MATH 251 – LECTURE 7
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.
Space curves - Arc length
The length of a space curve r(t) = hf (t), g(t), h(t)i as a ≤ t ≤ b is
Z bp
Z b
|r0(t)| dt =
L=
f 0(t)2 + g 0(t)2 + h0(t)2 dt
a
a
√
√
Exercise 1. Compute the length of the curve r(t) = h 2 t, 3 t2, 2 t3i as 0 ≤ t ≤ 1.
Functions of two variables
Definition 2. Let D ⊂ R2. A function f of two-variables is a rule that assigns to each point (x, y) in D a
value f (x, y) in R.
The set D is called the domain of f . The set of all values that f takes on, that is the set E = {f (x, y) | (x, y)
is contained in D} is called the range of f .
If a function is given by a formula, then the domain is understood to be the set D of all points (x, y) such that
the formula is well-defined.
Exercise 3. Find the range of the function f (x, y) =
p
x2 + y 2 + 1.
Functions of two variables
To visualize a function of two variables, we set z = f (x, y), and consider the graph of the function f .
Definition 4. The graph of f is the set
G = {(x, y, z) | z = f (x, y)}.
Exercise 5. Sketch the graph of the function f (x, y) = x2 − 2y 2.
Functions of two variables
Definition 6. A level curve of a function f of two variables is a curve defined by the equation f (x, y) = k,
where k is a constant.
Exercise 7. Draw the level curves of the function f (x, y) = x2 − 2y 2.
Limits of functions of two variables
Example 8. Consider f (x, y) = yx2/(x4 + y 2).
0.5
0.5
0.0
0.0
0.2
0.2
-0.5
-0.5
-0.2
-0.2
0.0
0.0
0.0
0.0
-0.2
-0.2
0.2
0.2
Limits of functions of two variables
Definition 9. We say that the function f (x, y) → L when (x, y) → (a, b) if for each positive numbers ε > 0
there exists a positive number δ = δ(ε) such that if |hx, yi − ha, bi| < δ then |f (x, y) − L| < ε.
Example 10. Show that xy → 0 as (x, y) → (0, 0).
Definition 11. A function f (x, y) is said to be continous at a point (a, b) if f (x, y) → f (a, b) when (x, y) →
(a, b).
Theorem 12. All elementary functions are continuous in their domain.
Partial derivatives
Definition 13. 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) =
∂
∂f
=
f (x, y) = Dxf
∂x ∂x
2
Exercise 14. Find the partial derivatives of f (x, y) = sin(x)ey+x + xy 2.
Partial derivatives
Exercise 15. 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
Implicit derivation
Exercise 16. Let y = f (x) be defined by that xy = log(x + y). Find y 0(x).
Implicit derivation
Exercise 17. Let z = f (x, y) be defined by that xyz = ex+z . Find z 0(y).
Functions of more than two variables
Exercise 18. Find the domain of the function w = f (x, y, z) = log(x2 + y 2 − z).
Functions of more than two variables
Exercise 19. Sketch the level curves of the function w = f (x, y, z) = log(x2 + y 2 − 2z 2).
Functions of more than two variables
Exercise 20. Find all partial derivatives of the function u = f (x, y, z, w) = x2 + xy 3 + eyz .