Consider the polar function r(0) = 2+ 2 cos(20), for 0 € [0, 27). (a) Write parametric equations for the polar curve r r(0), using 0 E [0, 27) as a parameter. 4. dy (b) Find a general expression for dx 2(x), the (implicit) derivative of y with respect to x, as a function of 0. (c) Find the tangent line to the curve at the point corresponding to 0 =

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Chapter2: Second-order Linear Odes
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4.
Consider the polar function r(0) = 2 + 2 cos(20), for 0 E [0, 27).
(a) Write parametric equations for the polar curve r = r
(0), using 0 E [0, 27) as a parameter.
dy
(b) Find a general expression for
(x), the (implicit) derivative of y with respect to x, as a function
dx
of 0.
(c) Find the tangent line to the curve at the point corresponding to 0
||
(d) For which values of 0 does the curve have a horizontal tangent line and for which values of 0 does
the curve have a vertical tangent line? Recall that you need to give exact values (i.e., you need to
do the exercises without using a calculator). In this exercise, this means that some of your answers
might need to be given in terms of an inverse trigonometric function, such as arccos(x), arcsin(x)
etc. (Note: recall that, if, for a certain value 0o, both x'(00) and y'(00) are zero, you need to study
the limit
y'(0)
lim
0→0, x'(0)
to understand the nature of the tangent line.)
Transcribed Image Text:4. Consider the polar function r(0) = 2 + 2 cos(20), for 0 E [0, 27). (a) Write parametric equations for the polar curve r = r (0), using 0 E [0, 27) as a parameter. dy (b) Find a general expression for (x), the (implicit) derivative of y with respect to x, as a function dx of 0. (c) Find the tangent line to the curve at the point corresponding to 0 || (d) For which values of 0 does the curve have a horizontal tangent line and for which values of 0 does the curve have a vertical tangent line? Recall that you need to give exact values (i.e., you need to do the exercises without using a calculator). In this exercise, this means that some of your answers might need to be given in terms of an inverse trigonometric function, such as arccos(x), arcsin(x) etc. (Note: recall that, if, for a certain value 0o, both x'(00) and y'(00) are zero, you need to study the limit y'(0) lim 0→0, x'(0) to understand the nature of the tangent line.)
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