + πr²·dh. At r = 3, h = 10: dV = 2π(3)(10)(0.1) + π(9)(0.2) = 6π + 1.8π = 7.8π ≈ 24.5 cm³. The partial derivatives act as sensitivity coefficients — r appears squared in V, so errors in r are amplified by 2πh = 62.8, while errors in h are amplified by πr² = 28.3. Options A and D multiply errors together, which gives second-order terms — negligible compared to the first-order differential.

Question 2

The total differential df = (∂f/∂x)dx + (∂f/∂y)dy represents:

• A. The exact change in f when x changes by dx and y changes by dy
• B. The area enclosed by the tangent plane to the surface z = f(x, y)
• C. The linear (first-order) approximation to the change in f — exact on the tangent plane and increasingly accurate as dx and dy approach zero
• D. The second-derivative correction that improves upon the tangent line approximation

Answer: (C)

The total differential is a linear approximation, not an exact computation. The actual change Δf = f(a+dx, b+dy) − f(a,b) differs from df by second-order terms (proportional to dx², dxdy, dy²). The approximation df ≈ Δf becomes more accurate as the displacement (dx, dy) shrinks. The key insight is that df computes the change predicted by the tangent plane — the best flat (linear) approximation to the surface. Option A is the most tempting wrong answer: confusing the linear approximation with the exact change.

Question 3

The total differential df = (∂f/∂x)dx + (∂f/∂y)dy can be written as the dot product ∇f · (dx, dy), revealing it as a special case of the directional derivative.

• T. True
• F. False

Answer: True

The gradient ∇f = (∂f/∂x, ∂f/∂y) and the displacement vector (dx, dy) have dot product (∂f/∂x)dx + (∂f/∂y)dy = df. This connection is geometrically meaningful: it says the total differential measures how much f changes when you step in the direction (dx, dy), with the gradient acting as the linear functional that performs this measurement. The tangent plane, the gradient, and the total differential are all the same first-order approximation viewed from different angles.

Question 4

If both partial derivatives ∂f/∂x and ∂f/∂y equal zero at a point, the total differential predicts no change, which means the function is locally constant near that point.

• T. True
• F. False

Answer: False

A zero total differential means the tangent plane is horizontal — the linear approximation predicts no change. But the function itself can still vary via second-order (quadratic) terms. The point could be a local maximum, minimum, or a saddle point — all have zero gradient but non-constant behavior in a neighborhood. The total differential captures only the linear part of the change; it says nothing about what the function does beyond first order. Equating a horizontal tangent plane with local constancy confuses the approximation with the function.

Question 5

In what sense is the total differential the 'best linear approximation' to Δf, and how does this connect geometrically to the tangent plane?

Model answer: The total differential df = (∂f/∂x)dx + (∂f/∂y)dy is 'best' in the sense that it is the unique linear function of (dx, dy) that agrees with Δf up to first order: the error Δf − df shrinks faster than |(dx, dy)| as the displacement approaches zero. Geometrically, df computes the vertical displacement along the tangent plane to z = f(x, y) at the base point. The tangent plane z = f(a,b) + (∂f/∂x)(x−a) + (∂f/∂y)(y−b) is the best flat approximation to the surface near (a, b), and df is the amount of height that plane gains for a step of size (dx, dy). The tangent plane, the gradient vector, and the total differential are three descriptions of the same linear object.

This unification is the key insight of multivariable differentiation: the derivative is not a slope but a linear map — the total differential — and the tangent plane is its graph.