Calc 3 Lecture Notes
Section 12.4
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Section 12.4: Tangent Planes and Linear Approximations
Big idea: The object tangent to a 2D surface embedded in three dimensions is a plane. The
equation of the tangent plane is similar in form to the equation of a tangent line to a curve in two
dimensions.
Big skill: You should be able to compute the tangent plane to a curve and the total differential of
a function.
Recall that in two dimensions, the equation for the tangent line to a curve described by y = f(x) at
x = a is given by:
y = f(a) + f (a)(x – a).
Now let’s use tangent lines to find the tangent plane to the surface z  4  x 2  y 2 at
(a, b, f(a, b)) = (1, 1, 2):
1. Find equations of tangent lines in the planes x = 1 and y = 1.
Calc 3 Lecture Notes
Section 12.4
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2. Find vectors parallel to the tangent lines; use P1P = ta
3. Take cross product to get a vector normal to the plane.
4. Get eqn. of plane from nP1P = 0.
Now let’s do this calculation for any surface z  f  x, y  : at the point (a, b, f(a, b))
1. Find equations of tangent lines in the planes x = a and y = b.
Calc 3 Lecture Notes
Section 12.4
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2. Find vectors parallel to the tangent lines; use P1P = ta
3. Take cross product to get a vector normal to the plane.
4. Get eqn. of plane from nP1P = 0.
Theorem 4.1: Equation of a Plane Tangent to a Surface
Suppose that f(x, y) has continuous first partial derivatives at (a, b). A normal vector to the
tangent plane at z = f(x, y) at (a, b) is then <fx(a, b), fy(a, b), -1>. Further, an equation of the
tangent plane is given by:
z = f(a, b) + fx(a, b)(x – a) + fy(a, b) (y – b).
Also, the equation of the normal line through the point (a, b, f(a, b)) is given by:
x = a + fx(a, b)t,
y = b + fy(a, b)t,
z = f(a, b) – t.
Practice:
1. Find the equation of the tangent plane and normal line to the surface z  x 2 y 2  1 for
{ (x, y) } = {(1, 1), (1, 0), (0, 1), (0, 0)}
Calc 3 Lecture Notes
Section 12.4
2. Find the equation of the tangent plane and tangent line to the surface


z  cos x 2  y 2 for (x, y) = (, 0), and (0, 0).
Notice: What is the equation of the tangent plane at extrema?
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Calc 3 Lecture Notes
Section 12.4
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Just as the tangent line is a linear approximation for a curve near the tangent point, the tangent
plane is a linear approximation for a surface near the tangent point. Thus, the linear
approximation L(x, y) to a function f(x, y) at the point (a, b) is:
L(x, y) = f(a, b) + fx(a, b)(x – a) + fy(a, b) (y – b).
Increments and Differentials
Review of the differential from functions of one variable:
dy
Recall that the expression
is not a ratio, but instead represents the derivative of y with respect
dx
to x. So, we introduce two new variables dx and dy with the property that if their ratio exists, then
it will be equal to the derivative.
Defintion: The Differential of a Univariate Function
Let y = f(x) be a differentiable function. The differential dx is an independent variable. The
differential dy is: dy = f (x)dx.
Geometrically, dy is the vertical change (shown as L in the image below) along the tangent line
of a curve when x = a changes by an amount dx = x. The quantity y = f(a + dx) – f(a) is called
the increment in y. In the limit x  0, y  dy. The error   0 as x  0 as well.
Defining differentials allows us to “re-write” a derivative in calculus in much the same way we
can re-write a slope in algebra:
dy
y
 f   x   dy  f   x  dx
m
 y  mx
dx
x
Every differentiation formula in calculus has a corresponding form using differentials:
d  u  v  du dv


 d  u  v   du  dv
dx
dx dx
d  uv  du
dv

vu
 d  uv   vdu  udv
dx
dx
dx
Calc 3 Lecture Notes
Section 12.4
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Now let’s expand this notion into three dimensions. We’ll begin with the increment z and the
error between the increment and the linear approximation as we move away from the point
(a, b, f(a, b)), which will lead us to the total differential.
The quantity z = f(a + x, b + y) – f(a, b) is called the increment in z.
Theorem 4.2: Increment for a Function of Two Variables in Terms of Partial Derivatives
Suppose that z = f(x, y) is defined on the rectangular region R = {(x, y) | x0 < x < x1 and
y0 < y < y1} and fx and fy are defined on R and are continuous at (a, b)  R. Then for
(a + x, b + y)  R,
z = fx(a, b)x + fy(a, b)y + 1x + 2y,
where 1 and 2 are functions of x and y that both tend to zero as x and y tend to zero.
Picture:
Definition 4.1: Differentiability of a Function of Two Variables
Let z = f(x, y). We say that z is differentiable at (a, b) if we can write
z = fx(a, b)x + fy(a, b)y + 1x + 2y,
where 1 and 2 are functions of both x and y and 1 and 2  0 as (x, y) 0. We say that f
is differentiable on a region R  R2 whenever f is differentiable at every point in R.
DEFINITION: Total Differential of a Function of Two Variables
dz  f x  x, y  dx  f y  x, y  dy
Practice:
3. Write the increment for the function z = f(x, y) = 5x2 – 3xy in the form of theorem 4.2,
then compute its total differential.
Calc 3 Lecture Notes
Section 12.4
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Definition 4.2: Linear Approximation for a Function of Three Variables
The linear approximation to f  x, y, z  at the point  a, b, c  is given by:
L  x, y, z   f  a, b, c   f x  a, b, c  x  a   f y  a, b, c  y  b   f z  a, b, c  z  c 
Practice:
4. Suppose that the sag S in a beam of length L, width w, and height h is given by
L4
S  L, w, h   0.0004 3 , and that the length is 36  1, the width is 2  0.4, and the height
wh
is 6  0.8. Use a linear approximation to estimate the range in sag of the beam.