Chapter 1.2, Problem 78E

Solution

(a)

To find: is it increasing, decreasing, constant, or none of these.

It is increasing.

Given information: 

The GPA $\left(y\right)$ as a function of percentage grade $\left(x\right)$

  

Calculation:

According to the table, the grade increases, the GPA also increases.

Hence, $y$ as a function of $x$ is increasing.

Therefore, it is increasing.

(b)

To find: a table showing the change $\left(\Delta y\right)$ in GPA as you move down the list.

The table

  

Given information: 

The GPA $\left(y\right)$ as a function of percentage grade $\left(x\right)$

  

Calculation:

Here,

  $\Delta y$ can be found by subtracting the consecutive $y$ values.

There will be $1$ less row now since we are finding differences.

The table is shown below:

x moves from	$\Delta x$	$\Delta y$
60 to 65	5	1.00
65 to 70	5	1.05
70 to 75	5	0.52
75 to 80	5	0.43
80 to 85	5	0.36
85 to 90	5	0.33
90 to 95	5	0.31
95 to 100	5	0.28

(c)

To find: is it increasing, decreasing, constant, or none of these.

It is decreasing.

Given information: 

The change $(\Delta y)$ in GPA as a function of percentage grade $\left(x\right)$ ,

  

Calculation:

Here,

  $\Delta \Delta y$ can be found by subtracting the consecutive $\Delta y$ values.

There will be $1$ less row now since we are finding differences.

The table is shown below:

$\Delta y$	$\Delta \Delta y$
1.00
1.05	0.05
0.52	-0.53
0.43	-0.09
0.36	-0.07
0.33	-0.03
0.31	-0.02
0.28	-0.03

Except from $1.00$ to $1.05,\Delta y$ as a function of $x$ is decreasing since the $\Delta y$ values decrease as $x$ increases.

Hence, it is decreasing.

(d)

To find: the shape of the graph and $y$ is an increasing function of $x$ and $\Delta y$ is a decreasing function of $x$ .

The shape of the graph is curved downward but increasing from left to right.

Given information: 

The change $(\Delta y)$ in GPA as a function of percentage grade $\left(x\right)$ ,

  

Calculation:

Graph the two sets of data, letting the $\Delta y$ values start at $x=65$ you will obtain the scatter plots below:

  

When $y$ is an increasing function of $x$ and $\Delta y$ is a decreasing function of $x$ , the shape of the graph is curved downward but increasing from left to right, like the graph of a square root function, $y=\sqrt{x}$ .

Therefore, the shape of the graph is curved downward but increasing from left to right.

(e)

To find: the graph of a function $y$ of $x$ such that $y$ is a decreasing function of $x$ and $\Delta y$ is an increasing function of $x$ .

The shape of the graph is curved upward but decreasing from left to right.

Given information: 

The change $(\Delta y)$ in GPA as a function of percentage grade $\left(x\right)$ ,

  

Calculation:

Using the same reasoning of the result from part ( $\text{d})$ : When $y$ is a decreasing function of $x$ and $\Delta y$ is an increasing function of $x$, the shape of the graph is curved upward, but decreasing:

  

Therefore, the shape of the graph is curved upward but decreasing from left to right.