Chapter 2: Space Curves
Section 2.3: Tangent Vectors
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Example 2.3.7
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Given the plane curve defined by ,
a)
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Obtain , the radius-vector form of the curve, by the parametrization .
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b)
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Obtain and , where and .
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c)
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Graph and the vectors and .
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d)
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Graph , and determine the point at which is a maximum in this interval.
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Solution
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Mathematical Solution
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If , then , = ,
, , and .
Figure 2.3.7(a) displays a graph of the curve , the vector (green arrow), and the vector (black arrow). Figure 2.3.7(b) shows a graph of , from which is inferred the existence of a single maximum in the interval .
>
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use plots, Student:-VectorCalculus in
module()
local p1,p2,p3,R,T,R1,T1;
R:=<p,(-p^2+1)/(p^2+1)>;
T:=TangentVector(R,p,normalized);
R1:=eval(R,p=1);
T1:=RootedVector(root=[1,0],<1,-1>/sqrt(2));
p1:=SpaceCurve(R,p=0..3,caption="");
p2:=PlotVector([R1,T1],color=[green,black]);
p3:=display(p1,p2,scaling=constrained,labels=[x,y],tickmarks=[3,3],view =[0..3,-1..1]);
print(p3);
end module:
end use:
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Figure 2.3.7(a) ; (green), (black)
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Figure 2.3.7(b) Graph of
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The critical values for are found by solving the equation
and obtaining , in which case determines the point .
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Maple Solution - Interactive
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Initialize
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Tools≻Load Package: Student Vector Calculus
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Loading Student:-MultivariateCalculus
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Part (a)
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Context Panel: Assign Name
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Part (b)
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Calculus palette: Differentiation operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Simplify
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Context Panel: Assign to a Name≻dR
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Keyboard the norm bars.
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Assuming Positive
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Context Panel: Assign to a Name≻rho
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻T
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Part (c)
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For and :
Expression palette: Evaluation template
Context Panel: Evaluate and Display Inline
Context Panel: Plots≻Arrow from origin (for ) and Arrow from point≻ (for )
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For :
Write R
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To List
Context Panel: Plots≻Plot Builder
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Copy and paste the arrows onto the graph of
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Part (d)
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Write .
Context Panel: Plots≻Plot Builder
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Write and press the Enter key.
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Context Panel: Differentiate≻With Respect To≻
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Context Panel: Conversions≻Equate to 0
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Context Panel: Solve≻Solve
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Expression palette: Evaluation template
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Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻
Conversions≻To List
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Maple Solution - Coded
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Initialize
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Install the Student VectorCalculus package.
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Part (a)
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Define as the position vector R.
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Part (b)
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Part (c)
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Use the eval command to make the substitution in each of R and T.
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Use the ConvertVector command to convert to a rooted vector.
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To graph the vectors and , use the PlotVector command.
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Use the display command (plots package) to join the graph of with the graph of the vectors.
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Part (d)
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Apply the solve and diff commands to obtain the critical values for .
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Apply the eval command to R to obtain the extreme point as a vector.
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Apply the convert command (with option list) to change the column vector to a list.
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