Derivatives

Let’s investigate derivatives by considering the function

f(x) = x^{4} / 4 - x^{3} / 2 - x^{2} + 2 x / 3 +
1

Turn on the **CASIO CFX-9850G** by pressing

Use the cursor arrows to highlight the GRAPH menu and press **EXE **or just press **5**
when at the main menu screen.

Enter X^4 ¸ 4 - X^3 ¸ 2 - X^{2}
+ 2 X ¸ 3 + 1 into Y1 and press **EXE** to store it in
memory.

Press **SHIFT** and **F3** (V-Window) to enter the viewing window screen, and
press **F1** (INIT) to select the initial viewing window. Press **EXIT** to return
to the function list.

Press **F6** (DRAW) to see the graph.

Obviously this is not the best viewing window for this graph. So let’s change the
viewing window by pressing **SHIFT** **F3** (V-Window) and entering -2 for Xmin and
3 for Xmax. Press **EXIT** and **F6** (DRAW) for a better view.

The first derivative of Y1 is f’(x) = x^{3} - 3x^{2} / 2 - 2 x + 2
/ 3. However, let’s make a mistake. Press **EXIT** and enter into Y2 X^3 - __2 X ^{2}
¸ 3__

Press **F6** (DRAW) to see the two graphs.

How do we know we made a mistake taking the derivative?

Let’s fix the mistake and change the color of Y2.

Press **F6** (G<->T) to return to the function list.

Scroll to Y2 and fix the problem so that Y2 is X^3 - 3X^{2} ¸
2 - 2 X + 2 ¸ 3. Press **EXE** to store the new Y2 into
memory.

Scroll back to Y2. Press **F4** (COLR) **F2** (Orng). Press **EXIT**.

If we turn on the derivative feature of the calculator we will be able to compare the
value of the numerical derivative of Y1 with the actual value of Y2 when we trace. We can
do this by pressing **SHIFT**, **SETUP** (MENU), scrolling down to Derivative, and
pressing **F1** (On). Press **EXIT** and **F6** (DRAW) to view both functions.

We can trace on these two functions by pressing **F1** (Trace). Tracing starts on
the left of the viewing window and on the first selected graph, in this case Y1. Use the
right cursor key to "walk on" Y1. See that the numerical derivative is also
displayed on the screen.

As an example, trace along Y1 to x = -1.325396... and y = 0.295335..., the derivative is dY/dX = -1.645. Use the down cursor arrow to verify that for Y2 y = -1.645... .

The second derivative of Y1 is f’’(x) = 3 x^{2} - 3 x - 2. We will
enter this function into Y3 and make it green.

Press **F6** (G<->T) to return to the function list.

Scroll to Y3 and enter 3 X^{2} - 3 X - 2. Press **EXE** to store the Y3 into
memory.

Scroll back to Y3, press **F4** (COLR) **F3** (Grn), and press **EXIT**.

Press **F6** (DRAW) to view all three graphs.

If you were given this picture with no additional information, would you be able to give some meaning to what you see?

Draw the numerical derivatives for f(x).

Press **F6** (G<->T) to return to the function list.

Deselect Y2 by scrolling to it and pressing **F1** (SEL).

Deselect Y3 by scrolling to it and pressing **F1** (SEL).

Scroll to Y4. To enter d/dx(Y1,X), press **OPTN** **F2** (CALC) **F1** (d/dx),
press **VARS** **F4** (GRPH) **F1** (Y) 1, X ), and press **EXE** to store
this function in memory. You make Y4 orange by scrolling to it and pressing **F4**
(COLR) **F2** (Orng) and **EXIT**.

Scroll to Y5. To enter d^{2} / dx^{2}(Y1,X), press **OPTN** **F2**
(CALC) **F2** (d^{2} / d x^{2}), press **VARS** **F4** (GRPH) **F1**
(Y) 1, X ), and press **EXE** to store this function in memory. You make Y5 green by
scrolling to it and pressing **F4** (COLR) **F3** (Grn) and **EXIT**.

Press **F6** (DRAW) to view all three functions. This will take some time since the
calculator is using a numerical algorithm to get a value for each pixel on the screen that
it is displaying.

Trace on these functions by pressing **F1** (Trace) and using the cursor keys. As an
example, use the right cursor arrow to trace on Y1 to x = -0.333333... and y =
0.688271...; the display screen also shows dY/dX = 1.1296. Use the down or up cursor arrow
to jump to Y4; the x value is the same but the y value is the value of the derivative of
f(x), i.e. y = 1.129629... and the derivative of Y4 at x = -0.333333... is also displayed
dY/dX = 250. Now use the up or down cursor arrow to jump to Y5. Here y = -0.666666... and
dY/dX = -4.997.

What does G-Solv do?

Deselect Y4 and Y5 by scrolling to them and pressing **F1** (SEL) and select again
Y2 and Y3. Draw the functions Y1, Y2, and Y3 by pressing **F6** (DRAW).

Press **SHIFT** **F5** (G-Solv) **F1** (Root). Use the up or down arrow key to
select the desired function, for example Y1 and press **EXE**. The calculator works
from left to right until it finds a zero of the selected function. After a root has been
found you may press the right or left arrow to find additional roots of the given function
in the current viewing window. The roots for Y1 are: (1.206637..., 0) and (2.898361, 0).
Practice finding the roots of Y2 and Y3.

Press **SHIFT** **F5** (G-Solv) **F2** (MAX). Use the up or down arrow key to
select the desired function, for example Y2 and press **EXE**. The calculator works
from left to right until it finds a local maximum of the selected function. After a MAX
has been found you may press the right or left arrow to find additional local maximums of
the given function in the current viewing window. The local maximum for Y2 is
(-0.457427..., 1.171949...). Practice finding the local maximums of Y1 and Y3.

Press **SHIFT** **F5** (G-Solv) **F3** (MIN). Use the up or down arrow key to
select the desired function, for example Y3 and press **EXE**. The calculator works
from left to right until it finds a local minimum of the selected function. After a MIN
has been found you may press the right or left arrow to find additional local minimums of
the given function in the current viewing window. The local minimum for Y3 is (0.5,
-2.75). Practice finding the local minimums of Y1 and Y2.

Press **SHIFT** **F5** (G-Solv) **F4** (Y-ICPT). Use the up or down arrow key
to select the desired function, for example Y1 and press **EXE**. The calculator works
from left to right until it finds the y-intercept of the selected function. The
y-intercept of Y1 is (0, 1). Practice finding the y-intercepts of Y2 and Y3.

We now want to find the intersection of two functions.

Press** SHIFT F5 **(G-Solv)** F5 **(ISCT).** **Use the up or down arrow key to
select the first desired function, for example Y1 and press **EXE**. Use the up or down
arrow key to select the second desired function, for example Y2 and press **EXE**. The
calculator works from left to right until it finds the first intersection point. Use the
right cursor key to find additional intersection points in this viewing window. The points
of intersection for Y1 and Y2 are (-1.020986..., 0.080730...), (-0.129383...,
0.898115...), and (1.935552..., -1.572787...). Practice finding intersection points of Y1
and Y3 or Y2 and Y3.

To find a y value for a given x value we press **SHIFT** **F5** (G-Solv) **F6**
( |> ) **F1** (Y-CALC). Use the up or down arrow key to select the desired function,
for example Y2 and press **EXE**. Enter a value for X at the screen prompt, for example
0.5, and press EXE. The calculator finds the y value corresponding to this x value, in
this case y = -0.583333... . Practice finding y values for other values of x using Y1, Y2
or Y3.

To find an x value for a given y value we press **SHIFT** **F5** (G-Solv) **F6**
( |> ) **F2** (X-CALC). Use the up or down arrow key to select the desired function,
for example Y3 and press **EXE**. Enter a value for Y at the screen prompt, for example
-0.25, and press **EXE**. The calculator finds the x value corresponding to this y
value, in this case x = -0.412870... . Practice finding x values for other values of y
using Y1, Y2, or Y3.

To find the graphical integral of a function on an interval we press **SHIFT** **F5**
(G-Solv) **F6** ( |> ) **F3** (?dx). Use the up or down arrow key to select the
desired function, for example Y1 and press **EXE**. Now use the right or left arrow
keys to move to a lower limit and press **EXE** to store this lower limit in memory,
for this example x = -0.333333... . Use the right or left arrows to move to an upper limit
and press **EXE** to store this upper limit in memory, for this example x = 0.5. The
calculator shades the appropriate area and displays a numerical integral, ?dx =
0.821116... . Practice by finding integrals using other intervals on function Y1, Y2, or
Y3.

Let’s store the current functions that we have in memory. Press **F6** (G
<-> T) **F5** (GMEM) **F1** (STO) **F2** (GM2). Also, keep the viewing
window in memory by pressing **SHIFT** **F3** (V-Window) **F4** (STO) **F2**
(V-W2) and **EXIT**.

Now recall the function stored in GM1 by pressing **F5** (GMEM) **F2** (RCL) **F1**
(GM1). Recall the viewing window we used with this function by pressing **SHIFT** **F3**
(V-Window) **F5** (RCL) **F1** (V-W1). Press **EXIT** and **F6** (DRAW).

Use Y-CALC in G-Solv to evaluate y1 at x=2. What happens and how do you explain this?

Press **EXIT**. Now recall the functions stored in GM2 by pressing **F5** (GMEM) **F2**
(RCL) **F2** (GM2). Recall the viewing window we used with these function by pressing **SHIFT**
**F3** (V-Window) **F5** (RCL) **F2** (V-W2). Press **EXIT**.

Press **MENU**. Use the cursor arrows to highlight the RUN menu and press **EXE**
or press 1 when at the main menu screen.

We will now review how to use the sketching capabilities of the calculator in RUN mode.
To see the sketching possibilities in RUN mode press **SHIFT** **F4** (Sketch).

To clear the drawing window press **F1** (Cls) **EXE**.

If we want to see the tangent line to Y1 at x = 1 press **F2** (Tang) **VARS** **F4**
(GRPH) **F1** (Y) 1 , 1 **EXE**.

Press **AC/ ^{ON}**. If we want to see the normal line to Y1 at x = 1 press

Press **AC/ ^{ON}**. If we want to see Y2 and its inverse press

Press **AC/ ^{ON}**. If we want to draw the inequality y = Y2 press

Press **AC/ ^{ON}**

Press **EXIT**.

Press **MENU**.

Turn off the **CASIO CFX-9850G** by pressing