MATH2263-Project 1 Solved

1           Part 1: Traces
The cone and paraboloid are two mathematical surfaces with distinct shapes that, at first glance, may appear to be similar. A cursory examination of both Figure 1 and Figure 2 reveals that both shapes exhibit an almost circular-like distribution. However, upon delving deeper into this project, I will demonstrate how significantly different these shapes are.

 
Upon observing the contour maps of the cone and paraboloid, a notable disparity in their growth patterns is immediately discernible. Specifically, the paraboloid exhibits a rate of growth that could be described as approaching an exponential function, evidenced by the discrepancy in growth between z = 4 and z = 3 when compared to the difference in growth between z = 2 and z = 3.

In contrast, the growth of the cone appears to be more uniform across its dimensions. This observation could be attributed to the paraboloid’s underlying quadratic equation, which imparts a degree of curvature and accentuates the rate of growth in the z direction. Such distinctions are critical to understanding the properties and applications of these shapes in various fields, including physics, engineering, and mathematics.

2           Part 2: Directional Derivatives
I began by calculating the directional derivatives and magnitude of both shapes:

2.1           Step 1
Find the Gradient Vector and Magnitude for z = x2+ y2:

 

Plugging in the point (1,0,1), we have:

 

Gradient Vector:  

 

√ 

Magnitude: = 5

2.2           Step 2
 

Find the Gradient Vector and Magnitude for z = px2+ y2:

 


Plugging in the point (1,0,1), we have:

 

Gradient Vector:  

 

√ 

Magnitude: = 2

2.3           Final Answer
  Upon analyzing the rate of change of the surface areas of a paraboloid and a cone, it is evident that the paraboloid has the highest rate of increase, with a√    √

value of 5, while the cone has a rate of increase of 2. The computation of the greatest increase highlights the substantial difference between the two shapes, revealing that the paraboloid increases at a much faster rate than the cone.

Furthermore, observing the contour maps for both shapes provides a graphical representation of their rate of change. From the contour map, it is clear that the paraboloid has a much larger increase as compared to the cone, while the latter seems to increase at a relatively constant rate.

3           Part 3: Tangent Places
 Suppose that f has a continuous partial derivative. An equation of the tangent plane to the surface. An equation of the tangent plane to the surface z = f(x,y) at the point P(x0,y0,z0) is given by:

3.1           Step 1
Find the Tangent Planes on P(0,0,0) for z = x2+ y2

 

Plugging in x = 0 and y = 0, we get:

z −0 = 2x(0,0)(x −0)+2y(0,0)(y −0)

so the equation of the tangent plane is:

z = 0

 

3.2           Step 2
 

Find the Tangent Planes on P(0,0,0) for Z = px2+ y2

 ,

Plugging in x = 0 and y = 0, we get:

 

so the equation of the tangent plane is:

z = 0.

 

3.3           Step 3
 

3D image of both Z = px2+ y2 and Z = x2+ y2

To provide a comprehensive understanding of the cone and paraboloid shapes, I decided to utilize the powerful visualization tool, GeoGebra 3D. This tool allows for an interactive 3D representation of the shapes, which provides a better insight into their structure and properties.

3.4           Final Answer
Therefore, in our effort to derive the equation of the tangent plane to the surface z = f(x,y) at the point P(0,0,0), we have obtained z = 0 as the equation of the tangent plane. This result indicates that the tangent line at the origin lies on the x-y plane. This finding can also be clearly observed in the accompanying graphs (figure 1 & 2), where the point P(0,0,0) is seen to be untouched by the surface.