Partial derivatives
2
2
1. Let f (x, y) = e(x +y ) + x2 + y 2 + xy + 2y + 3.
f
f
a) Compute
and
.
x
y
b) Show the second partials can be computed in any order. That is,
2f
2f
=
.
xy
yx
f
(1, 3).
x
f
2
2
Answer: a)
= 2xe(x +y ) + 2x + y,
x
c) Find
f
2
2
=
Tangent approximation
1. a) Find the equation tangent plane to the graph of z = x2 + y 2 at the point (2,1,5). b) Give the tangent approximation for z near the point (x0 , y0 ) = (2, 1). z z z z Answer: a) = 2x and = 2y (2, 1) = 4 and (2, 1) = 2. x y x y
Equation of a Plane
1. Later we will return to the topic of planes in more detail. Here we will content ourself
with one example.
Find the equation of the plane containing the three points P1 = (1, 3, 1), P2 = (1, 2, 2),
P3 = (2, 3, 3).
Answer:
The vector
Graphing a function
1. Draw the graph of z = y 2 x2 .
Answer: Here is the graph.
1. First we drew the trace in the yz-plane, which is an upward pointing parabola.
2. Then we drew the xz-traces with y = 0, y = 1 and y = 1.
3. Finally we drew the yz-traces
Equation of a plane
1. Find the equation of the plane containing the three points P1 = (1, 0, 1), P2 = (0, 1, 1),
P3 = (1, 1, 0).
Answer: This problem is identical (with changed numbers) to the worked example we just
saw.
The vectors P1 P2 and P1 P3 are i
Level Curves and Contour Plots
Level curves and contour plots are another way of visualizing functions of two variables. If
you have seen a topographic map then you have seen a contour plot.
Example: To illustrate this we rst draw the graph of z = x2 + y
Cross Product
The cross product is another way of multiplying two vectors. (The name comes from the
symbol used to indicate the product.) Because the result of this multiplication is another
vector it is also called the vector product.
As usual, there is
The Tangent approximation
4. Critique of the approximation formula.
First of all, the approximation formula for functions of two or three variables
w
w
x +
y,
if x 0, y 0 .
(6)
w
x 0
y 0
(7)
w
w
x
x +
0
w
y
y +
0
w
z
z,
if
x, yz 0 .
0
is not a precise m
The Tangent Approximation
1. The tangent plane.
w
For a function of one variable, w = f (x), the tangent line to its graph(
)
dw
.
at a point (x0 , w0 ) is the line passing through (x0 , w0 ) and having slope
dx 0
w=f(x,y)
w=f(x,y 0)
For a function of two
Partial derivatives
Partial derivatives
Let w = f (x, y) be a function of two variables. Its graph is a surface in xyz-space, as
w
pictured.
w=f(x,y)
Fix a value y = y0 and just let x vary. You get a function of one variable,
(1)
w = f (x, y0 ),
w=f(x,y 0
Guia do Estudo para O Teste Final
1. Dias do ms e meses do ano
a. Dias
i. Segunda-feira
ii. Tera-feira
iii. Quarta-feira
iv. Quinta-feira
v. Sexta-feira
vi. Sbado
vii. Domingo
b. Meses
i. Janeiro
ii. Fevereiro
iii. Maro
iv. Abril
v. Maio
vi. Junho
vii. Ju