Quiz:Differentiation rules - Revision history

Vipul: /* Predicting when things become zero */

2011-10-19T17:04:04Z

Predicting when things become zero

← Older revision		Revision as of 17:04, 19 October 2011
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	{Suppose <math>\! f,g</math> are everywhere twice differentiable functions on <math>\R</math> and <math>\! f'' = g'' = 0</math> everywhere. For which of the following functions can we ''not'' conclude that the second derivative is zero everywhere?		{Suppose <math>\! f,g</math> are everywhere twice differentiable functions on <math>\R</math> and <math>\! f'' = g'' = 0</math> everywhere. For which of the following functions can we ''not'' conclude that the second derivative is zero everywhere?
	\|type="()"}		\|type="()"}
	- <math>f + g</math>		- <math>\! f + g</math>
	+ <math>f \cdot g</math>		+ <math>f \cdot g</math>
	\|\| <math>f</math> and <math>g</math> must both be either constant or linear functions. However, the product of two linear functions is quadratic and hence not linear. For instance <math>f(x) = x</math> and <math>g(x) = x</math> are examples.		\|\| <math>f</math> and <math>g</math> must both be either constant or linear functions. However, the product of two linear functions is quadratic and hence not linear. For instance <math>f(x) = x</math> and <math>g(x) = x</math> are examples.

Vipul: /* Predicting when things become zero */

2011-10-19T17:03:26Z

Predicting when things become zero

← Older revision		Revision as of 17:03, 19 October 2011
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	<quiz display=simple>		<quiz display=simple>
	{Suppose <math>f,g</math> are everywhere differentiable functions on <math>\R</math> and <math>f' = g' = 0</math> everywhere. For which of the following functions can we ''not'' conclude that the derivative is zero everywhere?		{Suppose <math>\! f,g</math> are everywhere differentiable functions on <math>\R</math> and <math>\! f' = g' = 0</math> everywhere. For which of the following functions can we ''not'' conclude that the derivative is zero everywhere?
	\|type="()"}		\|type="()"}
	- <math>f + g</math>		- <math>\! f + g</math>
	- <math>f \cdot g</math>		- <math>f \cdot g</math>
	- <math>f \circ g</math>		- <math>f \circ g</math>
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	+ None of the above, i.e., the derivative is necessarily zero everywhere for each of these functions		+ None of the above, i.e., the derivative is necessarily zero everywhere for each of these functions

	{Suppose <math>f,g</math> are everywhere twice differentiable functions on <math>\R</math> and <math>f'' = g'' = 0</math> everywhere. For which of the following functions can we ''not'' conclude that the second derivative is zero everywhere?		{Suppose <math>\! f,g</math> are everywhere twice differentiable functions on <math>\R</math> and <math>\! f'' = g'' = 0</math> everywhere. For which of the following functions can we ''not'' conclude that the second derivative is zero everywhere?
	\|type="()"}		\|type="()"}
	- <math>f + g</math>		- <math>f + g</math>
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	- None of the above, i.e., the second derivative is necessarily zero everywhere for each of these functions		- None of the above, i.e., the second derivative is necessarily zero everywhere for each of these functions

	{Suppose <math>f,g</math> are everywhere thrice differentiable functions on <math>\R</math> and <math>f''' = g''' = 0</math> everywhere. For which of the following functions can we ''definitively'' conclude that the third derivative is zero everywhere?		{Suppose <math>\! f,g</math> are everywhere thrice differentiable functions on <math>\R</math> and <math>\! f''' = g''' = 0</math> everywhere. For which of the following functions can we ''definitively'' conclude that the third derivative is zero everywhere?
	\|type="()"}		\|type="()"}
	+ <math>f + g</math>		+ <math>\! f + g</math>
	- <math>f \cdot g</math>		- <math>f \cdot g</math>
	\|\| Both <math>f</math> and <math>g</math> are polynomials of degree at most 2, but their product need not be.		\|\| Both <math>f</math> and <math>g</math> are polynomials of degree at most 2, but their product need not be.

Vipul: /* Predicting when things become zero */

2011-10-19T17:02:32Z

Predicting when things become zero

← Older revision		Revision as of 17:02, 19 October 2011
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	\|\| Both <math>f</math> and <math>g</math> are polynomials of degree at most 2, but their product need not be.		\|\| Both <math>f</math> and <math>g</math> are polynomials of degree at most 2, but their product need not be.
	- All of the above, i.e., the third derivative is zero everywhere for each of these functions		- All of the above, i.e., the third derivative is zero everywhere for each of these functions
	- None of the above, i.e., the third derivative is necessarily zero everywhere for ~~each~~ of these functions		- None of the above, i.e., the third derivative is not necessarily zero everywhere for any of these functions

	</quiz>		</quiz>

Vipul: /* Generic point computation questions */

2011-10-19T17:01:48Z

Generic point computation questions

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 - None of the above, i.e., they are all valid as general rules.
-{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \cdot g)''</math>, the second derivative of the pointwise product?
+{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \cdot g)''</math>, the second derivative of the [[pointwise product of functions]]?
 |type="()"}
 - <math>f'' \cdot g + f \cdot g''</math>
@@ Line 57: / Line 57: @@
 - <math>f'' \cdot g - 2f' \cdot g' + f \cdot g''</math>
-{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \circ g)''</math>, the second derivative of the pointwise composite?
+{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \circ g)''</math>, the second derivative of the [[composite of two functions]]?
 |type="()"}
 - <math>(f'' \circ g) \cdot g''</math>
@@ Line 83: / Line 83: @@
 - <math>(f_1 \cdot f_2' \cdot f_3') \circ (f_2 \cdot f_3') \circ f_3</math>
 - <math>f_1' \circ f_2' \circ f_3'</math>
 </quiz>

Vipul: /* Generic point computation questions */

2011-10-19T16:54:20Z

Generic point computation questions

← Older revision		Revision as of 16:54, 19 October 2011
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	- <math>f'' \cdot g - 2f' \cdot g' + f \cdot g''</math>		- <math>f'' \cdot g - 2f' \cdot g' + f \cdot g''</math>

	{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \circ g)''</math>, the second derivative of the pointwise ~~product~~?		{Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \circ g)''</math>, the second derivative of the pointwise composite?
	\|type="()"}		\|type="()"}
	- <math>(f'' \circ g) \cdot g''</math>		- <math>(f'' \circ g) \cdot g''</math>

Vipul at 17:56, 15 October 2011

2011-10-15T17:56:34Z

← Older revision		Revision as of 17:56, 15 October 2011
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			Review the rules before or while attempting the quiz questions: <toggledisplay>{{#lst:derivative\|differentiation rules}}</toggledisplay>

			See also [[:Category:Differentiation rules]] for a list of all the differentiation rules pages (including pages for higher derivatives, which are not in this table).

	==Qualitative and existential questions==		==Qualitative and existential questions==

Vipul: /* Generic point computation questions */

2011-10-15T17:50:40Z

Generic point computation questions

← Older revision		Revision as of 17:50, 15 October 2011
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	- <math>(f_1 \circ f_2' \circ f_3') \cdot (f_2 \circ f_3') \cdot f_3</math>		- <math>(f_1 \circ f_2' \circ f_3') \cdot (f_2 \circ f_3') \cdot f_3</math>
	- <math>(f_1 \cdot f_2' \cdot f_3') \circ (f_2 \cdot f_3') \circ f_3</math>		- <math>(f_1 \cdot f_2' \cdot f_3') \circ (f_2 \cdot f_3') \circ f_3</math>
			- <math>f_1' \circ f_2' \circ f_3'</math>
	</quiz>		</quiz>

Vipul: /* Qualitative and existential questions */

2011-10-15T17:47:43Z

Qualitative and existential questions

← Older revision		Revision as of 17:47, 15 October 2011
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	- The difference <math>f - g</math>, i.e., the function <math>x \mapsto f(x) - g(x)</math>		- The difference <math>f - g</math>, i.e., the function <math>x \mapsto f(x) - g(x)</math>
	- The product <math>f \cdot g</math>, i.e., the function <math>x \mapsto f(x)g(x)</math>		- The product <math>f \cdot g</math>, i.e., the function <math>x \mapsto f(x)g(x)</math>
	- The composite <math>f \circ g</math>, i.e., the function <math>x \mapsto f(g(x))</math>		+ The composite <math>f \circ g</math>, i.e., the function <math>x \mapsto f(g(x))</math>
	+ None of the above, i.e., they are all guaranteed to have a left hand derivative on all of <math>\R</math>		- None of the above, i.e., they are all guaranteed to have a left hand derivative on all of <math>\R</math>

	</quiz>		</quiz>

Vipul: /* Qualitative and existential questions */

2011-10-15T17:47:00Z

Qualitative and existential questions

← Older revision		Revision as of 17:47, 15 October 2011
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	+ None of the above, i.e., they are all guaranteed to have a left hand derivative on all of <math>\R</math>		+ None of the above, i.e., they are all guaranteed to have a left hand derivative on all of <math>\R</math>

	<quiz>		</quiz>

	==Generic point computation questions==		==Generic point computation questions==

Vipul at 17:46, 15 October 2011

2011-10-15T17:46:27Z

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 <quiz display=simple>
 {Suppose <math>f</math> and <math>g</math> are both functions from <math>\R</math> to <math>\R</math>. Suppose further that <math>f</math> and <math>g</math> are both differentiable at a point <math>x_0 \in \R</math>. Which of the following functions can we ''not'' guarantee to be differentiable at <math>x_0</math>?
 |type="()"}
@@ Line 36: / Line 26: @@
 - The composite <math>f \circ g</math>, i.e., the function <math>x \mapsto f(g(x))</math>
 + None of the above, i.e., they are all guaranteed to have a left hand derivative on all of <math>\R</math>
 {Suppose <math>f</math> and <math>g</math> are both twice differentiable functions everywhere on <math>\R</math>. Which of the following is the correct formula for <math>(f \cdot g)''</math>, the second derivative of the pointwise product?