Quiz:Product rule for differentiation - Revision history

Vipul: /* Qualitative and existential significance */

2024-04-11T05:44:29Z

Qualitative and existential significance

← Older revision		Revision as of 05:44, 11 April 2024
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	- None of the above		- None of the above

	{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>a,b \in \R</math> are such that <math>1 < a < b</math>. Suppose <math>f</math> is known to be differentiable at <math>a</math> and <math>g</math> is known to be differentiable at <math>b</math>. We do not have information about where else <math>f</math or <math>g</math> is differentiable. What can we conclude about where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?		{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>a,b \in \R</math> are such that <math>1 < a < b</math>. Suppose <math>f</math> is known to be differentiable at <math>a</math> and <math>g</math> is known to be differentiable at <math>b</math>. We do not have information about where else <math>f</math> or <math>g</math> is differentiable. What can we conclude about where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
	\|type="()"}		\|type="()"}
	- <math>f \cdot g</math> is differentiable at <math>ab</math>		- <math>f \cdot g</math> is differentiable at <math>ab</math>

Vipul: /* Qualitative and existential significance */

2024-04-11T05:43:33Z

Qualitative and existential significance

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	\|type="()"}		\|type="()"}
	- <math>f \cdot g</math> is differentiable at <math>ab</math>		- <math>f \cdot g</math> is differentiable at <math>ab</math>
	- <math>f \cdot g</math> is differentiable at at least one of <math>a</math or <math>b</math> but not necessarily at both		- <math>f \cdot g</math> is differentiable at at least one of <math>a</math> or <math>b</math> but not necessarily at both
	- <math>f \cdot g</math> is differentiable at both <math>a</math> and <math>b</math>		- <math>f \cdot g</math> is differentiable at both <math>a</math> and <math>b</math>
	+ We don't have enough information to conclude anything about the set of points where <math>f \cdot g</math> is differentiable		+ We don't have enough information to conclude anything about the set of points where <math>f \cdot g</math> is differentiable

Vipul: /* Qualitative and existential significance */

2024-04-11T05:41:47Z

Qualitative and existential significance

← Older revision		Revision as of 05:41, 11 April 2024
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	- It contains the union <math>A \cup B</math>		- It contains the union <math>A \cup B</math>
	- None of the above		- None of the above

			{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>a,b \in \R</math> are such that <math>1 < a < b</math>. Suppose <math>f</math> is known to be differentiable at <math>a</math> and <math>g</math> is known to be differentiable at <math>b</math>. We do not have information about where else <math>f</math or <math>g</math> is differentiable. What can we conclude about where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
			\|type="()"}
			- <math>f \cdot g</math> is differentiable at <math>ab</math>
			- <math>f \cdot g</math> is differentiable at at least one of <math>a</math or <math>b</math> but not necessarily at both
			- <math>f \cdot g</math> is differentiable at both <math>a</math> and <math>b</math>
			+ We don't have enough information to conclude anything about the set of points where <math>f \cdot g</math> is differentiable

	{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>?		{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>?

Vipul: /* Computational results significance */

2024-04-11T05:30:54Z

Computational results significance

← Older revision		Revision as of 05:30, 11 April 2024
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	+ We cannot conclude from the given information whether <math>f \cdot g</math> or any of the derivatives of <math>f \cdot g</math> is periodic.		+ We cannot conclude from the given information whether <math>f \cdot g</math> or any of the derivatives of <math>f \cdot g</math> is periodic.

	{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>\! f'</math> and <math>\! g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?		{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>f'</math> and <math>g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?
	\|type="()"}		\|type="()"}
	- Both <math>f \cdot g</math> and <math>(f \cdot g)'</math> are expressible as rational functions.		- Both <math>f \cdot g</math> and <math>(f \cdot g)'</math> are expressible as rational functions.

Vipul: /* Qualitative and existential significance */

2024-04-11T05:29:41Z

Qualitative and existential significance

← Older revision		Revision as of 05:29, 11 April 2024
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	- None of the above		- None of the above

	{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>.		{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>?
	\|type="()"}		\|type="()"}
	- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition and scalar multiplication, i.e., it forms a [[vector space]] of functions.		- It is sufficient to ensure that <math>\mathcal{F}</math> is closed under addition and scalar multiplication, i.e., it forms a [[vector space]] of functions.

Vipul: /* Computational results significance */

2023-12-12T04:05:37Z

Computational results significance

← Older revision		Revision as of 04:05, 12 December 2023
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	- <math>(f \cdot g)''</math> must be periodic, but <math>(f \cdot g)'</math> may or may not be periodic.		- <math>(f \cdot g)''</math> must be periodic, but <math>(f \cdot g)'</math> may or may not be periodic.
	- <math>(f \cdot g)'''</math> must be periodic, but <math>(f \cdot g)''</math> may or may not be periodic.		- <math>(f \cdot g)'''</math> must be periodic, but <math>(f \cdot g)''</math> may or may not be periodic.
	+ We cannot conclude from the given information whether any of the derivatives of <math>f \cdot g</math> is periodic.		+ We cannot conclude from the given information whether <math>f \cdot g</math> or any of the derivatives of <math>f \cdot g</math> is periodic.

	{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>\! f'</math> and <math>\! g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?		{Suppose <math>f</math> and <math>g</math> are functions defined and differentiable on the open interval <math>(0,1)</math>. Suppose, further, that on <math>(0,1)</math>, the derivative functions <math>\! f'</math> and <math>\! g'</math> are both expressible as [[rational function]]s. What can we say about <math>f \cdot g</math> and <math>(f \cdot g)'</math> on <math>(0,1)</math>?

Vipul: /* Formulas */

2023-12-12T04:01:12Z

Formulas

← Older revision		Revision as of 04:01, 12 December 2023
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	+ <math>f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'</math>		+ <math>f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'</math>
	\|\| See [[product rule for differentiation#Statement for multiple functions]]		\|\| See [[product rule for differentiation#Statement for multiple functions]]
	- <math>f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1 \cdot f_2 \cdot f_3'</math>		- <math>f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1' \cdot f_2' \cdot f_3</math>
	- <math>f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1</math>		- <math>f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1</math>
	- <math>f_1'' \cdot f_2' \cdot f_3</math>		- <math>f_1'' \cdot f_2' \cdot f_3</math>

Vipul: /* Qualitative and existential significance */

2012-03-22T17:17:43Z

Qualitative and existential significance

← Older revision		Revision as of 17:17, 22 March 2012
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	+ If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, so is <math>f \cdot g</math>. However, differentiability of <math>f \cdot g</math> and <math>f</math> does not guarantee differentiability of <math>g</math>, and differentiability of <math>f \cdot g</math> and <math>g</math> does not guarantee differentiability of <math>f</math>.		+ If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, so is <math>f \cdot g</math>. However, differentiability of <math>f \cdot g</math> and <math>f</math> does not guarantee differentiability of <math>g</math>, and differentiability of <math>f \cdot g</math> and <math>g</math> does not guarantee differentiability of <math>f</math>.
	- We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.		- We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

			{Suppose <math>f</math> and <math>g</math> are continuous functions defined on all of <math>\R</math>. Suppose <math>A</math> is the subset of <math>\R</math> comprising those points where <math>f</math> is differentiable, and <matH>B</math> is the subset of <math>\R</math> comprising those points where <math>g</math> is differentiable. Then, what can we say is '''definitely true''' about the subset of <math>\R</math> comprising those points where the [[pointwise product of functions]] <math>f \cdot g</math> is differentiable?
			\|type="()"}
			- It is contained in the intersection <math>A \cap B</math>
			+ It contains the intersection <math>A \cap B</math>
			- It is contained in the union <math>A \cup B</math>
			- It contains the union <math>A \cup B</math>
			- None of the above

	{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>.		{Suppose <math>\mathcal{F}</math> is a collection of differentiable functions defined on all of <math>\R</math>. Further, suppose that there is a collection <math>\mathcal{B}</math> of functions such that every element of <math>\mathcal{F}</math> can be written as a polynomial in terms of the elements of <math>\mathcal{B}</math>, with constant coefficients. Suppose that the derivative of every element of <math>\mathcal{B}</math> is in <math>\mathcal{F}</math>. Which of the following conditions are sufficient to ensure that the derivative of every element of <math>\mathcal{F}</math> is in <math>\mathcal{F}</math>.

Vipul at 17:12, 22 March 2012

2012-03-22T17:12:00Z

← Older revision		Revision as of 17:12, 22 March 2012
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	==Practical==		==Practical==

			Corresponds to [[Practical:Product rule for differentiation]].

			General difficulty level of questions in this section: School level (unless otherwise specified).
	<quiz display=simple>		<quiz display=simple>
	{Suppose <math>f</math> and <math>g</math> are both defined and differentiable at the point 1. Suppose <math>\! f(1) = 2, g(1) = 5, f'(1) = 4, g'(1) = 11</math>. What is the value of <math>(f \cdot g)'(1)</math> where <math>f \cdot g</math> denotes the [[pointwise product of functions]]?		{Suppose <math>f</math> and <math>g</math> are both defined and differentiable at the point 1. Suppose <math>\! f(1) = 2, g(1) = 5, f'(1) = 4, g'(1) = 11</math>. What is the value of <math>(f \cdot g)'(1)</math> where <math>f \cdot g</math> denotes the [[pointwise product of functions]]?
	\|type="()"}		\|type="()"}
	+ 42		+ 42
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	==Formulas==		==Formulas==

			General difficulty level of questions in this section: College level (unless otherwise specified)

	<quiz display=simple>		<quiz display=simple>
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	===Qualitative and existential significance===		===Qualitative and existential significance===

			Corresponds to [[Product rule for differentiation#Qualitative and existential significance]].

			General difficulty level of questions in this section: College level (unless otherwise specified).

	<quiz display=simple>		<quiz display=simple>
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	===Computational results significance===		===Computational results significance===

			Corresponds to [[Product rule for differentiation#Computational results significance]].

			General difficulty level of questions in this section: College level (unless otherwise specified).

	<quiz display=simple>		<quiz display=simple>

Vipul: /* Practical */

2012-03-22T17:06:37Z

Practical

← Older revision		Revision as of 17:06, 22 March 2012
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	- The information given is insufficient to find <math>(f \cdot g)'(1)</math>.		- The information given is insufficient to find <math>(f \cdot g)'(1)</math>.

	{What is the derivative of the function <math>x \mapsto \exp(x) \sin x</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of ~~<math>\exp</math>~~ is ~~<math>\exp</math>~~, ~~and~~ derivative of ~~<math>\sin</math>~~ is ~~<math>\cos</math>~~.</toggledisplay>		{What is the derivative of the function <math>x \mapsto \exp(x) \sin x</math>? Hint for derivatives of individual functions: <toggledisplay>Derivative of exponential function is exponential function, derivative of sine function is cosine function.</toggledisplay>
	\|type="()"}		\|type="()"}
	+ <math>x \mapsto \exp(x)(\sin x + \cos x)</math>		+ <math>x \mapsto \exp(x)(\sin x + \cos x)</math>