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Chapter 2: Space Curves
Section 2.3: Tangent Vectors
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Example 2.3.9
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Given the two plane curves ,
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a)
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At , obtain the equation of the line tangent to .
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b)
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Find the coordinates of the intersection of and the tangent line found in Part (a).
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c)
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Construct a vector from to the point found in Part (b).
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d)
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Obtain , the natural tangent vector at .
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e)
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Show that the vectors in Parts (c) and (d) are parallel.
(Hint: Show their components are proportional.)
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f)
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Draw both curves, the tangent line (Part (a)), and the tangent vector (Part (d)).
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Solution
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Mathematical Solution
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At , the line tangent to the graph of is given by
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The intersection of this tangent line with the graph of is obtained by solving the equations and for the two points and . Because of the restriction , only the second solution is accepted.
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>
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use plots, VectorCalculus in
module()
local p1,p2,p3,T;
T:=RootedVector(root=[1,1],<1,2>);
p1:=PlotVector(T);
p2:=plot([x^2,8-(x/4)^2,2*x-1],x=0..5,y=0..8,color=[red,blue,green]);
p3:=display(p1,p2,scaling=constrained);
print(p3);
end module:
end use:
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Figure 2.3.9(a) Graph of and the tangent line (green)
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If is the radius-vector form of the curve defined by the graph of , then . Clearly, this tangent vector has the same direction as since these two vectors are proportional.
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Figure 2.3.9(a) shows the graph of in red, the graph of in blue, and of the tangent line in green. The vector is represented by the black arrow.
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Maple Solution - Interactive
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Initialize
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Tools≻Load Package: Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Context Panel: Assign Function
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Context Panel: Assign Function
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Part (a)
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Implement the point-slope form of the tangent line and press the Enter key.
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Part (b)
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Write the sequence of equations to be solved and press the Enter key.
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Context Panel: Solve≻Solve
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Part (c)
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Context Panel: Assign Name
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Context Panel: Evaluate and Display Inline
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Part (d)
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Context Panel: Assign Name
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Expression palette: Evaluation template
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Calculus palette: Differentiation operator
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Context Panel: Assign to a Name≻dR1
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Part (e)
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Write V and 3 dR1, and press the Enter key.
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Since these two vectors are obviously equal, V and dR1 are proportional.
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Part (f)
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Context Panel: Evaluate and Display inline
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Context Panel: Plots: Arrow from point
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Write the sequence of three functions shown to the right.
Press the Enter key.
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Context Panel: Plots≻Plot Builder
Set
Options:
Range from 0 to 8
Constrained Scaling
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Copy and paste the tangent vector.
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Maple Solution - Coded
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Initialize
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Part (a)
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Obtain, at , the equation of the line tangent to the graph of .
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Part (b)
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Obtain the intersection of the graph of and the line tangent to the graph of .
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Part (c)
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Obtain the vector from the point to , the point of intersection of the line tangent to the graph of and the graph of .
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Part (d)
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Use the notation for the position-vector representation of the graph of . Then use the map command to apply the diff command to each component of the vector. Finally, use the eval command to evaluate the tangent vector at .
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Part (e)
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Use the Equal command from the LinearAlgebra package to show that .
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Part (f)
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