Differentiation rule for piecewise definition by interval - Revision history

Vipul at 20:49, 20 September 2021

2021-09-20T20:49:48Z

← Older revision		Revision as of 20:49, 20 September 2021
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	where <math>u = f_1'(c) = f_2'(c)</math>. In particular, the value at <math>c</math> can be included in either the left side or the right side definition.		where <math>u = f_1'(c) = f_2'(c)</math>. In particular, the value at <math>c</math> can be included in either the left side or the right side definition.

	~~<center>{{#widget:YouTube\|id=TqVuosJMdOU}}</center>~~

	===Version for higher derivatives===		===Version for higher derivatives===
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	The above holds with the following modification: we only require <math>f</math> to be defined as <math>f_1</math> on the ''immediate'' left of <math>c</math> (i.e., on some interval of the form <math>(c - \delta,c)</math> for <math>\delta > 0</math> and as <math>f_2</math> on the ''immediate'' right of <math>c</math> (i.e., on some interval of the form <math>(c,c + \delta)</math> for <math>\delta > 0</math>). Further, we only require that <math>f_1</math> and <math>f_2</math> be defined and differentiable on [[open interval]]s containing <math>c</math>, not necessarily on all of <math>\R</math>.		The above holds with the following modification: we only require <math>f</math> to be defined as <math>f_1</math> on the ''immediate'' left of <math>c</math> (i.e., on some interval of the form <math>(c - \delta,c)</math> for <math>\delta > 0</math> and as <math>f_2</math> on the ''immediate'' right of <math>c</math> (i.e., on some interval of the form <math>(c,c + \delta)</math> for <math>\delta > 0</math>). Further, we only require that <math>f_1</math> and <math>f_2</math> be defined and differentiable on [[open interval]]s containing <math>c</math>, not necessarily on all of <math>\R</math>.

	~~<center>{{#widget:YouTube\|id=aURWLFjy5Qo}}</center>~~

	==Examples==		==Examples==
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	In situations where the definitions given on one side of a point ''do not'' extend naturally to the point, we ''cannot'' use the above methods. In most such cases, we need to go back to the original definition of the derivative as a limit of a difference quotient.		In situations where the definitions given on one side of a point ''do not'' extend naturally to the point, we ''cannot'' use the above methods. In most such cases, we need to go back to the original definition of the derivative as a limit of a difference quotient.

	~~<center>{{#widget:YouTube\|id=GuYPBvWZ5uI}}</center>~~

Vipul: /* Example of piecewise rational function */

2015-05-16T16:30:35Z

Example of piecewise rational function

← Older revision		Revision as of 16:30, 16 May 2015
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	Consider the function:		Consider the function:

	<math>f(x) := \lbrace \begin{array}{rl} \frac{1}{x - 1}, & x < 0 \\ \frac{-1}{x + 1}, & x > 0 \\ -1, & x = 0 \\\end{array}</math>		<math>f(x) := \left\lbrace \begin{array}{rl} \frac{1}{x - 1}, & x < 0 \\ \frac{-1}{x + 1}, & x > 0 \\ -1, & x = 0 \\\end{array}\right.</math>

	Note that here, in the notation we have used, we have:		Note that here, in the notation we have used, we have:

Vipul: /* Example of piecewise polynomial function: higher derivatives */

2012-04-25T02:39:47Z

Example of piecewise polynomial function: higher derivatives

← Older revision		Revision as of 02:39, 25 April 2012
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	<math>f''(x) = \left\lbrace\begin{array}{rl} 2, & x < 0 \\ 6x + 2, & x > 0 \\ 2, & x = 0 \\\end{array}\right.</math>		<math>f''(x) = \left\lbrace\begin{array}{rl} 2, & x < 0 \\ 6x + 2, & x > 0 \\ 2, & x = 0 \\\end{array}\right.</math>

	<math>f'''(x) = \left\lbrace\begin{array}{rl} 0, & x < 0 \\ 6, & x > 0 \\\end{array}</math>		<math>f'''(x) = \left\lbrace\begin{array}{rl} 0, & x < 0 \\ 6, & x > 0 \\\end{array} \right.</math>

	Note that <math>f'''</math> is not defined at 0.		Note that <math>f'''</math> is not defined at 0.
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	For <math>k \ge 4</math>, we have:		For <math>k \ge 4</math>, we have:

	<math>f^{(k)}(x) = \left\lbrace\begin{array}{rl} 0, & x < 0 \\ 0, & x > 0 \\\end{array}</math>		<math>f^{(k)}(x) = \left\lbrace\begin{array}{rl} 0, & x < 0 \\ 0, & x > 0 \\\end{array} \right.</math>

	But <math>f^{(k)}(0)</math> does not exist.		But <math>f^{(k)}(0)</math> does not exist.

Vipul: /* Example of piecewise polynomial function: higher derivatives */

2012-04-25T02:39:20Z

Example of piecewise polynomial function: higher derivatives

← Older revision		Revision as of 02:39, 25 April 2012
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	Consider the function:		Consider the function:

	<math>f(x) := \\~~leftlbrace~~ \begin{array}{rl} x^2, & x < 0 \\ x^3 + x^2, & x > 0 \\ 0, & x = 0 \\\end{array}\right.</math>		<math>f(x) := \left\lbrace \begin{array}{rl} x^2, & x < 0 \\ x^3 + x^2, & x > 0 \\ 0, & x = 0 \\\end{array}\right.</math>

	Here, <math>c = 0, v = 0, f_1(x) = x^2, f_2(x) = x^3 + x^2</math>. To keep track of what we're doing, we make a table:		Here, <math>c = 0, v = 0, f_1(x) = x^2, f_2(x) = x^3 + x^2</math>. To keep track of what we're doing, we make a table:

Vipul: /* Example of piecewise polynomial function: higher derivatives */

2012-04-25T02:38:40Z

Example of piecewise polynomial function: higher derivatives

@@ Line 104: / Line 104: @@
 Consider the function:
-<math>f(x) := \lbrace \begin{array}{rl} x^2, & x < 0 \\ x^3 + x^2, & x > 0 \\ 0, & x = 0 \\\end{array}</math>
+<math>f(x) := \\leftlbrace \begin{array}{rl} x^2, & x < 0 \\ x^3 + x^2, & x > 0 \\ 0, & x = 0 \\\end{array}\right.</math>
 Here, <math>c = 0, v = 0, f_1(x) = x^2, f_2(x) = x^3 + x^2</math>. To keep track of what we're doing, we make a table:
@@ Line 121: / Line 121: @@
 Note that ''no higher derivative'' of <math>f</math> exists at zero. For instance, we ''do'' have that <math>f_1^{(4)}(0) = 0 = f_2^{(4)}(0)</math>, but <math>f^{(4)}(0)</math> does not exist.
 ==Caveat==

Vipul: /* Example of piecewise polynomial function: higher derivatives */

2012-04-25T02:34:45Z

Example of piecewise polynomial function: higher derivatives

← Older revision		Revision as of 02:34, 25 April 2012
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	\| Function \|\| <math>f_1(x) = x^2</math> \|\| <math>f_1(0) = 0^2 = 0</math> \|\| <math>f_2(x) = x^3 + x^2</math> \|\| <math>f_2(0) = 0^3 + 0^2 = 0</math> \|\| <math>f</math> is continuous at 0 \|\| We are also given that the function value at 0 is 0. Thus, <math>f_1(0) = f_2(0) = v = 0</math>. So, <math>f</math> is continuous at 0.		\| Function \|\| <math>f_1(x) = x^2</math> \|\| <math>f_1(0) = 0^2 = 0</math> \|\| <math>f_2(x) = x^3 + x^2</math> \|\| <math>f_2(0) = 0^3 + 0^2 = 0</math> \|\| <math>f</math> is continuous at 0 \|\| We are also given that the function value at 0 is 0. Thus, <math>f_1(0) = f_2(0) = v = 0</math>. So, <math>f</math> is continuous at 0.
	\|-		\|-
	\| First derivative \|\| <math>f_1'(x) = 2x</math> \|\| <math>f_1'(0) = 2(0) = 0</math> \|\| <math>f_2'(x) = 3x^2 + 2x</math> \|\|<math>f_2'(0) = 3(0)^2 + 2(0) = 0</math> \|\| <math>f</math> is differentiable at 0, and <math>f'(0) = 0</math>. \|\| We already checked continuity, and we have now checked that <math>f_1'(0) = f_2'(0)</math>.		\| First derivative \|\| <math>f_1'(x) = 2x</math> \|\| <math>f_1'(0) = 2(0) = 0</math> \|\| <math>f_2'(x) = 3x^2 + 2x</math> \|\|<math>f_2'(0) = 3(0)^2 + 2(0) = 0</math> \|\| <math>f</math> is differentiable at 0, and <math>f'(0) = 0</math> \|\| We already checked continuity, and we have now checked that <math>f_1'(0) = f_2'(0)</math>.
	\|-		\|-
	\| Second derivative \|\| <math>f_1''(x) = 2</math> \|\| <math>f_1''(0) = 2</math> \|\| <math>f_2''(x) = 6x + 2</math> \|\| <math>f_2''(0) = 6(0) + 2 = 2</math> \|\| <math>f</matH> is twice differentiable at 0, and <math>f''(0)= 2</math>.. \|\| We already checked differentiability. Thus, it suffices to check that <math>f_1''(0) = f_2''(0)</math>, which is true since both equal 2.		\| Second derivative \|\| <math>f_1''(x) = 2</math> \|\| <math>f_1''(0) = 2</math> \|\| <math>f_2''(x) = 6x + 2</math> \|\| <math>f_2''(0) = 6(0) + 2 = 2</math> \|\| <math>f</matH> is twice differentiable at 0, and <math>f''(0)= 2</math> \|\| We already checked differentiability. Thus, it suffices to check that <math>f_1''(0) = f_2''(0)</math>, which is true since both equal 2.
	\|-		\|-
	\| Third derivative \|\| <math>f_1'''(x) = 0</math> \|\| <math>f_1'''(0) = 0</math> \|\| <math>f_2'''(x) = 6</math> \|\| <math>f_2'''(0) = 6</math> \|\| <math>f</math> is ''not'' thrice differentiable at 0. \|\| We have <math>f_1'''(0) = 0 \ne f_2'''(0) = 6</math>.		\| Third derivative \|\| <math>f_1'''(x) = 0</math> \|\| <math>f_1'''(0) = 0</math> \|\| <math>f_2'''(x) = 6</math> \|\| <math>f_2'''(0) = 6</math> \|\| <math>f</math> is ''not'' thrice differentiable at 0 \|\| We have <math>f_1'''(0) = 0 \ne f_2'''(0) = 6</math>.
	\|}		\|}

Vipul: /* Local generalization */

2012-04-25T02:27:20Z

Local generalization

← Older revision		Revision as of 02:27, 25 April 2012
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	<center>{{#widget:YouTube\|id=aURWLFjy5Qo}}</center>		<center>{{#widget:YouTube\|id=aURWLFjy5Qo}}</center>

			==Examples==

			===Example of piecewise rational function===

			This example is covered in the video embedded above.

			Consider the function:

			<math>f(x) := \lbrace \begin{array}{rl} \frac{1}{x - 1}, & x < 0 \\ \frac{-1}{x + 1}, & x > 0 \\ -1, & x = 0 \\\end{array}</math>

			Note that here, in the notation we have used, we have:

			<math>f_1(x) = \frac{1}{x - 1}, \qquad f_2(x) = \frac{-1}{x + 1}, \qquad c = 0, \qquad v = -1</math>

			Note that the function <math>f_1</math> is defined ''around'' zero, i.e., the definition extends to the point zero and the immediate right -- in fact, <math>f_1</math> is defined and infinitely differentiable on the interval <math>(-\infty,1)</math>.

			Similarly, <math>f_2</math> is defined ''around'' zero, i.e., i.e., the definition extends to the immediate left of zero -- in fact, <math>f_2</math> is defined and infinitely differentiable on the interval <math>(-1,\infty)</math>.

			Thus, we see that:

			* <math>f_1(0) = 1/(0 - 1) = -1</math>, <math>f_2(0) = -1/(0 + 1) = -1</math>, and <math>f(0) = v = -1</math>. Thus, we see that <math>f_1(0) = f_2(0) = v = -1</math>, so the function <math>f</math> is continuous at 0.
			* We have <math>f_1'(x) = -1/(x - 1)^2</math> and <math>f_2'(x) = 1/(x + 1)^2</math>. We see that <math>f_1'(0) = -1/(0-1)^2 = -1</math> and <math>f_2'(0) = 1/(0 + 1)^2 = 1</math>. We see that <math>f_1'(0) \ne f_2'(0)</math>, so <math>f</matH> is not differentiable at 0.

			This means that the first and higher derivatives of <math>f</math> do not exist at 0.

			===Example of piecewise polynomial function===

			This example is covered in the video embedded above.

			Consider the function:

			<math>f(x) := \lbrace \begin{array}{rl} x^2, & x < 0 \\ x^3 + x, & x > 0 \\ 0, & x = 0 \\\end{array}</math>

			Here:

			<math>f_1(x) := x^2, \qquad f_2(x) := x^3 + x, \qquad c = 0, \qquad v = 0</math>

			We see that:

			* <math>f_1(0) = 0^2 = 0</math>, <math>f_2(0) = 0^3 + 0 = 0</math>, and <math>v = 0</math>. Thus, <math>f_1(0) = f_2(0) = v</math>, so <math>f</math> is continuous at 0.
			* <math>f_1'(x) = 2x</math> and <math>f_2'(x) = 3x^2 + 1</math>. Evaluated at 0, we get <math>f_1'(0) = 0</math> and <math>f_2'(0) = 1</math>, so <math>f_1'(0) \ne f_2'(0)</math>. So, <math>f</math> is not differentiable at 0.

			===Example of piecewise polynomial function: higher derivatives===
			This example is covered in the video embedded above.

			Consider the function:

			<math>f(x) := \lbrace \begin{array}{rl} x^2, & x < 0 \\ x^3 + x^2, & x > 0 \\ 0, & x = 0 \\\end{array}</math>

			Here, <math>c = 0, v = 0, f_1(x) = x^2, f_2(x) = x^3 + x^2</math>. To keep track of what we're doing, we make a table:

			{\| class="sortable" border="1"
			! !! Expression for <math>f_1</math> !! Value for <math>f_1</math> at 0 !! Expression for <math>f_2</math> !! Value for <math>f_2</math> at 0 !! Conclusion !! Explanation
			\|-
			\| Function \|\| <math>f_1(x) = x^2</math> \|\| <math>f_1(0) = 0^2 = 0</math> \|\| <math>f_2(x) = x^3 + x^2</math> \|\| <math>f_2(0) = 0^3 + 0^2 = 0</math> \|\| <math>f</math> is continuous at 0 \|\| We are also given that the function value at 0 is 0. Thus, <math>f_1(0) = f_2(0) = v = 0</math>. So, <math>f</math> is continuous at 0.
			\|-
			\| First derivative \|\| <math>f_1'(x) = 2x</math> \|\| <math>f_1'(0) = 2(0) = 0</math> \|\| <math>f_2'(x) = 3x^2 + 2x</math> \|\|<math>f_2'(0) = 3(0)^2 + 2(0) = 0</math> \|\| <math>f</math> is differentiable at 0, and <math>f'(0) = 0</math>. \|\| We already checked continuity, and we have now checked that <math>f_1'(0) = f_2'(0)</math>.
			\|-
			\| Second derivative \|\| <math>f_1''(x) = 2</math> \|\| <math>f_1''(0) = 2</math> \|\| <math>f_2''(x) = 6x + 2</math> \|\| <math>f_2''(0) = 6(0) + 2 = 2</math> \|\| <math>f</matH> is twice differentiable at 0, and <math>f''(0)= 2</math>.. \|\| We already checked differentiability. Thus, it suffices to check that <math>f_1''(0) = f_2''(0)</math>, which is true since both equal 2.
			\|-
			\| Third derivative \|\| <math>f_1'''(x) = 0</math> \|\| <math>f_1'''(0) = 0</math> \|\| <math>f_2'''(x) = 6</math> \|\| <math>f_2'''(0) = 6</math> \|\| <math>f</math> is ''not'' thrice differentiable at 0. \|\| We have <math>f_1'''(0) = 0 \ne f_2'''(0) = 6</math>.
			\|}

			Note that ''no higher derivative'' of <math>f</math> exists at zero. For instance, we ''do'' have that <math>f_1^{(4)}(0) = 0 = f_2^{(4)}(0)</math>, but <math>f^{(4)}(0)</math> does not exist.

	==Caveat==		==Caveat==

Vipul: /* Local generalization */

2012-04-24T19:33:13Z

Local generalization

← Older revision		Revision as of 19:33, 24 April 2012
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	The above holds with the following modification: we only require <math>f</math> to be defined as <math>f_1</math> on the ''immediate'' left of <math>c</math> (i.e., on some interval of the form <math>(c - \delta,c)</math> for <math>\delta > 0</math> and as <math>f_2</math> on the ''immediate'' right of <math>c</math> (i.e., on some interval of the form <math>(c,c + \delta)</math> for <math>\delta > 0</math>). Further, we only require that <math>f_1</math> and <math>f_2</math> be defined and differentiable on [[open interval]]s containing <math>c</math>, not necessarily on all of <math>\R</math>.		The above holds with the following modification: we only require <math>f</math> to be defined as <math>f_1</math> on the ''immediate'' left of <math>c</math> (i.e., on some interval of the form <math>(c - \delta,c)</math> for <math>\delta > 0</math> and as <math>f_2</math> on the ''immediate'' right of <math>c</math> (i.e., on some interval of the form <math>(c,c + \delta)</math> for <math>\delta > 0</math>). Further, we only require that <math>f_1</math> and <math>f_2</math> be defined and differentiable on [[open interval]]s containing <math>c</math>, not necessarily on all of <math>\R</math>.

			<center>{{#widget:YouTube\|id=aURWLFjy5Qo}}</center>

	==Caveat==		==Caveat==

Vipul at 22:41, 20 April 2012

2012-04-20T22:41:27Z

← Older revision		Revision as of 22:41, 20 April 2012
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	The above holds with the following modification: we only require <math>f</math> to be defined as <math>f_1</math> on the ''immediate'' left of <math>c</math> (i.e., on some interval of the form <math>(c - \delta,c)</math> for <math>\delta > 0</math> and as <math>f_2</math> on the ''immediate'' right of <math>c</math> (i.e., on some interval of the form <math>(c,c + \delta)</math> for <math>\delta > 0</math>). Further, we only require that <math>f_1</math> and <math>f_2</math> be defined and differentiable on [[open interval]]s containing <math>c</math>, not necessarily on all of <math>\R</math>.		The above holds with the following modification: we only require <math>f</math> to be defined as <math>f_1</math> on the ''immediate'' left of <math>c</math> (i.e., on some interval of the form <math>(c - \delta,c)</math> for <math>\delta > 0</math> and as <math>f_2</math> on the ''immediate'' right of <math>c</math> (i.e., on some interval of the form <math>(c,c + \delta)</math> for <math>\delta > 0</math>). Further, we only require that <math>f_1</math> and <math>f_2</math> be defined and differentiable on [[open interval]]s containing <math>c</math>, not necessarily on all of <math>\R</math>.

			==Caveat==

			In situations where the definitions given on one side of a point ''do not'' extend naturally to the point, we ''cannot'' use the above methods. In most such cases, we need to go back to the original definition of the derivative as a limit of a difference quotient.

			<center>{{#widget:YouTube\|id=GuYPBvWZ5uI}}</center>

Vipul at 16:58, 20 April 2012

2012-04-20T16:58:35Z

← Older revision		Revision as of 16:58, 20 April 2012
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	where <math>u = f_1'(c) = f_2'(c)</math>. In particular, the value at <math>c</math> can be included in either the left side or the right side definition.		where <math>u = f_1'(c) = f_2'(c)</math>. In particular, the value at <math>c</math> can be included in either the left side or the right side definition.

			<center>{{#widget:YouTube\|id=TqVuosJMdOU}}</center>

	===Version for higher derivatives===		===Version for higher derivatives===