Explain why a curve that is symmetric about the x-axis is not the graph of a function, unless the function is y=0

Hi,
Let us consider this to be math problem. From the physics point of view the units are inconsistent:thus the problem is ill-posed.
What you need it to draw the graph pf the function y=cos(t^2), then the integration feature of the calculator to calculate the integral between 0 and some upper limit, which will be found by trial and error.
Here are some screen capture to help you along the way
Make sure the graph type is y=cos(X^2). Don<t worry the variable in this type of graph must be X
Choose the window to start at x=0

Here is your graph.

Press SHIFT F5 (G-Solv) You see the following image

Press F6 (Arrow pointing right) to access the other features, as on the following screen:

You see the integral sign above F3: Press F3 to do integration.


Use the left Arrow key to move the cross on the graph to X-0 or just enter 0. and press EXE. This is the lower bound.

You need to enter the upper bound. Since you do not know where to stop you must try different values and each time, read the value of the integral. For example if you enter Xupperlimit=0.5
the value of the integral is 0.45 (not 1)
Repeat the process changing the upper limit until (if you are lucky) you hit an integral of 1.
Now a few more pictures





Getting close. Now I use the 1st zero of the function, x=1.2533 approximate

Getting close. However from that zero to the next the area under the curve under the x-axis is negative and there will not be a solution until maybe somewhere in the second positive part.
So you can accept the value of the integral from X=0 to the 1st zero (1.2533) to be about 1. Or you will keep playing the game. But as the next picture shows, you will not get a better approximation than at the first zero.



Good Luck.

This is problem with graphing differential equation. Invalid Axes error occurs if Axes=TIME or if t(time) is set as a CUSTOM axis. This is important to noticed: if Fields=DIRFLD you cannot plot a time axis. See captured image:




This graph represented differential system of equations y1'(x)=y2(x), y2'(x)=-y1(x) for initial conditions y1(0)=0, y2(0)=-1.Finally, x axes=y1(x) and y axes=y2(x)

No need for a claculator to understand these concepts.
A linear function is a function of the form y=ax+b. It contains an independent variable,x, a dependent variable, y, and two constants, a and b,
The value of the constant a is the measure of the rate of change of the function, The value of the constant b is the ordinate (value along the y-axis) where the straight line cuts the y-axis. It is called the y-intercept, or the initial value.
If the value of b=0, the straight line passes through the origin O(0,0). The purists call this type of variation, a direct variation. If b is not equal to 0, it is called a partial variation or some other name.

If you have the graph drawn, b is read off the y-axis: You look at the graph and try to estimate the ordinate of the point where the line cuts (intersects) the y-axis.
To get the rate of change a, you select two points on the line. Let 1st point have coordinates (x1,y1) and the 2nd point have coordinates (x2,y2).
The rate of change is given by the ratio (y2-y1)/(x2-x1). You can also use a= (y1-y2)/(x1-x2). Respect the order in the two expression or you will get the negative value of the rate of change.

The vertical line is called an asymptote, it is not connecting the graph, it means that the function goes to infinity as the function nears the line. In this case the graph goes to infinity and negative infinity as the x nears 7. This is because the function becomes undefined at x = 7 , plug 7 into x and we get 1/0, dividing by zero is undefined. Thus, the calculator puts the vertical asymptote in the graph to symbolize the action of the function nearing infinity.

The intersection function on the t-nspire is meant to find the intersection of two graphs, not of the graph and an axis. The easiest way to do this is to set the equation equal to 0 and then use nSolve() to find the intercept.

Graph the function then use the trace function to locate the point(s) where two curves intersect. In your case one of the curves is the X-AXIS.

See the screen capture to learn how to use the trace functionality.

Hello,
The function Intersect from the CALCULATE menu finf the coordinates of a point at which two or more curves intersect.
To use it:
1. Draw the functions.

2. Press [2nd][CALC][5:Intersect]



The cursor is on one of the curves. Read the equation top of the screen. If it is one of the curves you want press [ENTER]. The cursor jumps to another curve (in this case the only other curve).

Read the equation on top of the screen to verify thst it is the correct one. Press [ENTER]. The calculator asks asks for a guess of the coordinates of the intersection point.


As the intersection point is to the left of the current cursor position, use the left arrow to move cursor closer to the point.


Press [ENTER], and wait for the solution. Here it is.


In your question you talk about y intercept. If you want to calculate the ordinate of the point where a curve intersects the Y-axis, it is more efficient to use the [2nd][CAL][1:Value] selection.

You enter X=0 and press [ENTER]. The cursor jumps on the first curve (Y1=) an gives you the y-intercept.

Notice the position of cursor on graph. The y-value at the bottom is its ordinate.
To get the y-intercept of the second curve, leave the cursor on y axis and press the DownArrow. Cursor jumps to tthe second curve.
Since the X=0 is still stored, the value of y is displayed directly.



Hope it helps.

Hello,
The second derivative indicates the rate of variation of the first derivative. Each point where the second derivative vanishes (has value 0) is an inflexion point of the curve.


How about using you calculator to graph that function? that is how you learn things. It will be my pleasure to help you learn how to graph functions.

Hello,
Press [Window]
Set the values of Xmin, Xmax, Ymin, Ymax.
If the x values that the function takes are all positive, you may set Xmin =0. If the values of the function are all positive, you can set Ymin=0.
If you are drawing a sine or a cosine function, you know that the function values are between -1 and +1, so you cant restict Ymin an Ymax to say -1.2 and 1.2. Similarly if you draw trigonometric function no need to have a range of X values between -10 and 10: the function being periodic you can restrict values (in radians to -pi to +pi, ) or to -90 to +90 degrees.
You set the limits of the window by making use of your knowledge of the functions you are drawing.

Hope it helps.

Instruction book for this calculator is shocking for sure. It would take too long to explain fully, here is an example of plotting a simple Y= graph.

In main menu, when you first turn it on, press 1
This is normal calculator mode.
Press [shift] [menu] to get the set up parameters.
Scroll curser down one to Func Type, and select Y= if this is not already set, by pressing F1
Continue scrolling down to set Coord, Grid, Axes and Label functions. Turn these on or off as desired. Turn them all off for this example. F1 and F2 select On or Off here.
Press [exit]
Press [shift] F3 to access the V-window parameters. Here you set the size of the axis in the graph
Enter 0 for Xmin, 360 for Xmax, 0.1 for scale
Enter -1 for Ymin, 1 for Ymax, and 0.1 for scale
Press [exit]
Press [shift] F4 to enter graph plot mode (Sketch mode)
Press F1 then [EXE] to clear any previous graphs
Press F5 then F1 to select Y= type graph
Display should show Graph Y=
press [sin] [ X,0,T] (this key just under red alpha key) then [EXE]
Calculator should draw a sine wave, showing 1 complete wave
Try setting the parameters in Set up, Axes On, Label On etc to see the effect.
Hope this helps a little. Page 123 in book deals with graphing.