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Use a graphing utility to determine the first three points with 020 at which the spiral r = 60 has a horizontal tangent line. Find the first three points with 020 at which the spiral r = 60 has a vertical tangent line. The first point (r,0), with 0≥0, at which the spiral has a horizontal tangent line is (0,0). (Type an ordered pair. Round to one decimal place as needed.) The second point (r,0), with 0≥0, at which the spiral has a horizontal tangent line is.

Use a graphing utility to determine the first three points with 020 at which the spiral r = 60 has a horizontal tangent line. Find the first three points with 020 at which the spiral r = 60 has a vertical tangent line. The first point (r,0), with 0≥0, at which the spiral has a horizontal tangent line is (0,0). (Type an ordered pair. Round to one decimal place as needed.) The second point (r,0), with 0≥0, at which the spiral has a horizontal tangent line is.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please find the third point too!

Use a graphing utility to determine the first three points with 020 at which the spiral r = 60 has a horizontal tangent line. Find the first three points with 020 at which the spiral r = 60 has a vertical tangent line. The first point (r,0), with 0≥0, at which the spiral has a horizontal tangent line is (0,0). (Type an ordered pair. Round to one decimal place as needed.) The second point (r,0), with 0≥0, at which the spiral has a horizontal tangent line is.
Transcribed Image Text:Use a graphing utility to determine the first three points with 020 at which the spiral r = 60 has a horizontal tangent line. Find the first three points with 020 at which the spiral r = 60 has a vertical tangent line. The first point (r,0), with 0≥0, at which the spiral has a horizontal tangent line is (0,0). (Type an ordered pair. Round to one decimal place as needed.) The second point (r,0), with 0≥0, at which the spiral has a horizontal tangent line is.
Expert Solution
Step 1: Equation of tamgent in polar

Let the equation of a curve be r=f(θ).

Relation between Cartesian and Polar coordinates:

x=rcos⁡θ ; y=rsin⁡θ

Then the slope of a tangent m=dydx=dydθdxdθ=r′sin⁡θ+rcos⁡θr′cos⁡θ−rsin⁡θ

For horizontal tangent

r′sin⁡θ+rcos⁡θ=0

and for vertical tangent 

r′cos⁡θ−rsin⁡θ=0

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