(e) Find the point(s) on C where the tangent line is vertical. Do not rely on the graph from (a) – show analytical work to support your findings. Your graph should show that part of the graph of C consists of a loop that begins and ends at (9,0). (f) Use the arc length formula to find the length of that loop. (Be careful when determining bounds for the integral.) (g) Find the area enclosed by the loop. (h) Find the area of the surface obtained by rotating the upper half of the loop about the r-axis. (i) Find a Cartesian equation of the curve C. Hint: square the equation for y and replace the expressions in t in terms of r. Discuss how difficult it would be to answer parts (f)-(h) without the given

(e) Find the point(s) on C where the tangent line is vertical. Do not rely on the graph from (a) – show analytical work to support your findings. Your graph should show that part of the graph of C consists of a loop that begins and ends at (9,0). (f) Use the arc length formula to find the length of that loop. (Be careful when determining bounds for the integral.) (g) Find the area enclosed by the loop. (h) Find the area of the surface obtained by rotating the upper half of the loop about the r-axis. (i) Find a Cartesian equation of the curve C. Hint: square the equation for y and replace the expressions in t in terms of r. Discuss how difficult it would be to answer parts (f)-(h) without the given

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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Concept explainers

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.

Question

Need help with parts e - i.

(e) Find the point(s) on C where the tangent line is vertical. Do not rely on the graph from (a) – show
analytical work to support your findings.
Your graph should show that part of the graph of C consists of a loop that begins and ends at (9,0).
(f) Use the arc length formula to find the length of that loop. (Be careful when determining bounds
for the integral.)
(g) Find the area enclosed by the loop.
(h) Find the area of the surface obtained by rotating the upper half of the loop about the r-axis.
(i) Find a Cartesian equation of the curve C. Hint: square the equation for y and replace the expressions
in t in terms of x. Discuss how difficult it would be to answer parts (f)-(h) without the given
parametrization of the curve C.

11. Let C be the curve with parametrization a =
3t2, y = 3t – t³.
(a) Use a graphing calculator or some computer graphing program (Desmos, wolframalpha.com, etc.)
to sketch the portion of C where -2 <t< 2.
dy
(b) Find
dr
(c) Find the equation(s) of the tangent line(s) to the graph at the point (9,0). (Hint: looking at the
graph from part (a), how many should there be?)
(d) Find the point(s) on C where the tangent line is horizontal. Do not rely on the graph from (a)
show analytical work to support your findings.

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