MATH 251 – LECTURE 8
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 11.6–7, 12.1–3
webAssign: 11.6, 12.1, 12.3, due 2/8 11:55 p.m.
Next week: 12.4–6
webAssign: 12.4–6, opens 2/8 12 a.m.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Functions of more than two variables
Exercise 1. Find the domain of the function w = f (x, y, z) = log(x2 + y 2 − z).
Functions of more than two variables
Exercise 2. Sketch the level surfaces of the function w = f (x, y, z) = log(x2 + y 2 − 2z 2).
Partial derivatives
Definition 3. The partial derivatives of f (x, y) are defined by
f (x, y + h) − f (x, y)
f (x + h, y) − f (x, y)
and fy0 (x, y) = lim
fx0 (x, y) = lim
h→0
h→0
h
h
Alternative notation:
fx0 (x, y) = fx(x, y) =
∂
∂f
=
f (x, y) = Dxf.
∂x ∂x
2
Exercise 4. Find the partial derivatives of f (x, y) = sin(x)ey+x + xy 2.
Partial derivatives
Exercise 5. Let f (x, y) = xny m where n and m are integers. Find the second order partial derivatives
∂ ∂
∂ 2f
=
f
∂x∂y ∂x ∂y
and
∂ 2f
∂ ∂
=
f
∂y∂x ∂y ∂x
Functions of more than two variables
Exercise 6. Find all partial derivatives of the function u = f (x, y, z, w) = x2 + xy 3 + eyz .
Implicit derivation
Exercise 7. Let y = f (x) be defined by that xy = log(x + y). Find y 0(x).
Implicit derivation
Exercise 8. Let z = f (x, y) be defined by that xyz = ex+z . Find z 0(y).
Tangent planes
Consider the surface defined by z = f (x, y). Let P = P (x0, y0, z0) be a point on this surface. We want to find
an equation for the tangent plane of the surface at the point P . We have the two space curves intersecting at
the point P .
r1(x) = hx, y0, f (x, y0)i and r2(y) = hx0, y, f (x0, y)i
They have tangent vectors at P given by
r01(x0) = h1, 0, fx0 (x0, y0)i and r02(y0) = h0, 1, fy0 (x0, y0)i.
Tangent planes
Therefor, a normal vector to the tangent space is given by
r01(x0) × r02(y0) = h1, 0, fx0 (x0, y0)i × h0, 1, fy0 (x0, y0)i =
Tangent planes
We can conclude that the equation of the tangent plane is of the form
(x − x0)fx0 (x0, y0) + (y − y0)fy0 (x0, y0) − (z − z0) = 0
where
z0 = f (x0, y0).
Exercise 9. Find the equation of the tangent plane to the surface f (x, y) = x2 + ey + 1 at the point (1, 1, 3).
Increments and differentials
Let z = f (x, y). If x and y are given increments ∆x and ∆y, then the increment of z is
∆z = f (x + ∆x, y + ∆y) − f (x, y).
That is, the increment ∆z gives the change of height of the surface z = f (x, y) between the points (x, y) and
(x + ∆x, y + ∆y).
Let z = f (x, y). The differential dz is defined as
dz = fx(x, y)dx + fy (x, y)dy.
If dx = ∆x and dy = ∆y, then dz represents the change of height of the tangent plane between the points
(x, y) and (x + ∆x, y + ∆y).
The function f (x, y) is said to be differentiable at the point (x, y) if it is well-approximated by its tangent
plane at that point. That is, if
dz ≈ ∆z.
All elementary functions are differentiable in their domains.
Increments and differentials
Exercise 10. Let z = f (x, y) = x2 + 2y. Compute the increment of f at the point (1, 2) if x and y are given
increments 1/10 and 1/100.
Exercise 11. Let z = f (x, y) = x2 + 2y. Compute the differential of f at the point (1, 2).
Increments and differentials
The approximation dz ≈ ∆z = f (x + ∆x, y + ∆y) − f (x, y), can be rewritten as
f (x + ∆x, y + ∆y) = f (x, y) + dz.
Exercise 12. Approximate the number
q
120
168
with rational numbers.
Increments and differentials
Exercise 13. Approximate the number
√
120 −
√
168 +
√
98 with rational numbers.