Find the change in \(f(x,y)=x^2y^3\) in the direction of \(\vec{v}=\hat{i}+\hat{j}\) at the point \((-2,1)\)

Calculate the derivative of \( f(x,y) = xe^y + \cos(xy) \) at the point \( (2,0) \) in the direction of \( \vec{v} = 3\hat{i} - 4\hat{j} \).

At the point \( (1,1) \), determine the directions in which \( f(x,y) = x^2/2 + y^2/2 \) (a) increases most rapidly; (b) decreases most rapidly; (c) has zero change

Calculate the derivative of \( f(x,y,z) = x^3 - xy^2-z \) at the point \( (1,1,0) \) in the direction \( \vec{v} = 2\hat{i} - 3\hat{j} + 6\hat{k}\).

Calculate the directional derivative of \( f(x,y) = 1 - x^2/2 - y^4/4 \) in the direction \(\hat{i} + \hat{j}\) at the point \( (1,2) \).

Calculate the directional derivative of \( f(x,y) = x^2 + 2xy + 3y^2 \) at the point \( (2,1) \) in the direction \(\hat{i}+\hat{j}\).

Calculate the maximum directional derivative and its direction for \( f(x,y) = 2x^2 + 3xy + 4y^2 \) at \( (1,1) \).

Calculate the directional derivative of \( f(x,y) = x^2y^3 - y^4 \) at \( (2,1) \) in the direction given by the angle \( \theta = \pi/4 \).

Calculate the value of the maximum rate of change of the function \( f(x,y) = \sin(xy) \) at the point \( (1,0) \).

Calculate the direction of the maximum rate of change of the scalar field \( \phi(x,y,z) = -2xy + x\ln(y+z) \) at \( P(1,3,-2) \) and the maximum value of the directional derivative at P.

Find the directional derivative \( D_x f(x,y) \) of the function \( f(x,y) = x^2 - 4y^2 + 2x \) at the point \((1,1)\) in the direction of the vector \( (1/2)\hat{i} - (\sqrt{3}/2)\hat{j} \)

Find the directional derivative \( D_u f(x,y) \) of the function \( f(x,y) = e^{xy}+y \) at \((2,0)\) in the direction of \(-\hat{i} + \hat{j}\sqrt{3}\)

Write the directional derivative \( D_u f(x,y) \) for the surface \( f(x,y) = xy^2 - 4x + 3y \) in the direction of \( 3\hat{i} - 4\hat{j} \)

Find \(D_u f(x,y)\) if we have the function \(f(x,y) = e^{x-1}+e^y + 2x\) and are at the point where \(x=1\) and \(y=0\) and want to move in the direction where \(\theta = 7\pi / 6\)

To get the unit vector in the direction of motion from the angle, use \( \vec{u} = \langle \cos \theta, \sin \theta \rangle \)

Find \(D_u f(x,y)\) if we have the function \(f(x,y) = xy^3-x^2\) and are at the point where \(x=1\) and \(y=0\) and we want to move toward the origin.

a. Find \(\nabla f(x,y)\) if we have the function \(f(x,y) = x^2y^2 - x^2+y^4 - y\).
b. Find \(\nabla f(1,0)\) from the same function.
c. Write the directional derivative \(D_uf(x,y)\) as the dot product of the function at \((1,0)\) and in the direction of the vector \(\langle -5, 12 \rangle\).

Take the given function and direction vector and convert the directional derivative \( D_u f(x,y) \) into a dot product.
\( f(x,y) = e^{xy} + y \) in the direction of \(-\hat{i} + \hat{j}\sqrt{3}\)
Write the appropriate notation for the directional derivative.

Given a flies motion through space defined by the equation \( f(x,y) = xy^2 -4x + 3y \), solve these problems.
a) If it's location is at the point \((1,1,0)\) and it wants to move toward the point \((1,0,-4)\) find the steepness of its flight.
b) If it arrives at the point \((1,0,-4)\) and wants to move back to the origin, find the steepness of the flight at that instant.
c) If the fly arrives at the origin and wants to move upward at the greatest possible rate, what direction should it go and what will the steepness be?

For the surface \( f(x,y) = 3x^3-y^2+2xy^2 \) find the following
a. Find the complete point when \(x=1\) and \(y=0\).
b. Find the partial derivatives of \(f\)
c. Find the gradient function \( \nabla f(x,y)\)
d. Find the equation for the directional derivative in the direction of the vector \( \langle \cos \theta, \sin \theta \rangle \) at the point found in part a.
e. If a particle is at point a and moves toward the point \((1,3,1)\) what will \(D_u f(x,y)\) be?
f. In what direction should the particle move if it wants \( D_uf(x,y) \) to be the smallest value possible?