Quiz:Product rule for differentiation: Difference between revisions

Revision as of 01:43, 28 November 2011

For a quiz that tests all the differentiation rules together, see Quiz:Differentiation rules.

For background, see product rule for differentiation and product rule for higher derivatives.

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 - <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>
+</quiz>
+==Qualitative and existential significance==
+<quiz display=simple>
+{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. Which of the following is ''true'' (see last two options!)?
+|type="()"}
+- If <math>f</math> and <math>g</math> are both left differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
+- If <math>f</math> and <math>g</math> are both right differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
+- If <math>f</math> and <math>g</math> are both differentiable at <math>x_0</math>, then so is <math>f \cdot g</math>.
++ All of the above are true
+- None of the above is true
+{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math> and <math>f \cdot g</math> is the [[pointwise product of functions]]. What is the relationship between the differentiability of <math>f</math>, <math>g</math>, and <math>f \cdot g</math> at <math>x_0</math>?
+|type="()"}
+- If any two of the three functions are differentiable at <math>x_0</math>, then so is the third.
+- If <math>f \cdot g</math> is differentiable at <math>x_0</math>, so are <math>f</math> and <math>g</math>.
+- If <math>f \cdot g</math> and <math>f</math> are differentiable at <math>x_0</math>, so is <math>g</math>. However, differentiability of <math>f</math> and <math>g</math> at <math>x_0</math> does not guarantee differentiability of <math>f \cdot g</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.
 </quiz>

	$f''\cdot g+f\cdot g''$
	$f''\cdot g+f'\cdot g'+f\cdot g''$
	$f''\cdot g+2f'\cdot g'+f\cdot g''$
	$f''\cdot g-f'\cdot g'+f\cdot g''$
	$f''\cdot g-2f'\cdot g'+f\cdot g''$

	$f_{1}'\cdot f_{2}'\cdot f_{3}'$
	$f_{1}'\cdot f_{2}\cdot f_{3}+f_{1}\cdot f_{2}'\cdot f_{3}+f_{1}\cdot f_{2}\cdot f_{3}'$
	$f_{1}\cdot f_{2}'\cdot f_{3}'+f_{1}'\cdot f_{2}\cdot f_{3}'+f_{1}\cdot f_{2}\cdot f_{3}'$
	$f_{1}'\cdot f_{2}+f_{2}'\cdot f_{3}+f_{3}'\cdot f_{1}$
	$f_{1}''\cdot f_{2}'\cdot f_{3}$

	If $f$ and $g$ are both left differentiable at $x_{0}$ , then so is $f\cdot g$ .
	If $f$ and $g$ are both right differentiable at $x_{0}$ , then so is $f\cdot g$ .
	If $f$ and $g$ are both differentiable at $x_{0}$ , then so is $f\cdot g$ .
	All of the above are true
	None of the above is true

	If any two of the three functions are differentiable at $x_{0}$ , then so is the third.
	If $f\cdot g$ is differentiable at $x_{0}$ , so are $f$ and $g$ .
	If $f\cdot g$ and $f$ are differentiable at $x_{0}$ , so is $g$ . However, differentiability of $f$ and $g$ at $x_{0}$ does not guarantee differentiability of $f\cdot g$ .
	If $f$ and $g$ are both differentiable at $x_{0}$ , so is $f\cdot g$ . However, differentiability of $f\cdot g$ and $f$ does not guarantee differentiability of $g$ , and differentiability of $f\cdot g$ and $g$ does not guarantee differentiability of $f$ .
	We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

Revision as of 01:43, 28 November 2011

Formulas

Qualitative and existential significance