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In Chapter 3 we developed the meaning of constant rate of change. In Chapter 4 we learned that a function's rate of change at a moment provides useful information about the function's behavior at that moment. Knowing a function's rate of change at all moments in an interval provides useful information about the function's behavior over that interval.
Suppose that $f'(k)=c$ when $x=k$. Recall that $f'(k)=c$ means that the function f has essentially a constant rate of change of c f-units per x-unit over a small interval containing $x=k$. The function f having an essentially constant rate of change over a small interval containing $x=k$ means that f is essentially linear over that interval, which means that $f(x) \doteq f'(k)(x-k)+f(k)$ for all x in that interval. If f has a rate of change at every moment within an interval, then it is locally linear at every moment within that interval.
Example 7.1.1
Let f be defined as $f(x)=0.5\sin(3x)$. Then $f'(x)=1.5\cos(3x)$. Examine the behavior of f at the moment that $x=2.45$.
Figure 7.1.1. Examining the behavior of $f(x)=0.5\sin(3x)$ at the moment that $x=2.45$. The behavior of f is essentially linear, with $f(x) \doteq r_f(k)(x-k)+f(k)$, over a suitably small interval containing $x=2.45$.
Reflection 7.1.1. The
slope of the graph of $y=r_f(k)(x-k)+f(k)$ appears to change in the last
three zooms. Does it actually change? If not, then why does it appear to
change?
There are exceptions to when you can use the rate of change function for a function f to conclude something about the behavior of f around a particular value of x. Activity 7.1.2 illustrates this.
Activity 7.1.2First, some terminology:
An interval is closed if it contains its
endpoints. It is customary to represent a closed interval with
brackets, e.g. [2, 3]. An interval is open if it excludes its endpoints. It is customary to represent an open interval with parentheses, e.g. (2, 3). You must rely on context to distinguish between (2,3) as a coordinate pair and (2,3) as an open interval. |
Let g be a function defined over a closed interval I.
g is increasing over I if for
all u and v in I, $g(u) \lt g(v)$
whenever $u \lt v$.^{Note
}g is decreasing over I if
for all u and v in I, $g(u) \gt g(v)$
whenever $u \lt v$. g has a maximum value in I at x = k if for all x in I, $x \neq k, g(x) \lt g(k)$. g has a minimum value in I at x = k if for all x in I, $x \neq k, g(x) \gt g(k)$. |
You saw in Activities 7.1.1 and 7.1.2 that when a function f has a rate of change at the moment $x=k$ that
The case of $f'(x)=0$ when $x=k$ is ambiguous with regard to the behavior of f around $x=k$. We know that f is essentially constant around $x=k$, but this can happen in any of four ways. Figure 7.1.2 shows the four possible ways that f can behave around $x=k$ when $f'(k)=0$.
(a) f decreases, then is momentarily constant, then increases. Notice that $r_f(x)$ is negative for $x<1$, then 0 at $x=1$, then positive for $x>1$. | (b) f increases, then is momentarily constant, then decreases. Notice that $r_f(x)$ is positive for $x<1$, then 0 at $x=1$, then negative for $x>1$. | (c) f increases, then is momentarily constant, then increases again. Notice that $r_f(x)$ is positive for $x<1$, then 0 at $x=1$, then positive for $x>1$. | (d) f decreases, then is momentarily constant, then decreases again. Notice that $r_f(x)$ is negative for $x<1$, then 0 at $x=1$, then negative for $x>1$. |
Figure 7.1.3. A function can have a rate of change of zero at the
moment x = 1 in four ways.
Figure 7.1.3 illustrates a general principle. When $r_f(k)=0$ and $r_f(x)$ changes sign around $x=k$, then f has a maximum or a minimum at $x=k$.
If $r_f(k)=0$ and $r_f(x)$ changes from negative to positive
around $x=k$, then f has a minimum value
at $x=k$. If $r_f(k)=0$ and $r_f(x)$ changes from positive to negative around $x=k$, then f has a maximum value at $x=k$. |
Activity 7.1.3.
Download this file. The slider named Function has four values. Each value gives one of the graphs in Figure 7.1.3.
We cannot say that all minima or maxima occur when $f'(k)=0$. Figure 7.1.2 gives an example. A minimum occurs at $x=-2.2$ and a maximum occurs at $x=2.2$ even though neither $f'(-2.2)$ nor $f'(2.2)$ are zero.
Another instance where an extremum might occur at $x=k$ and $f'(k)$ is undefined. $f'(k)$ can be undefined for a number of reasons, including when $f(k)$ is undefined, when the function is discontinuous at $x=k$ or when $f'(k)$ is undefined because f has a cusp at $x=k$. These cases are shown in Figure 7.1.4.
Reflection 7.1.2. Justify each statement in Figure 7.1.4. For example, explain why f has neither a min nor a max at $x=0$ and $x=6$, and explain how f has a min at $x=2$ and a min at $x=4$ even though $r_f(2)$ and $r_f(4)$ are not zero.
If g has a maximum at $x=k$ and $g(k) \gt g(x)$ for all x in I, $x \neq k$, then $g(k)$ is a global maximum for g in the interval I. If g has a maximum at $x=k$ and $g(k)$ is not a global maximum, then $g(k)$ is a local maximum for g in the interval I. If g has a minimum at $x=k$ and $g(k) \lt g(x)$ for all x in I, $x \neq k$, then $g(k)$ is a global minimum for g in the interval I. If g has a minimum at $x=k$ and $g(k)$ is not a
global minimum, then $g(k)$ is a local minimum
for g in the interval I. |
Explain, in your own words, the conditions that are necessary for an accumulation function to have:
A global maximum at $x=k$.
A global minimum at $x=k$
No local extremum anywhere in an interval.
A minimum at $x=k$.
Let f be a function and let c be a value in the domain of f. List the ways in which $f'(c)$ can fail to exist.
A student in a previous calculus class made the following statements. Say whether each statement is true or false. If you say "true", then justify your answer. If you say "false", then give an example to show that the statement is false.
The function $f(x)=1/x$ has a global maximum on the interval (0, 1].
All global maxima or global minima occur at a critical value or at an interval's endpoint.
Bill threw a ball into the air. The ball reached its peak height at 4.1 seconds and hit the ground after 10.5 seconds. The value of $g(t)$ gives the ball's height above ground in meters t seconds after Bill threw the ball.
Each of Part (a) to (d) might be answerable or not. If it can be answered, do so; if it cannot be answered with the given information, explain why.
$g(1)=3.2$. What information does this tell us about the ball’s height?
$g(1)=3.2$. What information does this tell us about how fast the ball’s height is changing at 1 second?
$g’(2) = 1.6$. What can this tell us about how high the ball is at 2 seconds?
$g(3.7)=4.5$ and $g’(3.7)=0.2$.
Sketch a graph of g for the interval $t=3.63$ to $t=3.77$.
Write an equation to describe your sketch for 2b.
What did you assume about the ROC of g for that interval? How is your assumption reflected in your graph?
During what intervals of time is:
g decreasing? Why?
During what intervals of time is:
g’ decreasing? Why?
For what value(s) of t is:
g’(t) = 0? Why?
For each of (a) and (b), define a rate of change function $r_g$ so that its accumulation function g satisfies the following conditions. Then use GC to graph $r_g$ in the left pane and g in the right pane. On a printout of the two, highlight the features on $r_g$'s graph that correspond to the given properties on g.
On the interval [-1,2], $r_g(0)=r_g(1)=r_g(2)=0$, g has a global maximum at $x=-1$, g has neither min nor max at $x=1$, and g has a global minimum at $x=2$.
Henry tried to communicate to Sandra a relationship between y
and x that he called f. Henry, playfully,
described this relationship to Sandra via text message, giving her
information about $f’$ instead of f. Here is what Henry told
Sandra: $f’(2)=1, f’(4)=-2,$ and f’ has local maximums at
x=-1 and x=3 and a global maximum at x=7.
Sandra replied, "Not enough info to get your exact graph!"
Sketch 3 different graphs for f that fit these criteria.
Henry texted Sandra again about a new function g, and told her only that $g(3)=0.7$ and $g’(3)=5.6$. This is enough information to write the definition for a linear function (call it h) that is a good approximation of $g(x)$ from $x=2.9$ to $x=3.1$. Sandra replied, "Got it! Your linear function is ... " and she was correct. Write Sandra's definition of h.
Some textbooks say that a function f is decreasing over an
interval $[a,b]$ if for all values of x and y
in $[a,b]$, $f(x) \ge f(y)$ whenever $x \lt y$.
The animation below shows how the value of tan(x) varies as
the value of x (in radians) varies.
Answer this question based solely on the graph of $y=\tan(x)$: "Does the value of tan(x) ever decrease as the value of x increases?"
If your answer to (a) is "Yes" and your answer to (b) is "No", then you have experienced a contradiction. Resolve the contradiction by doing either of the following. Explain your answer.
Revise your understanding of the definition of a decreasing function that was given at the beginning of this exercise.
The function v is defined as $v(x)=\sum\limits_{k=0}^{10} 2^{-k}\sin\left(2^kx\right).$
Use GC to estimate the value of $v(7)$. How did GC calculate this value?
The function j is defined as $$j(x) = \left\lbrace\begin{align} &\dfrac{\cos(x)-1}{x} &\text{ if }x \neq 0 \\ &0 &\text{ if }x = 0 \end{align}\right.$$.
According to your answer to (a), what should the graph of j look like around $x=0$?
Graph the function j in GC. Zoom in on the graph around $x=0$. Does it look as you expected? If not, why?
Does the value of $r(0)$ make sense relative to the graph of $y=j(x)$? Why?
What do you now think is the value of $j'(0)$? Explain.
A steel ball bearing is dropped vertically from a height of 2 feet
above a concrete floor. The ball bounces vertically repeatedly. Values
of the function h give the bearing's height above the ground
at each moment in time during the first 7 seconds after it was
released. The graph of $y=h(x)$ is shown below. Assume that the
bearing changes direction instantaneously each time it bounces.
At what values of x is $h(x)$ defined?
At what values of x is $r_h(x)$ defined?
At what moments in time does the bearing's height have a local maximum? A local minimum?
Does h have a global maximum? A global minimum? Explain.
At what moments in time does the bearing's height have a rate of change of 0?
Over what intervals of time is the bearing's height increasing? Decreasing?
Over what intervals of time is the bearing's rate of change of height with respect to time increasing? Decreasing?
A ball suspended by a 10-foot long rubber cord is at rest. It is given a sudden push downward; the cord stretches, then retracts, pulling the ball upward. The ball bobs up and down with time. (See the animation, below). The ball's displacement from rest is given by the function d, defined as $$d(t)=-e^{-0.0625t}4\sin(t), 0≤t≤18$$where d(t) is in feet and t is in seconds since the ball was pushed.
At what moments during the first 18 seconds does the ball's displacement reach a local minimum? A local maximum?
Does the ball's displacement from rest have a global maximum? A global minimum?
The amount of water in a Phoenix water tower varies according to time
of day. The function w gives the amount of water (in
thousands of gallons) in the tank t hours after noon on
March 1, 2011. The graph of $y=r_w(t)$ is displayed below.
What does it mean that the point (0.9, 2.4) is on the graph of $y=r_w(t)$?
What is the unit of $r_w(t)$?
Over what period(s) of time was the amount of water in the tank increasing? Explain.