GRAPHS

• Graph 1 — A graph of a function (an elliptic paraboloid) with contour lines and gradient vectors.
• Graph 2 — A graph of a function (another elliptic paraboloid) with contour lines and gradient vectors.
• Graph 3 — A graph of a function (a hyperbolic paraboloid) with contour lines and gradient vectors.

Background & Notes — Exercise 1

The gradient \(\text{grad }(f)\) or \(\vec{\nabla}f\) is the vector field whose coordinate functions are the partial derivatives of \(f\).

If \(f(x,y)\) is a function of two variables:

\[\vec{\nabla} f(x,y) = \left< \frac{\partial f}{\partial x} , \frac{\partial f}{\partial y} \right> = \frac{\partial f}{\partial x} \hat{\imath} + \frac{\partial f}{\partial y} \hat{\jmath}\]

EXERCISE 1 — Properties of the Gradient

Assumptions

Let \(f(x,y)\), \(g(x,y)\), and \(h(x,y)\) be the functions:

\[\begin{align*}
f(x,y) &= \frac{x^2}{9} + \frac{y^2}{4}\\
g(x,y) &= 10 - \left(\frac{x^2}{9} + \frac{y^2}{4}\right)\\
h(x,y) &= \frac{x^2}{9} - \frac{y^2}{4}
\end{align*}\]

Instructions

1. Compute the Gradients

Compute \(\vec{\nabla} f(x,y)\), \(\vec{\nabla} g(x,y)\), and \(\vec{\nabla} h(x,y)\).

2. Graph & Compare

Compare the graphs, gradients and, level curves of the functions:

• \(f(x,y) = \frac{x^2}{9} + \frac{y^2}{4}\) — graph 1
• \(g(x,y) = 10 - \left(\frac{x^2}{9} + \frac{y^2}{4}\right)\) — graph 2
• \(h(x,y) = \frac{x^2}{9} - \frac{y^2}{4}\) — graph 3

Questions:

1. What appears to be the angle between a gradient vector and the level curve at which it’s based?
2. Does the graph of the function increase or decrease in the direction of the gradient?
3. Does the length of the gradient vector increase or decrease as the graph becomes steeper?

Big Ideas — Properties of the Gradient

If \(f(x,y)\) is differentiable (has a tangent plane at each point of its graph), then:

• The gradient vector \(\vec{\nabla} f(a,b)\) is orthogonal to the level curve of \(f\) at the point \((a,b)\).
• The direction of the gradient vector \(\vec{\nabla} f(a,b)\) is the direction in which \(f\) increases most rapidly at the point \((a,b)\).

Background & Notes — Exercise 2

A directional derivative of a differentiable function \(f\) at a point \(P\) is the slope of a tangent line to the graph of \(f\) at point \(P\).

Directional derivatives depend on both the point \(P\) and the “direction” of the tangent line, indicated by a unit vector \(\hat{u}\).

Directional derivatives are computed as the dot product of the gradient \(\vec{\nabla} f(P)\) and a directional unit vector \(\hat{u}\):

\[D_{\hat{u}}f(P) = \vec{\nabla} f(P) \cdot \hat{u}\]

EXERCISE 2 — “Special” Directional Derivatives

Assumptions

Let \(f(x,y)\) be differentiable at the point \(P(a,b)\).

Instructions

1. Maximizing & Minimizing the Directional Derivative

The directional derivative can be expressed using the geometric definition of the dot product:

\[D_{\hat{u}}f(x,y) = \vec{\nabla}f(x,y) \cdot \hat{u} = |\vec{\nabla}f(x,y)| | \hat{u} | \cos\theta\]

Use this geometric definition to show that:

1. The magnitude of the gradient is a directional derivative — in fact, \(|\vec{\nabla}f(x,y)|\) is the maximum value of \(D_{\hat{u}}f(x,y)\) over all unit vectors \(\hat{u}\).
2. The direction of the gradient is the direction in which this maximum derivative occurs.
3. \(-|\vec{\nabla}f(x,y)|\) is the smallest value of \(D_{\hat{u}}f(x,y)\) over all unit vectors, and it occurs in the direction \(-\vec{\nabla}f(x,y)\).

2. Directional Derivative & Partial Derivatives

Use the algebraic definition of the dot product to show that:

1. \(\frac{\partial f}{\partial x}\) is the directional derivative of \(f(x,y)\) in the direction of the unit vector \(\hat{\imath}\).
2. \(\frac{\partial f}{\partial y}\) is the directional derivative of \(f(x,y)\) in the direction of the unit vector \(\hat{\jmath}\).

Big Ideas — “Special” Directional Derivatives

\(|\vec{\nabla}f(P)|\) is the directional derivative in the direction of the gradient, and the maximum of all directional derivatives at \(P\).

\(-|\vec{\nabla}f(P)|\) is the directional derivative in the direction opposite to the gradient, and the minimum of all directional derivatives at \(P\).

Partial derivatives are directional derivatives in the directions of the coordinate axes.