Math 2204 Multivariable Calculus – Chapter 14: Partial Derivatives
Sec. 14.4: Tangent Planes and Linear Approximations
I.
Tangent Planes
A. Definitions
Let
C1 and C2 be the curves obtained by intersecting the planes y = y0 and x = x0
z = f (x, y) . Let T1 and T2 be the tangent lines to the curves
C1 and C2 at the pt P0 = (x0 , y0 , z0 ) where z0 = f (x0 , y0 ) . The tangent plane
to the surface z = f (x, y) at P0 is defined to be the plane that contains both
with the surface
lines
T1 and T2 .
B. Equation
Tangent Plane to
is given by
z = f (x, y) at the pt P0 (x0 , y0 , z0 ) where z0 = f (x0 , y0 )
z = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )
C. Examples
1. Find the plane tangent to the surface
z = x cos(y) − ye x at the point P(0, 0, 0) .
2. Find the equation of the tangent plane to the surface
P(1, 0, −2) .
xe y z 2 + ln x = 4 at the point
III.
Linearization
A. Definitions
1. The increment of
z (the change in the value of f ) is Δz = f (x0 + Δx, y0 + Δy) − f (x0 , y0 )
2. The equation of the tangent plane to the graph of
f (x, y) at a point ( x0 , y0 , f (x0 , y0 ) )
is z = f (x0 , y0 ) + f x (x0 , y0 )(x − x0 ) + f y (x0 , y0 )(y − y0 ) . The linear function whose graph is
this tangent plane,
linearization of
L(x, y) = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) is called the
f (x, y) at the point ( x0 , y0 ) . The approximation f (x, y) ≈ L(x, y) is the
linear approximation or the tangent plane approximation of
f (x, y) at (x0 , y0 ) .
3. Note: Linearization approximates the function value
B.
Example
1. Find the linearization of
f (x, y) = x y at the point (1,4). Estimate f (1.1, 3.9) and
f (0,2) using the linearization and compare your answer to the true value.
2. Find the linearization of
f (x, y, z) = xz − 3yz + 2 at the point (1,1,2).
II.
Differentials
A. Theorem
If the partial derivatives
fx and fy exist near (x0 , y0 ) and are continuous at (x0 , y0 ) , then
f (x, y) is differentiable at (x0 , y0 ) .
B. Definitions
1. For a differentiable function of two variables,
and
z = f (x, y) , we define the differentials dx
dy to be independent variables. The differential dz is defined by
⎛∂f ⎞
⎛∂f ⎞
dz = fx (x, y) dx + fy (x, y) dy = ⎜ ⎟ dx + ⎜ ⎟ dy .
⎝ ∂x ⎠
⎝ ∂y ⎠
The differential is also called the total differential.
2. If
z = f (x, y) is a differentiable function and dx = Δx and dy = Δy ,
dz = fx (x0 , y0 ) dx + fy (x0 , y0 ) dy
= fx (x0 , y0 ) Δx + fy (x0 , y0 ) Δy
= fx (x0 , y0 ) (x − x0 ) + fy (x0 , y0 ) (y − y0 )
3. Note: Differential approximates the change in the function value
C. Examples
f (x, y) = x 2 + 3xy − y 2 will change
if (x,y) changes from (2,3) to (2.05, 2.96). Compare the values of Δz and dz .
1. Use differentials to estimate how much the value of
2. The dimensions of a rectangular box are measured to be
each measurement is correct to within
75cm, 60cm, and 40cm, and
0.2cm. Use differentials to estimate the largest
possible error in the volume of the box.
3. Circular cylindrical tanks are manufactured with a height of
3 ft and a radius of 1 ft.
Approximate the change in the tank’s volume if the radius changes by
height changes by
the radius?
1
8
and the
− 18 ? Is the volume more sensitive to the changes in the height or
4. Suppose that
T is to be calculated from the formula T = x(e y + e− y ) , where x and y
are found to be
2 and ln(2) with maximum possible errors of dx = 0.1 and dy = 0.02 .
Use differentials to estimate the maximum error in the calculated value of
T.