Difficulty: Intermediate | Prerequisites: Partial derivatives (MATH 124/125), familiarity with equations of planes in 3D.
Tangent planes and level curves are two core ways of understanding the geometry of multivariable functions. Tangent planes generalise the idea of a tangent line to surfaces in three dimensions, and are built directly from partial derivatives. Level curves slice a surface horizontally at constant output values, giving a two-dimensional map of a three-dimensional landscape. Both topics appear early in MATH 126 and underpin later work on optimisation, gradients, and double integrals.
The tangent plane to a surface z = f(x, y) at a point is the flat plane that best approximates the surface near that point, and its equation comes straight from the partial derivatives f_x and f_y. Level curves are the contour lines you get by setting f(x, y) = c for different constants c. Recognising which function produces which level-curve pattern is a standard exam skill.
Tangent plane
The unique plane that touches a surface z = f(x, y) at a given point and has the same slope as the surface in every direction at that point. Think of it as the best flat approximation to the surface right at the point of contact.
Partial derivative (f_x, f_y)
The rate of change of f with respect to one variable while the other is held constant. In simple terms, f_x tells you how steep the surface is in the x-direction, and f_y tells you the steepness in the y-direction.
Level curve (contour line)
The set of all points (x, y) in the domain where f(x, y) equals a fixed constant c. Think of it as a horizontal slice through the surface at height c, projected down onto the xy-plane, like contour lines on a topographic map.
Level set
The general term for the collection of inputs that produce the same output value. For functions of two variables, level sets are curves. For three variables, they are surfaces.
Finding the tangent plane to z = f(x, y) at a point (a, b, f(a,b))
Compute the partial derivatives f_x(x, y) and f_y(x, y).
Evaluate both at the point (a, b) to get the slopes.
Plug into the tangent plane formula: z - f(a,b) = f_x(a,b)(x - a) + f_y(a,b)(y - b).
Worked example from the exam
Given z = x²y - x, find the tangent plane at (2, 3, 10).
f_x = 2xy - 1. At (2, 3): f_x = 2(2)(3) - 1 = 11.
f_y = x². At (2, 3): f_y = 4.
Check: f(2, 3) = 4(3) - 2 = 10. Confirmed.
Tangent plane: z - 10 = 11(x - 2) + 4(y - 3).
Expanded: z = 11x + 4y - 24.
What level curves look like for common functions
f(x, y) = x + y (Andrea): Level curves are the family of parallel lines x + y = c, i.e. lines with slope -1. Equally spaced for equal increments of c.
f(x, y) = x - y (Dorian): Also parallel lines, but with slope +1. The lines x - y = c run in the other diagonal direction.
f(x, y) = x² + y² (Barry): Level curves are concentric circles centred at the origin. The circles get closer together as you move outward because the function grows faster.
f(x, y) = √(x² + y²) (Chantal): Also concentric circles, but equally spaced because the function grows linearly with distance from the origin (this is just the distance function).
f(x, y) = ⁴√(x² + y²) (Fernand): Concentric circles again, but now the spacing between circles increases outward because the fourth root grows more and more slowly.
f(x, y) = x² - 2xy + y² = (x - y)² (Erin): Level curves are parallel lines of the form x - y = ±√c (slope +1). The same family of parallel lines as Dorian, but with unequal spacing because of the squaring.
How to identify level curve families on an exam
Set f(x, y) = c and solve for the relationship between x and y.
Parallel straight lines indicate a linear function or a function that depends only on a linear combination of x and y.
Concentric circles indicate a function that depends only on x² + y² (radial symmetry).
Spacing tells you about the rate of growth: evenly spaced contours mean linear growth in that direction; contours bunching together mean the function is steepening.
Tangent plane equation
z - f(a, b) = f_x(a, b) · (x - a) + f_y(a, b) · (y - b)
Equivalently: z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)
This is the linearisation (first-order Taylor approximation) of f near the point (a, b).
Level curve equation
f(x, y) = c, where c is a constant. Each value of c gives a different curve in the xy-plane.
Tangent planes are the foundation of linear approximation in engineering and physics. When you linearise a system around an operating point (temperature, pressure, velocity), you are building a tangent plane. Level curves show up as contour maps in geography, pressure isobars in meteorology, and equipotential lines in electrostatics.
Students often confuse the tangent plane formula with the equation of the normal line. The tangent plane uses f_x and f_y as coefficients of (x - a) and (y - b). The normal line points in the direction of the gradient.
Forgetting to verify that the given point actually lies on the surface. Always check that f(a, b) equals the stated z-value before writing the tangent plane.
Assuming that equally spaced level curves mean the function is linear. Only certain radial functions (like √(x² + y²)) produce equally spaced circles. For x² + y², the circles bunch together.
Mixing up √(x² + y²) and x² + y² when looking at level curve spacing. The square root version has evenly spaced circles; the squared version has circles that get closer together as you move outward.
Tangent plane problems are among the most straightforward on a MATH 126 midterm, worth full marks if you execute the partial derivatives cleanly and plug in correctly.
Level curve matching (function to contour plot) is a favourite exam question because it tests geometric intuition rather than computation.
Expect to see functions that look similar but differ in their level curve spacing (e.g. x² + y² vs √(x² + y²) vs ⁴√(x² + y²)). Know the spacing signatures.
The tangent plane formula reappears later as the linear approximation and differential: dz = f_x dx + f_y dy.
True or false: The tangent plane to z = f(x, y) at (a, b) passes through the point (a, b, f(a, b)). (True.)
Fill in the blank: The level curves of f(x, y) = x + y are parallel ______ with slope ______. (Lines; -1.)
True or false: If the level curves of a function are concentric circles, the function must be x² + y². (False. It could be any function of x² + y².)
Fill in the blank: To find the tangent plane, you need the two ______ ______ evaluated at the point. (Partial derivatives.)
True or false: Level curves can cross each other for a well-defined function. (False. Each point has only one output value.)
Q: Find the equation of the tangent plane to z = x³ - xy² at the point (1, 2, -3).
A: f_x = 3x² - y² = 3 - 4 = -1 at (1, 2). f_y = -2xy = -4 at (1, 2). Tangent plane: z - (-3) = -1(x - 1) + (-4)(y - 2), i.e. z = -x - 4y + 6.
Q: Describe the level curves of f(x, y) = x² + y². What happens to the spacing as c increases?
A: The level curves are circles of radius √c centred at the origin. As c increases, the radius grows like √c, so successive circles get closer together (the gap between radius √c and √(c+1) shrinks).
Q: Two functions both have concentric circular level curves. One is f = x² + y², the other is f = √(x² + y²). How can you tell them apart from the contour plot alone?
A: Look at the spacing. For f = √(x² + y²), the circles are equally spaced because the radius equals c directly. For f = x² + y², the circles bunch together as you move outward because the radius is √c.
Q: If f(x, y) = (x - y)², what do the level curves look like?
A: Setting (x - y)² = c gives x - y = ±√c, which are parallel lines with slope 1. The spacing between lines increases as c grows because √c grows more slowly than c.
The tangent plane formula is the two-variable version of the linear approximation L(x) = f(a) + f'(a)(x - a) from single-variable calculus. It extends directly into the gradient vector, where ∇f = (f_x, f_y) is perpendicular to level curves. This connection between gradients and level curves is central to Lagrange multipliers and constrained optimisation later in the course.
Tangent plane, linearisation, linear approximation, first-order approximation, partial derivatives, level curves, contour lines, contour plot, equipotential lines, isobars, topographic map, concentric circles, radial symmetry, gradient direction, MATH 126, multivariable calculus, Calculus III, surfaces in 3D