Examples for Sections 12.3, 12.4, and 12.5

EXAMPLES FOR SECTIONS 13.3, 13.4, AND 13.5

Maple Worksheet on Partial Derivatives

Section 13.3 #40 (7e 36)

DPGraph Picture of the surface and the plane.

Graph of the curve of intersection in the xz-coordinate plane

Click on the picture to see a 3-D blowup with a tangent line

Maple Picture "from the back"

DPGraph Picture of the surface and the plane.

Graph of the curve of intersection in the yz-coordinate plane

Click on the picture to see a 3-D blowup with a tangent line

Maple Picture (different view)

This DPGraph Picture shows the surface intersecting the graph of the plane y = (4/3)x. Maple Picture What might this lead to?

Here is a Maple Worksheet related to the example above. In the worksheet the point and the tangent line at the point have been added to each picture.

Total Differential Example

The two pictures at the right are both of the surface described by the function given above. Notice the window given for each picture.

Implicit Differentiation Example DPGraph of the Surface

We can evaluate the partial derivative computed on the left at a point on the surface,

for example point (1, -2, 1).

This would give us the "slope" at the point (1, -2, 1) of the curve of intersection of the surface defined on the left and the plane whose equation is y = -2.

Click here to view a DPGraph presentation of the surface, the plane y = -2, and the point (1, -2, 1).

Click here to view a DPGraph presentation of the surface, the plane y = -2, and the point (1, -2, 1) that zooms in on the point. Here we zoom in even more.

Click Picture To Enlarge Click Here To Zoom

Section 13.5 # 42

Section 13.5 #55 Chain Rule Application

DPGraph Picture 1 DPGraph Picture 2

Chain Rule Application

DPGraph Picture of the expanding cylinder

The radius of a cylinder is increasing at the rate of 2 meters/sec and the height is increasing at the rate of 3 meters/sec. How fast is the volume of the cylinder increasing when the radius is 8 meters and the height is 11 meters?

Chain Rule Application

Here is a DPGraph Picture of the expanding and contracting frustum.

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu