Parametric derivative - Revision history

Vipul: /* Limiting behavior on undefined cases */

2021-09-27T14:46:20Z

Limiting behavior on undefined cases

← Older revision		Revision as of 14:46, 27 September 2021
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	When <math>t = \pi /2</math> or <math>t = 3\pi / 2</math>, the parametric derivative does not have a defined limit, with the left-hand and right-hand limits being opposite signs of infinity. Geometrically, we have vertical cusps at these points.		When <math>t = \pi /2</math> or <math>t = 3\pi / 2</math>, the parametric derivative does not have a defined limit, with the left-hand and right-hand limits being opposite signs of infinity. Geometrically, we have vertical cusps at these points.

	NOTE: In general, when the parametric derivative is not defined at a point, but there ''is'' a limit for it at that point, then the ordinary derivative, if defined, ''must'' equal that value. Moreover, if the limit is a nonzero value, then the ordinary derivative ''is'' defined and equals that value. However, if the limit is zero, then we ''don't know'' offhand whether the ordinary derivative exists; further work is needed to determine that.		NOTE: In general, when the parametric derivative is not defined at a point, but there ''is'' a limit for it at that point, then the ordinary derivative, if defined, ''must'' equal that value.

Vipul: /* The parametric derivative makes sense and is the correct expression when it is defined */

2021-09-25T02:19:52Z

The parametric derivative makes sense and is the correct expression when it is defined

← Older revision		Revision as of 02:19, 25 September 2021
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	=== The parametric derivative makes sense and is the correct expression when it is defined ===		=== The parametric derivative makes sense and is the correct expression when it is defined ===

	We have now come full circle. In the case that <math>f'(~~t_0~~) \ne 0</math>, <math>y</math> is locally a function of <math>x</math> around <math>t_0</math>. In such cases, if <math>g'(t_0)</math> also exists, the expression <math>\frac{g'(t_0)}{f'(t_0)}</math> gives the parametric derivative.		We have now come full circle. In the case that <math>f'(t) \ne 0</math> around <math>t_0</math>, <math>y</math> is locally a function of <math>x</math> around <math>t_0</math>. In such cases, if <math>g'(t_0)</math> also exists, the expression <math>\frac{g'(t_0)}{f'(t_0)}</math> gives the parametric derivative.

	The proof that the parametric derivative expression is correct follows from the [[chain rule for differentiation]]. As established above, <math>y</math> is locally a function of <math>x</math> around <math>t_0</math>. Let's call this local function <math>h</math>. Locally around <math>t_0</math>, we have:		The proof that the parametric derivative expression is correct follows from the [[chain rule for differentiation]]. As established above, <math>y</math> is locally a function of <math>x</math> around <math>t_0</math>. Let's call this local function <math>h</math>. Locally around <math>t_0</math>, we have:

Vipul: /* If f'(t_0) \ne 0, the parametric expression defines a function locally around t_0 */

2021-09-25T02:19:23Z

If f'(t_0) \ne 0, the parametric expression defines a function locally around t_0

← Older revision		Revision as of 02:19, 25 September 2021
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	In the circle example, the only points at which <math>y</math> is locally not a function of <math>x</math> are the ones at the far left and far right: <math>x = 1</math> and <math>x = -1</math> (ironically, these are the only points with unique ''global'' <math>y</math>-values).		In the circle example, the only points at which <math>y</math> is locally not a function of <math>x</math> are the ones at the far left and far right: <math>x = 1</math> and <math>x = -1</math> (ironically, these are the only points with unique ''global'' <math>y</math>-values).

	=== If <math>f'(t_0~~) \ne 0~~</math>, the parametric expression defines a function locally around <math>t_0</math> ===		=== If <math>f'(t)</math> is nonzero (and of constant sign) for <math>t</math> around <math>t_0</math>, the parametric expression defines a function locally around <math>t_0</math> ===

	If <math>f'(~~t_0~~)</math> is nonzero~~, then (depending on the~~ sign ~~of <math>f'</math>), for a small enough open interval~~ around <math>t_0</math>, <math>f</math> is an increasing or decreasing function. In particular, within that open interval, <math>f</math> is a one-one function, so no value of <math>x</math> repeats. Thus locally <math>y</math> is a function of <math>x</math> on that interval.		If <math>f'(t)</math> is nonzero and of constant sign around <math>t_0</math>, then (depending on the sign)<math>f</math> is an increasing or decreasing function around <mth>t_0</math>. In particular, within that open interval, <math>f</math> is a one-one function, so no value of <math>x</math> repeats. Thus locally <math>y</math> is a function of <math>x</math> on that interval.

			NOTE: To deduce increasing/decreasing behavior, it is '''not''' enough to assume that <math>f'(t_0) \ne 0</math>. For a counterexample, see [[positive derivative at a point not implies increasing around the point]].

			NOTE 2: The assumption of constant sign is not necessary; it can be deduced from the derivative being nonzero. That's because [[derivative of differentiable function satisfies intermediate value property]].

	=== The parametric derivative makes sense and is the correct expression when it is defined ===		=== The parametric derivative makes sense and is the correct expression when it is defined ===

Vipul: /* Algebraic definition */

2021-09-25T02:15:10Z

Algebraic definition

← Older revision		Revision as of 02:15, 25 September 2021
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	===Algebraic definition===		===Algebraic definition===

	The '''parametric derivative''' <math>dy/dx</math> for a parametric curve <math>x = f(t), y = g(t)</math> at a point <math>t = t_0</math> is given as follows, where <math>f'(t_0)</math> and <math>g'(t_0)</math> both exist and <math>f'(t_0~~) \ne 0~~</math>:		The '''parametric derivative''' <math>dy/dx</math> for a parametric curve <math>x = f(t), y = g(t)</math> at a point <math>t = t_0</math> is given as follows, where <math>f'(t_0)</math> and <math>g'(t_0)</math> both exist and <math>f'(t)</math> is nonzero and of constant sign for an open interval around <math>t_0</math>:

	<math>\! \frac{dy}{dx} \|_{t = t_0} = \frac{g'(t_0)}{f'(t_0)}</math>		<math>\! \frac{dy}{dx} \|_{t = t_0} = \frac{g'(t_0)}{f'(t_0)}</math>

Vipul: /* Algebraic definition */

2021-09-25T01:09:23Z

Algebraic definition

← Older revision		Revision as of 01:09, 25 September 2021
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	As a general function of <math>t</math>, the parametric derivative <math>dy/dx</math> is defined as <math>g'(t)/f'(t)</math>.		As a general function of <math>t</math>, the parametric derivative <math>dy/dx</math> is defined as <math>g'(t)/f'(t)</math>.

	'''NOTE''': When calculating the general expression for the parametric derivative, before canceling any factors between <math>g'(t)</math> and <math>f'(t)</math>, it is important to separate out the cases where that common value is zero. For any points where <math>f'(t) = 0</math>, the parametric derivative is not defined (the ordinary derivative ''may'' still be defined, but we would need another method to calculate it).		'''NOTE''': When calculating the general expression for the parametric derivative, before canceling any factors between <math>g'(t)</math> and <math>f'(t)</math>, it is important to separate out the cases where that common value is zero. For any points where <math>f'(t) = 0</math>, the parametric derivative is not defined (the [[derivative\|ordinary derivative]] ''may'' still be defined, but we would need another method to calculate it).

	{{generic point specific point confusion}}		{{generic point specific point confusion}}

Vipul: /* Limiting behavior on undefined cases */

2021-09-25T01:08:15Z

Limiting behavior on undefined cases

← Older revision		Revision as of 01:08, 25 September 2021
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	==== Limiting behavior on undefined cases ====		==== Limiting behavior on undefined cases ====

	When <math>t = 0</math> or <math>t = \pi</math>, the limit of the parametric derivative expression (<math>- \tan t</math>) equals zero. However, despite the limit existing, the parametric derivative is undefined at the point. In fact, the parametric curve is ''not'' locally a function at these points: geometrically, the <math>x<math>-coordinate doubles back, forming a horizontal cusp.		When <math>t = 0</math> or <math>t = \pi</math>, the limit of the parametric derivative expression (<math>- \tan t</math>) equals zero. However, despite the limit existing, the parametric derivative is undefined at the point. In fact, the parametric curve is ''not'' locally a function at these points: geometrically, the <math>x</math>-coordinate doubles back, forming a horizontal cusp.

	When <math>t = \pi /2</math> or <math>t = 3\pi / 2</math>, the parametric derivative does not have a defined limit, with the left-hand and right-hand limits being opposite signs of infinity. Geometrically, we have vertical cusps at these points.		When <math>t = \pi /2</math> or <math>t = 3\pi / 2</math>, the parametric derivative does not have a defined limit, with the left-hand and right-hand limits being opposite signs of infinity. Geometrically, we have vertical cusps at these points.

	NOTE: In general, when the parametric derivative is not defined at a point, but there ''is'' a limit for it at that point, then the ordinary derivative, if defined, ''must'' equal that value. Moreover, if the limit is a nonzero value, then the ordinary derivative ''is'' defined and equals that value. However, if the limit is zero, then we ''don't know'' offhand whether the ordinary derivative exists; further work is needed to determine that.		NOTE: In general, when the parametric derivative is not defined at a point, but there ''is'' a limit for it at that point, then the ordinary derivative, if defined, ''must'' equal that value. Moreover, if the limit is a nonzero value, then the ordinary derivative ''is'' defined and equals that value. However, if the limit is zero, then we ''don't know'' offhand whether the ordinary derivative exists; further work is needed to determine that.

Vipul: /* Relation with ordinary derivative */

2021-09-25T01:08:03Z

Relation with ordinary derivative

← Older revision		Revision as of 01:08, 25 September 2021
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	The left side is the expression we are trying to calculate, and the right side is the expression that we want to prove it to be.		The left side is the expression we are trying to calculate, and the right side is the expression that we want to prove it to be.

			=== When the parametric derivative is undefined, the ordinary derivative may or may not be defined ===

			Consider the example of the [[astroid]] as a parametric curve:

			<math>x = f(t) = \cos^3t, y = g(t) = \sin^3t</math>

			The derivatives are as follows:

			<math>f'(t) = -3\cos^2t \sin t, g'(t) = 3\sin^2t \cos t</math>

			The parametric derivative is therefore:

			<math>\frac{3\sin^2 t \cos t}{-3\cos^2 t \sin t}</math>

			We can cancel the 3:

			<math>\frac{\sin^2 t \cos t}{-\cos^2 t \sin t}</math>

			However, we should ''not'' cancel the <math>\sin t</math> and <math>\cos t</math> until we have isolated the cases where either of them is zero.

			==== Cases of undefined parametric derivative ====

			The cases where the derivative is undefined correspond precisely to the cases where <math>\cos t = 0</math> or <math>\sin t = 0</math>, which correspond precisely to the integer multiples of <math>\pi / 2</math>. At these points, the parametric derivative is ''not'' defined. The actual point coordinates are (1,0), (0,1), (-1,0), and (0,-1).

			==== Cases of defined parametric derivative ====

			When the parametric derivative is defined, we can cancel the common factors from the numerator and denominator. After simplifying, we get:

			<math>\frac{\sin^2 t \cos t}{-\cos^2 t \sin t} = \frac{\sin t}{-\cos t} = - \tan t</math>

			==== Limiting behavior on undefined cases ====

			When <math>t = 0</math> or <math>t = \pi</math>, the limit of the parametric derivative expression (<math>- \tan t</math>) equals zero. However, despite the limit existing, the parametric derivative is undefined at the point. In fact, the parametric curve is ''not'' locally a function at these points: geometrically, the <math>x<math>-coordinate doubles back, forming a horizontal cusp.

			When <math>t = \pi /2</math> or <math>t = 3\pi / 2</math>, the parametric derivative does not have a defined limit, with the left-hand and right-hand limits being opposite signs of infinity. Geometrically, we have vertical cusps at these points.

			NOTE: In general, when the parametric derivative is not defined at a point, but there ''is'' a limit for it at that point, then the ordinary derivative, if defined, ''must'' equal that value. Moreover, if the limit is a nonzero value, then the ordinary derivative ''is'' defined and equals that value. However, if the limit is zero, then we ''don't know'' offhand whether the ordinary derivative exists; further work is needed to determine that.

Vipul: /* The parametric derivative makes sense when it is defined */

2021-09-25T00:53:54Z

The parametric derivative makes sense when it is defined

← Older revision		Revision as of 00:53, 25 September 2021
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	If <math>f'(t_0)</math> is nonzero, then (depending on the sign of <math>f'</math>), for a small enough open interval around <math>t_0</math>, <math>f</math> is an increasing or decreasing function. In particular, within that open interval, <math>f</math> is a one-one function, so no value of <math>x</math> repeats. Thus locally <math>y</math> is a function of <math>x</math> on that interval.		If <math>f'(t_0)</math> is nonzero, then (depending on the sign of <math>f'</math>), for a small enough open interval around <math>t_0</math>, <math>f</math> is an increasing or decreasing function. In particular, within that open interval, <math>f</math> is a one-one function, so no value of <math>x</math> repeats. Thus locally <math>y</math> is a function of <math>x</math> on that interval.

	=== The parametric derivative makes sense when it is defined ===		=== The parametric derivative makes sense and is the correct expression when it is defined ===

	We have now come full circle. In the case that <math>f'(t_0) \ne 0</math>, <math>y</math> is locally a function of <math>x</math> around <math>t_0</math>. In such cases, if <math>g'(t_0)</math> also exists, the expression <math>\frac{g'(t_0)}{f'(t_0)}</math> gives the parametric derivative.		We have now come full circle. In the case that <math>f'(t_0) \ne 0</math>, <math>y</math> is locally a function of <math>x</math> around <math>t_0</math>. In such cases, if <math>g'(t_0)</math> also exists, the expression <math>\frac{g'(t_0)}{f'(t_0)}</math> gives the parametric derivative.

Vipul: /* Algebraic definition */

2021-09-25T00:52:55Z

Algebraic definition

← Older revision		Revision as of 00:52, 25 September 2021
Line 3:		Line 3:
	===Algebraic definition===		===Algebraic definition===

	The '''parametric derivative''' <math>dy/dx</math> for a parametric curve <math>x = f(t), y = g(t)</math> at a point <math>t = t_0</math> is given as:		The '''parametric derivative''' <math>dy/dx</math> for a parametric curve <math>x = f(t), y = g(t)</math> at a point <math>t = t_0</math> is given as follows, where <math>f'(t_0)</math> and <math>g'(t_0)</math> both exist and <math>f'(t_0) \ne 0</math>:

	<math>\! \frac{dy}{dx} \|_{t = t_0} = \frac{g'(t_0)}{f'(t_0)}</math>		<math>\! \frac{dy}{dx} \|_{t = t_0} = \frac{g'(t_0)}{f'(t_0)}</math>

Vipul: /* If f'(t_0) \ne 0, the parametric expression defines a function locally around t_0 */

2021-09-25T00:52:14Z

If f'(t_0) \ne 0, the parametric expression defines a function locally around t_0

← Older revision		Revision as of 00:52, 25 September 2021
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	=== If <math>f'(t_0) \ne 0</math>, the parametric expression defines a function locally around <math>t_0</math> ===		=== If <math>f'(t_0) \ne 0</math>, the parametric expression defines a function locally around <math>t_0</math> ===

			If <math>f'(t_0)</math> is nonzero, then (depending on the sign of <math>f'</math>), for a small enough open interval around <math>t_0</math>, <math>f</math> is an increasing or decreasing function. In particular, within that open interval, <math>f</math> is a one-one function, so no value of <math>x</math> repeats. Thus locally <math>y</math> is a function of <math>x</math> on that interval.

			=== The parametric derivative makes sense when it is defined ===

			We have now come full circle. In the case that <math>f'(t_0) \ne 0</math>, <math>y</math> is locally a function of <math>x</math> around <math>t_0</math>. In such cases, if <math>g'(t_0)</math> also exists, the expression <math>\frac{g'(t_0)}{f'(t_0)}</math> gives the parametric derivative.

			The proof that the parametric derivative expression is correct follows from the [[chain rule for differentiation]]. As established above, <math>y</math> is locally a function of <math>x</math> around <math>t_0</math>. Let's call this local function <math>h</math>. Locally around <math>t_0</math>, we have:

			<math>g = h \circ f</math>

			By the [[chain rule for differentiation]]:

			<math>g'(t) = h'(f(t))f'(t)</math>

			At <math>t = t_0</math>, we get:

			<math>g'(t_0) = h'(f(t_0))f'(t_0)</math>

			Rearranging, we get:

			<math>h'(f(t_0)) = \frac{g'(t_0)}{f'(t_0)}</math>

			The left side is the expression we are trying to calculate, and the right side is the expression that we want to prove it to be.