Parametrize the sine curve y = sinx using the parametrization r(t)=(t,sint), t ∈ R.(a) show that this curve is smooth.(b) Compute the curvature, and find all points where the curvature is zero. What geometric property do all those points share?(c) Without doing any computations, explain why the torsion of this curve must be identically zero.(d) Rotating the plane 30◦ is an isometry, which transforms the original sine curve into the curve parametrized by r(t) = √3t −sint 2 , t + √3 sint 2 , t ∈ R. Compute the speed, curvature, and torsion of this curve, and compare them to those of the original curve.(e) By graphing the curve, decide if it is the graph of some function y = f(x).(f) In single-variable calculus, you studied the qualitative properties of the graphs of functions y = f(x). In particular, you characterized the maxima, minima, and inflection points in terms of the vanishing of certain derivatives of the function f(x). Using the earlier parts of this problem to supply examples, write a paragraph explaining why the notion of an inflection point is an intrinsic geometric property of the curve, but the notion of a maximum or minimum point is not. Problem 3.4. Parametrize the sine curve y= sinx using the parametrizationr(t) (r, sinr),ItER.(a) Using the definition in this chapter, show that this curve is smooth.(b) Compute the curvature, and find all points where the curvature is zero. Whatgeometric property do all those points share?(c) Without doing any computations, explain why the torsion of this curve mustbe identically zero.(d) Rotating the plane 30° is an isometry, which transforms the original sinecurve into the curve parametrized byV31-sint t-+V3 sintr(r) –TER.2.2.Compute the speed, curvature, and torsion of this curve, and compare them tothose of the original curve.(e) By graphing the curve, decide if it is the graph of some function y= f(x).(f) In single variable calculus, you studicd the qualitative properties of thegraphs of functions y /(x), In particular, you characterized the maxima, min-ima, and inflection points in terms of the vanishing of certain derivatives of thefunction f(x). Using the carlier parts of this problem to supply examples. writea paragraph explaining why the notion of an inflection point is an intrinsic geo-metric property of the curve, but the notion of a maximumorminimum point isnot

Answered: Parametrize the sine curve y = sinx…

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Parametrize the sine curve y = sinx using the parametrization r(t)=(t,sint), t ∈ R.

(a) show that this curve is smooth.

(b) Compute the curvature, and find all points where the curvature is zero. What geometric property do all those points share?

(d) Rotating the plane 30◦ is an isometry, which transforms the original sine curve into the curve parametrized by r(t) = √3t −sint 2 , t + √3 sint 2 , t ∈ R. Compute the speed, curvature, and torsion of this curve, and compare them to those of the original curve.

(e) By graphing the curve, decide if it is the graph of some function y = f(x).

(f) In single-variable calculus, you studied the qualitative properties of the graphs of functions y = f(x). In particular, you characterized the maxima, minima, and inflection points in terms of the vanishing of certain derivatives of the function f(x). Using the earlier parts of this problem to supply examples, write a paragraph explaining why the notion of an inflection point is an intrinsic geometric property of the curve, but the notion of a maximum or minimum point is not.

Problem 3.4. Parametrize the sine curve y= sinx using the parametrization
r(t) (r, sinr),
I
tER.
(a) Using the definition in this chapter, show that this curve is smooth.
(b) Compute the curvature, and find all points where the curvature is zero. What
geometric property do all those points share?
(c) Without doing any computations, explain why the torsion of this curve must
be identically zero.
(d) Rotating the plane 30° is an isometry, which transforms the original sine
curve into the curve parametrized by
V31-sint t-+V3 sint
r(r) –
TER.
2.
2.
Compute the speed, curvature, and torsion of this curve, and compare them to
those of the original curve.
(e) By graphing the curve, decide if it is the graph of some function y= f(x).
(f) In single variable calculus, you studicd the qualitative properties of the
graphs of functions y /(x), In particular, you characterized the maxima, min-
ima, and inflection points in terms of the vanishing of certain derivatives of the
function f(x). Using the carlier parts of this problem to supply examples. write
a paragraph explaining why the notion of an inflection point is an intrinsic geo-
metric property of the curve, but the notion of a maximumorminimum point is
not

Expert Solution

Step 1: Showing the curve is smooth

“Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts for you. To get the remaining sub-part solved please repost the complete question and mention the sub-parts to be solved.”

Given the sine curve $y = \sin x$ parametrized as $r (t) = (t, \sin t), t \in R$ .

(a)

For a parametrized curve $r (t) = (f (t), g (t))$ is smooth if $f^{'}$ and $g^{'}$ are continuous and not simultaneously zero.

In our case, $r apostrophe open parentheses t close parentheses equals open parentheses 1 comma cos t close parentheses comma space t element of straight real numbers$ .

Both components of the derivative is continuous and is not simultaneously zero since $f^{'} (t) = 1$ which is non-zero for any $t$ .

Therefore, the curve is smooth.

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