Chapter 2: Differentiation
Section 2.4: The Chain Rule
Example 2.4.8
If C is the ellipse defined parametrically by x=uθ=5 cosθ,y=vθ=3 sinθ,0≤θ≤2 π, graph C and obtain the slope of the line tangent to C at the point corresponding to θ=π/6.
Solution
Define uθ and vθ
Write the equation uθ=… Context Panel: Assign Function
uθ=5 cosθ→assign as functionu
Write the equation vθ=… Context Panel: Assign Function
vθ=3 sinθ→assign as functionv
Graph C
Type the list as shown to the right, and press the Enter key.
Context Panel: Plots≻Plot Builder≻2-d plot (parametric) Set: 0≤θ≤2 π 2-D Options: scaling →constrained
uθ,vθ
5cosθ,3sinθ
→
Obtain the slope
Form the ratio of derivatives. Context Panel: Evaluate and Display Inline
Context Panel: Evaluate at a Point≻θ=π/6
v′θu′θ = −35cosθsinθ→evaluate at point−353
Corroboration
Determine x,y at θ=π/6
Write the list of two function-values. Context Panel: Evaluate and Display Inline
uπ/6,vπ/6 = 523,32
Eliminate θ to obtain an implicit representation of the top of the ellipse
Write the sequence of equations x=uθ,y=vθ Press the Enter key.
Context Panel: Solve≻Solve for Variables≻θ,y
x=uθ,y=vθ
x=5cosθ,y=3sinθ
→solve (specified)
y=3525−x2,θ=arccos15x
Obtain y^′x and evaluate it at x=uπ/6=53/2
Control-drag y^x, the right-hand side of y=… Press the Enter key.
Context Panel: Differentiate≻With Respect To≻x
Context Panel: Evaluate at a Point≻x=523
Context Panel: Simplify≻Simplify
3525−x2
→differentiate w.r.t. x
−35x25−x2
→evaluate at point
−3502543
= simplify
−353
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