Description
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 di- mensions. This observation could be attributed to the paraboloid’s underlying quadratic equation, which imparts a degree of curvature and accentuates the
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rate of growth in the z direction. Such distinctions are critical to understand- ing 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: RecalltheGradientVectorFormula: ∇f=fx,fy,fz=∂fi+∂fj+∂fk
∂x ∂y ∂z
∂f =2x, ∂f =2y, ∂f =−1−→∇f =2x,2y,−1 ∂x ∂y ∂z
Plugging in the point (1, 0, 1), we have: ∇f(1,0,1) = 2,0,−1
Gradient Vector: 2,0,−1 √
2,0,−1 = 22 +02 +(−1)2 = 5 √
Magnitude: = 5
2.2 Step 2
Find the Gradient Vector and Magnitude for z = x2 + y2: RecalltheGradientVectorFormula: ∇f=fx,fy,fz=∂fi+∂fj+∂fk
∂x ∂y ∂z
∂f x ∂f y ∂f x y ∂x = x2 +y2, ∂y = x2 +y2, ∂z =−1−→∇f = x2 +y2,x2 +y2,−1
Plugging in the point (1, 0, 1), we have:
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2.3 Final Answer
∇f(1,0,1) = 1,0,−1 Gradient Vector: 1,0,−1
√ 1,0,−1 = 12 +02 +(−1)2 = 2
√
Magnitude: = 2
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 graph- ical 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 deriva- tive. 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:
∂ f ∂ f
z−z0 = ∂x(x0,y0)(x−x0)+ ∂y(x0,y0)(y−y0)
3.1 Step 1
Find the Tangent Planes on P (0, 0, 0) for z = x2 + y2 ∂f =2x, ∂f =2y,
∂x ∂y Plugging in x = 0 and y = 0, we get:
z − 0 = 2x(0, 0)(x − 0) + 2y(0, 0)(y − 0) 3
so the equation of the tangent plane is:
z=0
3.2 Step 2
∂x=x2+y2, ∂y=x2+y2, Plugging in x = 0 and y = 0, we get:
xy
z − 0 = x2 + y2 (0, 0)(x − 0) + x2 + y2 (0, 0)(y − 0)
so the equation of the tangent plane is:
z = 0.
3.3 Step 3
Plotting the surface and the tangent
3DimageofbothZ=x2+y2 andZ=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 al- lows for an interactive 3D representation of the shapes, which provides a better insight into their structure and properties.
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Find the Tangent Planes on P (0, 0, 0) for Z = x2 + y2 ∂f x ∂f y
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.
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