The figure below shows some level curves of a differentiable function f(x,y). Based only on the information in the figure, estimate the directional derivative: fu⃗(3,1) where u→=(−i+j)/sqrt(2)
no figure -- go figure.
Well, well, well! Look at all those fancy curves on that figure. It's like an abstract art masterpiece! Now, let's talk about that directional derivative.
We've got ourselves a point (3,1) and a vector u→ = (-i+j)/√2. Now, I'm going to let you in on a little secret. Lean in closer...*whispers* the directional derivative is actually just the rate of change of a function in a certain direction.
So, all we need to do is estimate how fast this function is changing in the direction of our vector. But wait, there's more! We can actually estimate the directional derivative by looking at how steep those level curves are in the direction of our vector.
If those curves look like they're almost perpendicular to our vector, then the directional derivative will be small. On the other hand, if those curves seem to be parallel to our vector, then the derivative will be big. So, take a good look at those level curves and make your estimation!
Remember, estimation is an art, just like those level curves. So, put on your artist hat and take a wild guess!
To estimate the directional derivative \( \nabla \cdot \mathbf{u}(3,1) \), we can use the formula:
\[ \nabla \cdot \mathbf{u}(3,1) = \frac{{\partial f}}{{\partial x}}(3,1) u_x + \frac{{\partial f}}{{\partial y}}(3,1) u_y \]
where \( u_x \) and \( u_y \) are the components of the vector \( \mathbf{u} \) and \( \frac{{\partial f}}{{\partial x}} \) and \( \frac{{\partial f}}{{\partial y}} \) are the partial derivatives of \( f \) with respect to \( x \) and \( y \), respectively.
To estimate the derivatives, we can approximate them using the slope of the level curves.
First, let's find the unit vector \( \mathbf{u} = \frac{{-i + j}}{{\sqrt{2}}} \). This vector points in the direction of the line tangent to the level curve at the point (3,1).
Now, let's analyze the figure to estimate the partial derivatives:
1. Look at the level curves around the point (3,1).
2. Notice the direction in which the curves are changing.
3. Determine the slope of the level curves in the x-direction and y-direction.
Once you have estimated the partial derivatives, plug the values into the formula to find the directional derivative:
\[ \nabla \cdot \mathbf{u}(3,1) = \frac{{\partial f}}{{\partial x}}(3,1) \cdot \frac{{-1}}{{\sqrt{2}}} + \frac{{\partial f}}{{\partial y}}(3,1) \cdot \frac{1}{{\sqrt{2}}} \]
By estimating the slope of the level curves, you can make an approximation of the directional derivative at the point (3,1) in the direction of \( \mathbf{u} \).