Difficulty: Intermediate | Prerequisites: Partial derivatives, basic integration (MATH 124/125)
Tangent planes are the multivariable version of tangent lines: they give you the best flat approximation to a surface near a point. Linear approximation uses that tangent plane to estimate function values at nearby points without recomputing the full function, which is especially useful when the function is complicated. Double integrals extend single-variable integration to two dimensions, letting you compute volumes, areas, and accumulated quantities over regions in the plane. Reversing the order of integration is a technical skill that makes otherwise impossible integrals tractable.
The tangent plane to z = f(x, y) at a point uses the partial derivatives to build a linear approximation. Double integrals are set up by choosing an order of integration and writing the correct limits. Reversing that order means re-describing the region of integration so the inner and outer variables swap.
Tangent plane
The plane that just 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 laying a perfectly flat sheet of paper on a curved surface so that it makes contact at exactly one point.
Linear approximation (linearisation)
Using the tangent plane to estimate f(x, y) at points near (a, b). The tangent plane value is easy to compute (it is just a linear function), and it is close to the true value when (x, y) is close to (a, b). In simple terms, you replace a complicated function with its best linear substitute near a known point.
Double integral
An integral of the form ∬_D f(x, y) dA, which sums the values of f over a region D in the xy-plane. When f ≥ 0, this gives the volume under the surface z = f(x, y) and above D.
Iterated integral
A double integral written as two nested single integrals, e.g. ∫∫ f(x, y) dy dx, where you integrate with respect to one variable first (holding the other fixed), then integrate the result with respect to the second variable.
Reversing the order of integration
Rewriting an iterated integral so that the variable you integrate first and the variable you integrate second are swapped. This requires re-describing the region D with the limits expressed the other way round.
Given z = f(x, y) and a point (a, b, f(a, b)):
Compute f_x(a, b) and f_y(a, b).
The tangent plane equation is: z = f(a, b) + f_x(a, b)(x − a) + f_y(a, b)(y − b).
Partial derivatives:
f_x = y² − 1/(2√x). At (4, 2): f_x = 4 − 1/4 = 15/4.
f_y = 2xy − 3 cos(y − 2). At (4, 2): f_y = 16 − 3 cos(0) = 16 − 3 = 13.
Tangent plane:
z = 14 + (15/4)(x − 4) + 13(y − 2)
The exam then asked: find an approximate y satisfying 14.22 = 3.92y² − √3.92 − 3 sin(y − 2).
The key observation is that this equation is f(3.92, y) = 14.22, which is the original function evaluated at x = 3.92 (close to 4) with z = 14.22 (close to 14). So you use the tangent plane as an approximation:
14.22 ≈ 14 + (15/4)(3.92 − 4) + 13(y − 2)
14.22 ≈ 14 + (15/4)(−0.08) + 13(y − 2)
14.22 ≈ 14 − 0.3 + 13(y − 2)
14.22 ≈ 13.7 + 13(y − 2)
0.52 = 13(y − 2)
y − 2 = 0.04, so y ≈ 2.04.
To find the volume between two surfaces z = f(x, y) (top) and z = g(x, y) (bottom) over a region R in the xy-plane:
∬_R [f(x, y) − g(x, y)] dA
For a rectangle [a, b] × [c, d], this becomes:
∫_a^b ∫_c^d [f(x, y) − g(x, y)] dy dx (or dx dy).
∫_3^5 ∫_2^4 [(eˣ + y³) − (1 + sin y)] dy dx
The order does not matter here because the region is a rectangle.
This is a four-step process:
Step 1: Identify the region D from the original limits. The outer integral gives the range of one variable. The inner limits (which may depend on the outer variable) describe the bounds for the other.
Step 2: Sketch the region D in the xy-plane. Mark the boundary curves.
Step 3: Re-describe the region with the roles of x and y swapped. Determine the new outer limits (now constants for the other variable) and the new inner limits (now functions of that variable).
Step 4: Write the new iterated integral with the reversed order.
From the original limits: x runs from 1 to 64. For each x, y runs from (x − 1)/21 to log₄ x.
The region D is bounded by the curves y = (x − 1)/21 and y = log₄ x.
Solving for x in terms of y:
From y = log₄ x: x = 4^y.
From y = (x − 1)/21: x = 21y + 1.
The two curves intersect where 4^y = 21y + 1. At y = 0: 4⁰ = 1 and 21(0) + 1 = 1, so they meet. At y = 3: 4³ = 64 and 21(3) + 1 = 64, so they meet again.
So y runs from 0 to 3. For each y, x runs from 21y + 1 (the left/lower curve) to 4^y (the right/upper curve).
The reversed integral is:
∫0^3 ∫{21y+1}^{4^y} x² sin(y³) dx dy
z = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b)(Tangent plane at (a, b, f(a, b)))
\iint_R \bigl[f(x,y) - g(x,y)\bigr]\,dA(Volume between two surfaces over region R)
Linear approximation is how GPS devices estimate your position between satellite readings: a tangent-plane-style correction based on the most recent known position and velocity. Double integrals are used to compute total mass from a density function, total charge on a surface, and accumulated rainfall over a geographic region.
Students often confuse the tangent plane equation with the equation of the surface. The tangent plane is linear in x and y; if your answer has x², sin, √ or similar terms, something has gone wrong.
When using the tangent plane for approximation, students sometimes plug in the exact point (a, b) instead of the nearby point they are trying to estimate. The whole purpose is to evaluate at the nearby point.
When reversing the order of integration, students sometimes swap the limits without re-solving for x in terms of y (or vice versa). The curves that bound the region need to be expressed in the new variable.
Students sometimes assume the integrand changes when reversing the order. The integrand stays the same; only the limits and the order of dx, dy change.
⚠️ Tangent plane questions typically come in two parts: find the plane, then use it for a linear approximation. Both parts are tested together and the second part builds on the first.
⚠️ For double integrals, the exam asks you to set them up, not evaluate them. Read the instructions carefully. Setting up with incorrect limits costs full marks.
⚠️ Reversing the order of integration requires a correct sketch of the region. Practise sketching regions bounded by log, exponential, and polynomial curves.
⚠️ Finding intersection points of the boundary curves is essential for reversing the order. If you cannot find where they meet, you cannot write the new limits.
True or False: The tangent plane equation always contains √ or sin terms from the original function. (False, it is always linear in x and y.)
Fill in the blank: Linear approximation works best when the point (x, y) is ____ to (a, b). (Close.)
True or False: When you reverse the order of integration, the integrand changes. (False, only the limits and order of dx/dy change.)
Fill in the blank: To set up ∬_R [f − g] dA for volume, f is the ____ surface and g is the ____ surface. (Top; bottom.)
True or False: You need to find where the boundary curves intersect when reversing the order of integration. (True.)
Q: Find the tangent plane to z = x² + y² at the point (1, 2, 5).
A: f_x = 2x, so f_x(1, 2) = 2. f_y = 2y, so f_y(1, 2) = 4. Tangent plane: z = 5 + 2(x − 1) + 4(y − 2).
Q: Use the tangent plane from the previous question to approximate f(1.1, 1.9).
A: z ≈ 5 + 2(0.1) + 4(−0.1) = 5 + 0.2 − 0.4 = 4.8. (The exact value is 1.21 + 3.61 = 4.82, so the approximation is close.)
Q: Set up an iterated integral for the volume below z = x + y and above z = 0 over the rectangle [0, 1] × [0, 2].
A: ∫_0^1 ∫_0^2 (x + y) dy dx.
Q: In the reversed-order problem, why did y run from 0 to 3?
A: Because the two boundary curves y = (x − 1)/21 and y = log₄ x intersect at y = 0 (where both give x = 1) and y = 3 (where both give x = 64). These intersection points set the outer limits for y.
Tangent planes connect to gradient vectors (the normal to the tangent plane of g(x, y, z) = 0 is ∇g) and to differentiability of multivariable functions. Double integrals lead directly into polar coordinates and triple integrals, and reversing the order of integration is a prerequisite skill for applications of double integrals (mass, centre of mass, probability).
tangent plane, linear approximation, linearisation, multivariable differentials, double integral, iterated integral, reversing order of integration, Fubini's theorem, volume between surfaces, MATH 126, Calculus III, partial derivatives, log base 4, parametric limits, region of integration