Let's start by determining the level curve and the gradient vector for the function $f(x, y) = 4x - 2y + 3$ at the point $P(1, 2)$.

Step 1: Find the level curve

The level curve of $f(x, y)$ that passes through the point $P(1, 2)$ is the set of points $(x, y)$ where $f(x, y)$ has the same value as it does at $P$.

At $P(1, 2)$, calculate the function value:

$f(1, 2) = 4(1) - 2(2) + 3 = 4 - 4 + 3 = 3$

So, the level curve we need to sketch is the set of points where:

$f(x, y) = 3$

This gives the equation:

$4x - 2y + 3 = 3$

Simplifying the equation:

$4x - 2y = 0$

Which simplifies further to:

$2x = y$

This is the equation of a straight line, $y = 2x$. Therefore, the level curve is a straight line passing through the origin with a slope of 2.

Step 2: Find the gradient vector

The gradient vector $\nabla f(x, y)$ of a function $f(x, y)$ is given by the vector of partial derivatives:

$\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$

For $f(x, y) = 4x - 2y + 3$, we calculate the partial derivatives:

$\frac{\partial f}{\partial x} = 4$ $\frac{\partial f}{\partial y} = -2$

Thus, the gradient vector is:

$\nabla f(x, y) = (4, -2)$

Step 3: Evaluate the gradient at $P(1, 2)$

Since the gradient does not depend on $x$ or $y$, the gradient vector at any point, including $P(1, 2)$, is:

$\nabla f(1, 2) = (4, -2)$

This vector points in the direction of the greatest rate of increase of $f(x, y)$.

Step 4: Sketch the level curve and gradient vector

• The level curve is the line $y = 2x$.
• The gradient vector at $P(1, 2)$ is the vector $(4, -2)$, which is a vector starting at $P(1, 2)$ and pointing in the direction $4$ units along the x-axis and $-2$ units along the y-axis.

Sketch:

1. Draw the line $y = 2x$ (this is the level curve).
2. Plot the point $P(1, 2)$.
3. Draw a vector from $P(1, 2)$ pointing in the direction $(4, -2)$.

Would you like me to generate a visual of this?