Calculus: Early Transcendentals, 8th Edition
Authors: James Stewart
ISBN-13: 978-1285741550
See our solution for Question 3E from Chapter 2.8 from Stewart's Calculus, 8th Edition.
Problem 3E
Chapter:
Problem:
Match the graph of each function in (a)–(d) with the graph of its derivative in I–IV. Give reasons for your choices.
Step-by-Step Solution
Step 1
We are given with some graphs that we have to match with their derivative graphs.
Step 2: (a)
Consider the Graph of function in part (a):
https://imgur.com/Ctec6TF
The correct graph of its derivative is given by function (II).
https://imgur.com/PDvfHMX
There are some reasons that we can state for that:
(1) The function changes sign at origin hence derivative must have zero slope at that point.
(2) There are two points symmetric about y axis where function value has local minima and maxima, thus at these two points, derivative must be zero (as shown in (ii)).
(3) We can also match for increasing and decreasing function with positive and negative values of derivative.
Therefore, \[(a)\,\,\, \Leftrightarrow \,\,(II)\]
Step 3: (b)
Consider the Graph of function in part (b):
https://imgur.com/8rYajfO
The correct graph of its derivative is given by function (IV).
https://imgur.com/WsU5D3S
There are some reasons that we can state for that:
(1) The function has sharp turns at two points symmetric about origin, at those corresponding points, derivative does not exist (as shown in (IV)
(2) The function is a piecewise linear type, hence derivative should be piecewise constant The function linearly increases, then decreases and then again increases. Hence, the derivative is first positive, then stays negative and finally becomes positive again. .
Therefore, \[(b)\,\,\, \Leftrightarrow \,\,(IV)\]
Step 4: (c)
Consider the Graph of function in part (c):
The correct graph of its derivative is given by function (I).
https://imgur.com/n4HCitN
There are some reasons that we can state for that:
(1) The function is symmetric about y axis, hence the derivative is symmetric about origin.
(2) The function is continuous throughout and smooth, hence the derivative is also throughout possible .
(3) As function approaches infinity, the slope becomes zero, so the derivative becomes o as the derivative approaches infinity .
Therefore, \[(c)\,\,\, \Leftrightarrow \,\,(I)\]
Step 5: (d)
Consider the Graph of function in part (d):
https://imgur.com/XqJ0dSE
The correct graph of its derivative is given by function (III).
https://imgur.com/6ku1k6f
There are some reasons that we can state for that:
(1) The function is symmetric about y axis, hence the derivative is symmetric about origin.
(2) The function is continuous throughout and smooth, hence the derivative is also throughout possible .
(3) The function has two local maxima and one local minima, so the derivative is zero at these three corresponding points. .
Therefore, \[(d)\,\,\, \Leftrightarrow \,\,(III)\]
We are given with some graphs that we have to match with their derivative graphs.
Step 2: (a)
Consider the Graph of function in part (a):
https://imgur.com/Ctec6TF
The correct graph of its derivative is given by function (II).
https://imgur.com/PDvfHMX
There are some reasons that we can state for that:
(1) The function changes sign at origin hence derivative must have zero slope at that point.
(2) There are two points symmetric about y axis where function value has local minima and maxima, thus at these two points, derivative must be zero (as shown in (ii)).
(3) We can also match for increasing and decreasing function with positive and negative values of derivative.
Therefore, \[(a)\,\,\, \Leftrightarrow \,\,(II)\]
Step 3: (b)
Consider the Graph of function in part (b):
https://imgur.com/8rYajfO
The correct graph of its derivative is given by function (IV).
https://imgur.com/WsU5D3S
There are some reasons that we can state for that:
(1) The function has sharp turns at two points symmetric about origin, at those corresponding points, derivative does not exist (as shown in (IV)
(2) The function is a piecewise linear type, hence derivative should be piecewise constant The function linearly increases, then decreases and then again increases. Hence, the derivative is first positive, then stays negative and finally becomes positive again. .
Therefore, \[(b)\,\,\, \Leftrightarrow \,\,(IV)\]
Step 4: (c)
Consider the Graph of function in part (c):
The correct graph of its derivative is given by function (I).
https://imgur.com/n4HCitN
There are some reasons that we can state for that:
(1) The function is symmetric about y axis, hence the derivative is symmetric about origin.
(2) The function is continuous throughout and smooth, hence the derivative is also throughout possible .
(3) As function approaches infinity, the slope becomes zero, so the derivative becomes o as the derivative approaches infinity .
Therefore, \[(c)\,\,\, \Leftrightarrow \,\,(I)\]
Step 5: (d)
Consider the Graph of function in part (d):
https://imgur.com/XqJ0dSE
The correct graph of its derivative is given by function (III).
https://imgur.com/6ku1k6f
There are some reasons that we can state for that:
(1) The function is symmetric about y axis, hence the derivative is symmetric about origin.
(2) The function is continuous throughout and smooth, hence the derivative is also throughout possible .
(3) The function has two local maxima and one local minima, so the derivative is zero at these three corresponding points. .
Therefore, \[(d)\,\,\, \Leftrightarrow \,\,(III)\]