Derivatives

 

Let’s investigate derivatives by considering the function

f(x) = x4 / 4 - x3 / 2 - x2 + 2 x / 3 + 1

Turn on the CASIO CFX-9850G by pressing AC/ON.

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 - X2 + 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) = x3 - 3x2 / 2 - 2 x + 2 / 3. However, let’s make a mistake. Press EXIT and enter into Y2 X^3 - 2 X2 3 - 2 X + 2 3, press EXE to store it in memory.

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 - 3X2 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 x2 - 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 X2 - 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 d2 / dx2(Y1,X), press OPTN F2 (CALC) F2 (d2 / d x2), 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 SHIFT F4 (Sketch) F3 (Norm) VARS F4 (GRPH) F1 (Y) 1 , 1 EXE.

Press AC/ON. If we want to see Y2 and its inverse press SHIFT F4 (Sketch) F1 (Cls) EXE F4 (Inv) VARS F4 (GRPH) F1 (Y) 2 EXE.

Press AC/ON. If we want to draw the inequality y = Y2 press SHIFT F4 (Sketch) F1 (Cls) EXE F5 (GRPH) F6 ( |> ) F3 (Y=) VARS F4 (GRPH) F1 (Y) 2 EXE. As another example, we can find the integral of Y3 between -2 and 1 by pressing SHIFT F4 (Sketch) F1 (Cls) EXE F5 (GRPH) F5 (G ?dx) VARS F4 (GRPH) F1 (Y) 3, -2, 1 EXE. This result is ?dx = 7.5.

Press AC/ON SHIFT F4 (Sketch) F6 ( |> ) and we see five additional sketching tools: F1 (PLOT) allows us to plot points, F2 (LINE), allows us to have a line drawn between 2 points, F3 (Crcl), allows us to draw a circle given its center and its radius, F4 (Vert), allows us to draws a vertical line passing through a given x value, and F5 (Hztl), allows us to draws a horizontal line passing through a given y value.

Press EXIT.

Press MENU.

Turn off the CASIO CFX-9850G by pressing SHIFT AC/ON (OFF).