12.4 Tangent Planes and Dierentials
Suppose a surface S has equation z = f (x, y), and let P (x0 , y0 , z0 ) be a point on S .
Let C1 and C2 be the curves obtained by intersecting the vertical planes y = y0 and
x = x0 with the surface S . Then the point P lies on both C1 and C2 .
Let T1 and T2 be the tangent lines to the curves C1 and C2 at the point P . Then
the tangent plane to the surface at the point is dened to be the plane that contains
both tangent lines T1 and T2 .
Theorem. An equation of the tangent plane to the graph of the function z = f (x, y)
at the point (x0 , y0 , z0 ) is
z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )
Note. A normal vector to the surface z = f (x, y) at the point P (x0, y0, z0) is
n = h fx (x0 , y0 ), fy (x0 , y0 ), −1 i
Example 1. Find the equation of the tangent plane to the surface z = ln(2x + y) at
the point (−1, 3, 0).
Dierentials
Consider a function of two variables z = f (x, y). If x and y are given increments ∆x
and ∆y , then the corresponding increment of z is
∆z = f (x + ∆x, y + ∆y) − f (x, y)
which represents the change in the value of f when (x, y) changes to (x + ∆x, y + ∆y).
The dierentials dx = ∆x and dy = ∆y are independent variables. The dierential dz (or the total dierential), is dened by
dz = fx (x, y)dx + fy (x, y)dy
FACT: ∆z ≈ dz
If we take dx = ∆x = x − a and dy = ∆y = y − b, then
f (a + ∆x, b + ∆y) ≈ f (a, b) + dz(a, b)
or
f (a + ∆x, b + ∆y) ≈ f (a, b) + fx (a, b)∆x + fy (a, b)∆y
Example 2. Use dierentials to approximate the number
√
1.032 + 1.983 .
Functions of three or more variables.
If u = f (x, y, z), then the increment of u is
∆u = f (x + ∆x, y + ∆y, z + ∆z) − f (x, y, z)
The dierential du is dened in terms of the dierentials dx, dy , and dz by
du = fx (x, y, z)dx + fy (x, y, z)dy + fz (x, y, z)dz
If dx = ∆x = x − a, dy = ∆y = y − b and dz = ∆z = z − c, then ∆u ≈ du and
f (a + ∆x, b + ∆y, c + ∆z) ≈ f (a, b, c) + fx (a, b, c)∆x + fy (a, b, c)∆y + fz (a, b, c)∆z
Example 3. The dimensions of a closed rectangular box are measured as 80 cm, 60
cm and 50 cm, respectively, with a possible error of 0.2 cm in each dimension. Use
dierentials to estimate the maximum error in calculating the surface area of the box.