Question

In Exercises \(43-46,\) find a normal vector to the level curve \(f(x, y)=c\) at \(P.\) $$ \begin{array}{l} f(x, y)=x^{2}+y^{2} \\ c=25, \quad P(3,4) \end{array} $$

Short answer

The normal vector to the level curve at point \(P(3,4)\) is \(\nabla f(3,4) = (6,8)\).

Step-by-step solution

Calculate the Gradient of f

Compute the gradient of \(f(x, y)=x^{2}+y^{2}\) with respect to x and y. These are the partial derivatives of \(f(x, y)\).The partial derivative with respect to 'x' is given by \(f_{x} = 2x\), and the partial derivative with respect to 'y' is given by \(f_{y} = 2y\).

Find the Gradient at P

Calculate the value of the gradient at the point \(P(3,4)\). Substituting \(x=3\), \(y=4\) into the gradient partial derivatives, you get respectively \(f_{x}(3,4) = 2*3 = 6\) and \(f_{y}(3,4) = 2*4 = 8\). Therefore, the gradient of f at P is \(\nabla f(3,4) = (6,8)\).

Normal Vector to the Curve

The gradient of the function gives the direction of the highest rate of increase of the function. This means that the normal vector to the curve is the same as the gradient of the function at that point. Thus, the normal vector is \(\nabla f(3,4) = (6,8)\).

Key concepts

Gradient of a Function

The gradient of a function is a vector that points in the direction of the greatest increase of the function and its magnitude is the rate of the increase. For a function with two variables, like our example function f(x, y) = x² + y², the gradient is composed of the partial derivatives with respect to each variable.

In simpler terms, imagine you are on a hill, and the height of the hill at any point is given by f(x, y). The gradient vector at any point would point straight uphill and tell you the steepest direction you could go if you started walking from that point. Computing the gradient involves finding how much the function changes as you make small steps in either the x or y directions (or in the direction of more variables if they exist), which leads us to our next concept: partial derivatives.

Partial Derivatives

When dealing with functions of multiple variables, partial derivatives tell us how the function changes as one specific variable changes whilst keeping all others constant. In the context of the given exercise, we calculated the partial derivative of the function f(x, y) with respect to x, denoted as f_x by treating y as a constant, and similarly with y for f_y.

It's like examining how the hill's slope changes when walking east, with respect to x, and then how it changes when walking north, with respect to y. When we calculate these derivatives for our function f(x, y) = x² + y², we get 2x and 2y respectively. Knowing these allows us to find the gradient at any point, which leads to finding which direction is 'uphill' from any location on our metaphorical hill.

Level Curves

Level curves are like contours on a map. They connect points where the function has the same value. In our problem, the level curve we are interested in is where f(x, y) equals a constant value, namely c = 25. You can think of this level curve as a path that traces out a constant elevation on a hill.

When we find a normal vector to a level curve at a point, we're essentially finding a straight line that is perpendicular to the path at that point. It's like standing on the contour line on a hill and facing directly uphill or downhill. The gradient vector at a point P on the level curve is always perpendicular to the level curve at P, hence giving us the normal vector to the curve at that point.