MATH 2210 In-Class Notes Level Curves and Slopes
Directions: Answer the following questions.
Note: Regarding changes along a surface
),( yxfz
, references to direction always
refer to the domain of f in the xy plane. For example, the partial derivative
x
f
is in the
positive x direction and its value represents the slope along the surface in that direction.
1. At the points A, B, C, D and E on the given contour map, determine the sign of
the partial derivatives
x
f
and
y
f
.
:)( Af x
:)( Bf
x
:)(Cf
x
:)(Df
x
:)( Ef
x
:)( Af
y
:)(Bf
y
:)(Cf
y
:)( Df
y
:)( Ef
x
2. At the points A, B, C, D and E indicate on the map the direction in which the
slope of the surface is increasing greatest.
3. At each point A, B, C, D and E, how is the direction of greatest decrease related
to the direction of greatest increase?
4. How is the direction of greatest increase related to the corresponding level
curve?
5. What is the rate of change at any point along a level curve?
6. At what point is
0
x
f
? What is the corresponding direction of greatest increase?
7. At what point is
0
y
f
? What is the corresponding direction of greatest increase?
8. At point A, how do the values of
x
f
and
y
f
compare to the slope of greatest
increase?
9. On the contour map, plot ALL points on each level curve at which
0
x
f
.
Identify a basic function
)( xgy
whose graph includes this set of points.
