Difficulty: Intermediate | Prerequisites: §13.2 (parametric curves), single-variable differentiation, familiarity with functions of two variables.
This section marks the transition from curves to surfaces. You are learning what it means to differentiate a function of two (or more) variables, how to read and estimate derivatives from tables and contour maps, and how to build the tangent plane, the best linear approximation to a surface at a point. If §13.2 on parametric curves is still shaky, revisit it first, because the tangent plane derivation uses velocity vectors of parametric traces on a surface.
Partial derivatives measure how a multivariable function changes when you vary one input while holding the others fixed. The tangent plane at a point on a surface z = f(x, y) is the plane that best approximates the surface near that point, built from the two partial derivatives fₓ and f_y. Linear approximation uses that tangent plane to estimate function values near a known point.
Partial derivative (∂f/∂x, fₓ)
The rate of change of f(x, y) with respect to x, holding y constant. Computed by differentiating f with respect to x while treating y as a constant. Think of it as slicing the surface with a plane of fixed y and measuring the slope of the resulting curve.
Partial derivative (∂f/∂y, f_y)
Same idea, but varying y while x is held fixed.
Tangent plane
The plane that touches the surface z = f(x, y) at a point (a, b, f(a, b)) and best approximates it nearby. Its equation is z − f(a, b) = fₓ(a, b)(x − a) + f_y(a, b)(y − b). In simple terms, it is the flat surface that "kisses" the curved surface at that point.
Linear approximation (linearisation)
Using the tangent plane to estimate f(x, y) near (a, b): f(x, y) ≈ f(a, b) + fₓ(a, b)(x − a) + f_y(a, b)(y − b). Think of it as the multivariable version of the tangent line approximation from single-variable calculus.
Contour map (level curves)
A picture showing curves where f(x, y) = c for various constants c. Each curve is a "level curve." Closely spaced contours mean the function is changing rapidly (steep surface); widely spaced contours mean it is changing slowly (gentle slope).
Trace (cross-section)
The curve you get by slicing a surface with a plane where one variable is held constant. For z = f(x, y), setting x = a gives the trace in the plane x = a, parametrised as r(t) = ⟨a, t, f(a, t)⟩.
When you have a table of values for h = f(v, t), you estimate partial derivatives using finite differences.
To estimate ∂f/∂v at (40, 15): hold t = 15 fixed and look at how f changes as v changes around 40. Using the values at v = 30 and v = 50: fᵥ(40, 15) ≈ (f(50, 15) − f(30, 15)) / (50 − 30) = (36 − 16) / 20 = 1.0.
To estimate ∂f/∂t at (40, 15): hold v = 40 fixed and look at how f changes as t changes around 15. Using t = 10 and t = 20: fₜ(40, 15) ≈ (f(40, 20) − f(40, 10)) / (20 − 10) = (28 − 21) / 10 = 0.7.
Physical interpretation: fᵥ(40, 15) ≈ 1.0 means that when wind has been blowing at 40 knots for 15 hours, increasing the wind speed by 1 knot raises the wave height by roughly 1 foot. fₜ(40, 15) ≈ 0.7 means each additional hour of blowing adds about 0.7 feet to wave height.
Both ∂h/∂v and ∂h/∂t should be positive: stronger wind and longer duration both produce higher waves. The table confirms this, as values increase along both rows (increasing v) and columns (increasing t).
This technique connects §13.2 (curves) to §14.3 (tangent planes). For a surface z = f(x, y):
Fix x = a and parametrise the trace: r₁(t) = ⟨a, t, f(a, t)⟩. The velocity r₁′(t) = ⟨0, 1, f_y(a, t)⟩ is tangent to the surface.
Fix y = b and parametrise the trace: r₂(s) = ⟨s, b, f(s, b)⟩. The velocity r₂′(s) = ⟨1, 0, fₓ(s, b)⟩ is tangent to the surface.
At the point (a, b, f(a, b)), both r₁′ and r₂′ lie in the tangent plane. The normal to the tangent plane is their cross product: n = r₁′ × r₂′.
Worked example for z = 2x² + y² at the point (1, 3, 11):
r₁(t) = ⟨1, t, 2 + t²⟩ (trace at x = 1). At t = 3: r₁′(3) = ⟨0, 1, 6⟩.
r₂(s) = ⟨s, 3, 2s² + 9⟩ (trace at y = 3). At s = 1: r₂′(1) = ⟨1, 0, 4⟩.
Normal: ⟨0, 1, 6⟩ × ⟨1, 0, 4⟩ = ⟨(1)(4) − (6)(0), (6)(1) − (0)(4), (0)(0) − (1)(1)⟩ = ⟨4, 6, −1⟩.
Tangent plane: 4(x − 1) + 6(y − 3) − 1(z − 11) = 0, i.e. 4x + 6y − z = 11.
The tangent plane at (a, b) gives the linear approximation:
f(x, y) ≈ f(a, b) + fₓ(a, b)(x − a) + f_y(a, b)(y − b)
Worked example: estimate (0.99)³ + (2.01)³ − 6(0.99)(2.01).
Let f(x, y) = x³ + y³ − 6xy. The nearby "nice" point is (a, b) = (1, 2).
f(1, 2) = 1 + 8 − 12 = −3.
fₓ = 3x² − 6y, so fₓ(1, 2) = 3 − 12 = −9.
f_y = 3y² − 6x, so f_y(1, 2) = 12 − 6 = 6.
f(0.99, 2.01) ≈ −3 + (−9)(−0.01) + (6)(0.01) = −3 + 0.09 + 0.06 = −2.85.
To draw a contour map of g(x, y) = √(9 − 9x² − y²):
Set g = c, so c² = 9 − 9x² − y², i.e. 9x² + y² = 9 − c². This is an ellipse (when 0 < c < 3), centred at the origin, with semi-axes a = √((9 − c²)/9) along x and b = √(9 − c²) along y.
As c increases from 0 to 3, the ellipses shrink to a point. The surface is the top half of an ellipsoid.
For m(x, y) = x/(x² + 3y²):
Set m = c, so x = c(x² + 3y²). Rearrange to cx² − x + 3cy² = 0, which (for c ≠ 0) is a circle or ellipse in transformed coordinates. The level curves are closed curves around a peak and a trough on the x-axis.
Partial derivatives: fₓ(a, b) = lim(h→0) [f(a + h, b) − f(a, b)] / h, and similarly for f_y.
Tangent plane to z = f(x, y) at (a, b): z − f(a, b) = fₓ(a, b)(x − a) + f_y(a, b)(y − b)
Normal vector to the tangent plane: n = ⟨fₓ(a, b), f_y(a, b), −1⟩ (read directly from the tangent plane equation rearranged as fₓ(x − a) + f_y(y − b) − (z − f(a, b)) = 0).
Linear approximation: f(x, y) ≈ f(a, b) + fₓ(a, b)(x − a) + f_y(a, b)(y − b)
Finite difference approximation: fₓ(a, b) ≈ [f(a + Δx, b) − f(a − Δx, b)] / (2Δx) (central difference, preferred when data is available on both sides).
Normal from parametric traces: If r₁′ and r₂′ are velocity vectors of two traces through a point on a surface, then n = r₁′ × r₂′ is normal to the surface at that point.
The wave height table in this homework is a real-world dataset. Oceanographers and sailors use exactly this kind of model to predict sea conditions from wind speed and duration. The partial derivatives tell you the sensitivity of wave height to each factor, which matters for safety decisions at sea.
Linear approximation is the basis of error analysis in engineering and the sciences. When you measure inputs with small uncertainties, the tangent plane tells you approximately how those uncertainties propagate into the output.
Students often think ∂f/∂x means "the derivative of f." It is the derivative with respect to x only, holding all other variables constant. Forgetting to hold y constant leads to wrong answers.
When estimating partial derivatives from a table, students sometimes use a one-sided difference (only values to one side of the target point) when a central difference is available. Central differences are more accurate.
A common error on tangent plane problems: students compute fₓ and f_y correctly but forget to evaluate them at the specific point (a, b). The partial derivatives are functions; you need their values at the point of tangency.
Students sometimes confuse the normal vector to the tangent plane with the tangent vector to a curve. They are perpendicular to each other. The tangent plane has a normal; curves on the surface have tangent vectors that lie in the tangent plane.
⚠️ Estimating partial derivatives from a table is a very common exam question. Know how to use central differences and interpret the result in physical terms.
⚠️ The tangent plane formula z − f(a, b) = fₓ(a, b)(x − a) + f_y(a, b)(y − b) must be memorised cold. Be ready to apply it to explicit surfaces (polynomials, exponentials) and to read off the normal vector.
⚠️ Linear approximation problems typically give a function with a "nice" nearby point. Identify the point, compute f and both partial derivatives there, then plug in the small deviations.
⚠️ The parametric-traces method for finding tangent planes (Problem 5) is a synthesis question testing §13.2 and §14.3 together. Expect it on exams where the course emphasises connections between topics.
⚠️ Contour map sketching: know how to identify the shape of level curves (circles, ellipses, hyperbolas, lines) from the function's algebraic form.
True or false: ∂f/∂x at (a, b) measures the slope of the surface in the y-direction. (False. It measures the slope in the x-direction.)
Fill in the blank: The tangent plane to z = f(x, y) at (a, b) has normal vector n = ____. (⟨fₓ(a, b), f_y(a, b), −1⟩)
True or false: If fₓ(a, b) = 0 and f_y(a, b) = 0, the tangent plane is horizontal. (True.)
Fill in the blank: A central difference estimate of fₓ(a, b) uses the formula ____. (fₓ ≈ [f(a + h, b) − f(a − h, b)] / (2h))
True or false: Level curves of f(x, y) = x² + y² are hyperbolas. (False. They are circles centred at the origin.)
Q: Find the equation of the tangent plane to z = x²y + xy³ at the point (2, 1, 6).
A: fₓ = 2xy + y³, so fₓ(2, 1) = 4 + 1 = 5. f_y = x² + 3xy², so f_y(2, 1) = 4 + 6 = 10. Tangent plane: z − 6 = 5(x − 2) + 10(y − 1), i.e. z = 5x + 10y − 14.
Q: Use linear approximation to estimate f(3.02, 0.97) where f(x, y) = x² − 3xy. Use the base point (3, 1).
A: f(3, 1) = 9 − 9 = 0. fₓ = 2x − 3y, so fₓ(3, 1) = 3. f_y = −3x, so f_y(3, 1) = −9. f(3.02, 0.97) ≈ 0 + 3(0.02) + (−9)(−0.03) = 0.06 + 0.27 = 0.33.
Q: From the wave height table, estimate ∂f/∂v at (30, 20). Interpret your answer.
A: fᵥ(30, 20) ≈ (f(40, 20) − f(20, 20)) / (40 − 20) = (28 − 8) / 20 = 1.0. At wind speed 30 knots after 20 hours, increasing wind speed by 1 knot raises wave height by about 1 foot.
Q: The surface z = x² − y² has a tangent plane at some point that is parallel to the plane 4x + 6y − z = 11. Find that point.
A: The normal to z = x² − y² at (a, b) is n = ⟨2a, −2b, −1⟩. For it to be parallel to the plane 4x + 6y − z = 11 (normal ⟨4, 6, −1⟩), we need 2a = 4 and −2b = 6, giving a = 2, b = −3. The point is (2, −3, 4 − 9) = (2, −3, −5).
Partial derivatives are the foundation for everything that follows in MATH 215: the gradient vector (§14.6), the chain rule for multivariable functions (§14.5), directional derivatives, and optimisation (§14.7). If partial derivatives are not solid, those topics will be very difficult.
The tangent plane reappears when you study differentiability (§14.4) and again in surface integrals (§16.6), where the normal vector to the tangent plane becomes the integrand's direction.
Linear approximation generalises to the total differential df = fₓ dx + f_y dy, which is the starting point for the multivariable chain rule.
Partial derivatives, partial differentiation, tangent plane, linearisation, linear approximation, contour map, level curves, level surfaces, traces, cross-sections, normal vector, finite differences, central difference, wave height table, multivariable calculus, MATH 215, §14.1, §14.3, Stewart Multivariable Calculus, gradient, differentiability