Problem 1: Linear Approximation
If \( w(x, y) = x^3 - 3xy + 2y^2 \), \( x, y \in \mathbb{R} \), find the linear approximation for \( w \) at \((1, -1)\).
1 Find the value of the function at the point (1, -1)
\( w(1, -1) = (1)^3 - 3(1)(-1) + 2(-1)^2 = 1 + 3 + 2 = 6 \)
2 Find the partial derivatives
\( \frac{\partial w}{\partial x} = 3x^2 - 3y \)
\( \frac{\partial w}{\partial y} = -3x + 4y \)
3 Evaluate partial derivatives at (1, -1)
\( \frac{\partial w}{\partial x}(1, -1) = 3(1)^2 - 3(-1) = 3 + 3 = 6 \)
\( \frac{\partial w}{\partial y}(1, -1) = -3(1) + 4(-1) = -3 - 4 = -7 \)
4 Write the linear approximation formula
\( L(x, y) = w(a, b) + \frac{\partial w}{\partial x}(a, b)(x-a) + \frac{\partial w}{\partial y}(a, b)(y-b) \)
5 Plug in the values
\( L(x, y) = 6 + 6(x - 1) - 7(y - (-1)) \)
\( L(x, y) = 6 + 6(x - 1) - 7(y + 1) \)
The linear approximation is: \( L(x, y) = 6 + 6(x - 1) - 7(y + 1) \)
Problem 2: Linear Approximation
Let \( z(x, y) = x^2 y + 3xy^4 \), \( x, y \in \mathbb{R} \). Find the linear approximation for \( z \) at \((2, -1)\).
1 Find \( z(2, -1) \)
\( z(2, -1) = (2)^2(-1) + 3(2)(-1)^4 = 4(-1) + 6(1) = -4 + 6 = 2 \)
2 Find partial derivatives
\( \frac{\partial z}{\partial x} = 2xy + 3y^4 \)
\( \frac{\partial z}{\partial y} = x^2 + 12xy^3 \)
3 Evaluate at (2, -1)
\( \frac{\partial z}{\partial x}(2, -1) = 2(2)(-1) + 3(-1)^4 = -4 + 3 = -1 \)
\( \frac{\partial z}{\partial y}(2, -1) = (2)^2 + 12(2)(-1)^3 = 4 + 12(2)(-1) = 4 - 24 = -20 \)
4 Write the linear approximation
\( L(x, y) = 2 - 1(x - 2) - 20(y - (-1)) \)
\( L(x, y) = 2 - (x - 2) - 20(y + 1) \)
The linear approximation is: \( L(x, y) = 2 - (x - 2) - 20(y + 1) \)
Problem 3: Differential
If \( v(x, y) = x^2 - xy + \frac{1}{4} y^2 + 7 \), \( x, y \in \mathbb{R} \), find the differential \( dv \).
1 Find partial derivatives
\( \frac{\partial v}{\partial x} = 2x - y \)
\( \frac{\partial v}{\partial y} = -x + \frac{1}{2}y \)
2 Write the differential formula
\( dv = \frac{\partial v}{\partial x}dx + \frac{\partial v}{\partial y}dy \)
3 Plug in the partial derivatives
\( dv = (2x - y)dx + (-x + \frac{1}{2}y)dy \)
The differential is: \( dv = (2x - y)dx + (-x + \frac{1}{2}y)dy \)
Problem 4: Differential
Let \( V(x,y,z) = xy + ye^z + zx \), \( x,y,z \in \mathbb{R} \). Find the differential \( dV \).
1 Find partial derivatives
\( \frac{\partial V}{\partial x} = y + z \)
\( \frac{\partial V}{\partial y} = x + e^z \)
\( \frac{\partial V}{\partial z} = ye^z + x \)
2 Write the differential formula for three variables
\( dV = \frac{\partial V}{\partial x}dx + \frac{\partial V}{\partial y}dy + \frac{\partial V}{\partial z}dz \)
3 Plug in the partial derivatives
\( dV = (y + z)dx + (x + e^z)dy + (ye^z + x)dz \)
The differential is: \( dV = (y + z)dx + (x + e^z)dy + (ye^z + x)dz \)