Activity: The Hillside

Prerequisite concepts

Differential Calculus The magnitude of the derivative at a point is the slope of a tangent line at that point

The value of the derivative at a point on a curve is equivalent to the slope of a line tangent to the curve at that point. The tangent line can be thought of as the limit of a secant line between two points infinitesimally close together

The Heater II

Derivatives can be found while holding one or more variables constant

Derivatives of multivariable functions can be found by holding one or more variables constant (subject to physical limitations).

There is a partial derivative in every direction at any point

For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.

The Hot Plate

There is a tangent line in every direction at every point

For a smooth, two dimensional function, a tangent line exists at any point and in any direction.

There is a partial derivative in every direction at any point The value of a partial derivative depends on the value(s) of what is held constant The magnitude of the gradient is proportional to the density of contour lines

The Master Formula

The gradient is a vector

For an $n-$dimensional function $f$, the gradient of $f$ at a point is an $n-$dimensional vector

*The Direction of the Gradient *The Magnitude of the Gradient

Instructor guide mvhillside

The Hillside

This small group activity using surfaces introduces the gradient. Students work in small groups to determine the steepest slope and direction at a given point, then compare these results with the computed gradient. The whole class wrap-up discussion emphasizes the geometric properties of the gradient.

Representations used

$\frac{\partial f}{\partial x}$

Surface

Inclinometer

Concepts taught

The direction of the gradient is perpendicular to contour lines

Contour Maps

As the gradient lies along the direction of greatest change for a function, and contour lines lie along the direction of no change, these two objects (the gradient and contour lines) are perpendicular.

The direction of the gradient shows what way is 'uphill'

Surface

The gradient at any point on the surface gives the direction in which the height of the surface is increasing the most. The gradient neither points towards the highest point on the surface (the global maximum) nor has a vertical component (it lies in the plane of the table).

The magnitude of the gradient is proportional to the density of contour lines

Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.

The components of the gradient are partial derivatives

FIXME