To approximate the directional derivative of the function f at point p in the direction of q, we first need to understand the values involved. We have:
- f(p) = 15
- f(q) = 20
- p = (3, 4)
- q = (3, 3.03, 3.96)
The directional derivative is typically found using the following formula:
D_u f(p) =,
abla f(p) ullet u
where
abla f(p) is the gradient of f at p and u is the unit vector in the direction of q – p.
First, we need to compute the direction vector q – p:
q - p = (3, 3.03, 3.96) - (3, 4) = (0, -0.97, 3.96)
Next, we calculate the magnitude of this vector:
||q - p|| =
ext{sqrt}(0^2 + (-0.97)^2 + (3.96)^2) \
= ext{sqrt}(0 + 0.9409 + 15.6816) = ext{sqrt}(16.6225) \
≈ 4.08
Now, we find the unit vector u:
u = rac{q - p}{||q - p||} = rac{(0, -0.97, 3.96)}{4.08} \
≈ (0, -0.238, 0.970)
Now, we assume we can approximate the gradient
abla f(p) as:
abla f(p) ≈ (f(q) – f(p)) / (||q – p||) = (20 – 15) / 4.08 = 1.225
Finally, we can calculate the directional derivative:
D_u f(p) =
abla f(p) ullet u \
≈ 1.225 imes (0 + (-0.238) + (0.970) ) \
= 1.225 imes 0.732 \
≈ 0.897
Thus, the approximate directional derivative of f at point p in the direction of q is approximately 0.897.