Quiz:Differentiation rules: Difference between revisions

Revision as of 17:46, 15 October 2011

Qualitative and existential questions

Syntax error

1 Suppose $f$ and $g$ are both functions from $\mathbb {R}$ to $\mathbb {R}$ . Suppose further that $f$ and $g$ are both differentiable at a point $x_{0}\in \mathbb {R}$ . Which of the following functions can we not guarantee to be differentiable at $x_{0}$ ?

	The sum $f+g$ , i.e., the function $x\mapsto f(x)+g(x)$
	The difference $f-g$ , i.e., the function $x\mapsto f(x)-g(x)$
	The product $f\cdot g$ , i.e., the function $x\mapsto f(x)g(x)$
	The composite $f\circ g$ , i.e., the function $x\mapsto f(g(x))$
	None of the above, i.e., they are all guaranteed to be differentiable.

	The sum $f+g$ , i.e., the function $x\mapsto f(x)+g(x)$
	The difference $f-g$ , i.e., the function $x\mapsto f(x)-g(x)$
	The product $f\cdot g$ , i.e., the function $x\mapsto f(x)g(x)$
	The composite $f\circ g$ , i.e., the function $x\mapsto f(g(x))$
	None of the above, i.e., they are all guaranteed to be everywhere differentiable

3 Suppose $f$ and $g$ are both functions from $\mathbb {R}$ to $\mathbb {R}$ and the left hand derivatives for $f$ and $g$ exist on all of $R$ . For which of the following functions can we not guarantee that the left hand derivative exists on all of $\mathbb {R}$ ?

4 Suppose $f$ and $g$ are both twice differentiable functions everywhere on $\mathbb {R}$ . Which of the following is the correct formula for $(f\cdot g)''$ , the second derivative of the pointwise product?

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

5 Suppose $f$ and $g$ are both twice differentiable functions everywhere on $\mathbb {R}$ . Which of the following is the correct formula for $(f\circ g)''$ , the second derivative of the pointwise product?

	$(f''\circ g)\cdot g''$
	$(f''\circ g)\cdot (f'\circ g')\cdot g''$
	$(f''\circ g)\cdot (f'\circ g')\cdot (f\circ g'')$
	$(f''\circ g)\cdot (g')^{2}+(f'\circ g)\cdot g''$
	$(f'\circ g')\cdot (f\circ g)+(f''\circ g'')$

6 Suppose $f_{1},f_{2},f_{3}$ are everywhere differentiable functions from $\mathbb {R}$ to $\mathbb {R}$ . What is the derivative $(f_{1}\cdot f_{2}\cdot f_{3})'$ , where $f_{1}\cdot f_{2}\cdot f_{3}$ denotes the pointwise product of functions?

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

7 Suppose $f_{1},f_{2},f_{3}$ are everywhere differentiable functions from $\mathbb {R}$ to $\mathbb {R}$ . What is the derivative $(f_{1}\circ f_{2}\circ f_{3})'$ where $\circ$ denotes the composite of two functions? In other words, $(f_{1}\circ f_{2}\circ f_{3})(x):=f_{1}(f_{2}(f_{3}(x)))$ .

	$(f_{1}'\circ f_{2}\circ f_{3})\cdot (f_{2}'\circ f_{3})\cdot f_{3}'$
	$(f_{1}'\cdot f_{2}\cdot f_{3})\circ (f_{2}'\cdot f_{3})\circ f_{3}'$
	$(f_{1}\circ f_{2}'\circ f_{3}')\cdot (f_{2}\circ f_{3}')\cdot f_{3}$
	$(f_{1}\cdot f_{2}'\cdot f_{3}')\circ (f_{2}\cdot f_{3}')\circ f_{3}$

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+==Qualitative and existential questions==
 <quiz display=simple>
-{Which of the following verbal statements is ''not valid as a general rule''?
-|type="()"}
-- The derivative of the sum of two functions is the sum of the derivatives of the functions.
-|| This ''is'' valid as a general rule. See [[differentiation is linear]].
-- The derivative of the difference of two functions is the difference of the derivatives of the functions.
-|| This ''is'' valid as a general rule. See [[differentiation is linear]].
-- The derivative of a constant times a function is the same constant times the derivative of the function.
-|| This ''is'' valid as a general rule. See [[differentiation is linear]].
-+ The derivative of the product of two functions is the product of the derivatives of the functions.
-|| This is a ''wrong'' rule that goes by the name of the [[freshman product rule]]. The correct version is the [[product rule for differentiation]].
-- None of the above, i.e., they are all valid as general rules.
 {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>
+<quiz>
+==Generic point computation questions==
+<quiz display=simple>
+{Which of the following verbal statements is ''not valid as a general rule''?
+|type="()"}
+- The derivative of the sum of two functions is the sum of the derivatives of the functions.
+|| This ''is'' valid as a general rule. See [[differentiation is linear]].
+- The derivative of the difference of two functions is the difference of the derivatives of the functions.
+|| This ''is'' valid as a general rule. See [[differentiation is linear]].
+- The derivative of a constant times a function is the same constant times the derivative of the function.
+|| This ''is'' valid as a general rule. See [[differentiation is linear]].
++ The derivative of the product of two functions is the product of the derivatives of the functions.
+|| This is a ''wrong'' rule that goes by the name of the [[freshman product rule]]. The correct version is the [[product rule for differentiation]].
+- 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?

	The sum $f+g$ , i.e., the function $x\mapsto f(x)+g(x)$
	The difference $f-g$ , i.e., the function $x\mapsto f(x)-g(x)$
	The product $f\cdot g$ , i.e., the function $x\mapsto f(x)g(x)$
	The composite $f\circ g$ , i.e., the function $x\mapsto f(g(x))$
	None of the above, i.e., they are all guaranteed to have a left hand derivative on all of $\mathbb {R}$
	The derivative of the sum of two functions is the sum of the derivatives of the functions.
	The derivative of the difference of two functions is the difference of the derivatives of the functions.
	The derivative of a constant times a function is the same constant times the derivative of the function.
	The derivative of the product of two functions is the product of the derivatives of the functions.
	None of the above, i.e., they are all valid as general rules.