now we'll look at two examples of
estimating the value of a partial
derivative using a contour map a contour
map of f of X comma Y is shown we want
to use it to estimate the partial
derivative of F with respect to Y when X
is 3 and Y is 20 this represents the
slope of the tangent line in the y
direction at this point we can estimate
the value of a partial derivative which
represents the slope of a tangent line
by using the slope of a secant line just
made the partial derivative of F with
respect to X we would use the slope of a
secant line in the X Direction where we
would find the change in the function
value and divide by the change of X and
to estimate the partial derivative of F
with respect to Y we'd use the slope of
a secant line in the Y direction so we
determine the change in the function
value and divide by the change of Y so
going back to our first example let's
first locate the point 3 comma 30 which
would be here notice how the function
value is 360 let's go ahead and record
that F of 3 comma 30 equals 360 and now
to find the slope of the secant line to
approximate this partial derivative we
want to find another point in the Y
direction so let's sketch a vertical
line parallel to the y axis here so the
second point must be somewhere on this
vertical line and of course the closer
it is the better let's go ahead and use
this point here because we can easily
determine the function value at this
point notice how the x coordinate is 3
and the y coordinate looks like it'd be
about 21 so f of 3 comma 21 would be
equal to 270 and now we can use this
information to find the slope of the
secant line to make our approximation
for this partial derivative partial of F
with respect to Y will be approximately
equal to the change in F divided by the
change of Y of the secant line
so the change of F would be 360 minus
270 and the change of Y would be 30
minus 21 so this gives us what ninety
divided by nine which is equal to ten so our
approximation for the partial of F with
respect to Y will be approximately ten
let's take a look at a second example we
have the same contour map now we want to
estimate the partial derivative of F
with respect to X at the point 2 comma 5
so here's the point 2 comma 5 notice how
the function value is 90 so we know F of
2 comma 5 equals 90 now because we're
finding the partial with respect to X
let's sketch a horizontal line that's
parallel to the x-axis and select the
second point on this horizontal line
let's go and select this point here
notice how the x coordinate is 4 the y
coordinate is 5 and the function value
looks like it'd be somewhere in between
90 and 180 let's estimate the function
value to be 135 so we'll say f of 4
comma 5 is equal to 135 and we use the
slope of the secant line in the
x-direction to approximate this partial
derivative so the partial of F with
respect to X would be approximately
equal to the change in the function
value divided by the change of X using
these two points so the change in the
function value would be 135 minus 90 and
the change in the x value would be 4
minus 2 so we have what 45 divided by 2
which is equal to twenty two point five
which we'll use for our approximation
for the partial derivative of F with
respect to X
I hope you found this helpful
