1-2. Enter and plot (with
SLOW plot on) y = ax.
Set a = 2. Place a point ON the curve,
draw a tangent and move the point up
and down to discuss the slope. Draw the gradient
function (slope).
Use zooms
and the constant controller to vary ‘a’ to
find a value that gives a gradient function
the same as y = ax.
3-4. Draw y = 1/x (slowly!).
Place and select points on the curve at x = 1 and x
= 2. Find the area using 500 rectangles.
Drag the right point until x = 2.718, when the area is
close to 1 (an underestimate – see the zoom! 5. Mark the points (1,
0) and (e, 1) as ‘targets’,
so that the (slow) plot of y = lnx can
be correctly anticipated, and can subsequently
shown to be the function whose gradient is y
= 1/x (for x>0).
1. Plot y = sinx in
Radians (slowly). Use the ‘default’ scales.
Then plot the Gradient function (slowly) – a roaming
tangent is drawn and the 1st derivative is created. It
pauses automatically at all max, min and points of inflexion.
2-3. Doing the same in Degrees is
instructive (‘default’ scales this time offer –90° to
360°) a good application of the Chain Rule, since
y = sinx° = sin(p/180.x), so dy/dx = p/180. sin(p/180.x)
4. Plot y = asin(bx+c),
with c = 0 and d = 0. The gradient function (slope) is
a ‘dependent’ object, so if a, b, c or d
is varied the gradient will vary accordingly, offering
an excellent summary test of understanding of the Chain
Rule.
5. There are many opportunities to
show the Trig Formulae graphically.
Here sin²x = ½(1 – cos2x)
is shown on the axes, and also in the Help file.
7. A nice illustration
of inverse Trig functions, showing that
the 2 angles in a right angled triangle add to p/2.
There are over 200 pages in the Autograph HELP file, including
a long list of formulae for pure mathematics and statistics.
1.Five selected points, construct
a conic – a dynamic object that will
recalculate if any of the points is moved.
Select each branch in turn and construct the asymptote.
2. A family of conics in polar
form: 1/r= 1 – kcosq. Polar graphs
can also be entered in the form r = f(q), r² =
f(q)
3. A construction of a parabola
from first principles, with directrix x
= –2, and focus (1, 0). Tutorial
recording||Autograph
file
Plot y = x and y
= acosx. Put a point ON y = x, select it
and y = acosx, and use the right-click option “x
= g(x) iteration”. This generates
a dependent object that will update if anything is
changed (eg the position of the point, or the value
of ‘a’).
Both 1st and 2nd order differential equations can be
entered implicitly in Autograph, as dynamic objects.
1st Order Differential Equations
In this case , enter yy' + x = 0 [where
y' is thenotation for dy/dx]. The 'slope field' is drawn,
and you can select a set of starting points on the y-axis.
Having noticed that the solutions are circles, centre
(0,0), Plot the general circle y² + x² =
r² for r = 1, 2, 3, 4,5 .
Differentiate this equation (implicitly) and divide by
2.
Another tack is to re-arrange the original equation as
y' = –x/y,noting that every value of the
slope is perpendicular to y/x, the slope ot the straight
line from (x, y) to (0,0).
2nd Order Differential Equations
Try a careful build-up from first principles: y" = 1, y" = –10, y" =
x, y" = y,y" = –y, y" +
y = 0, y" + 2ky' + y = 0, and finally y" +
2ky' + y = f(x).
1. A Binomial distribution with
any parameters can be tested. Use the option "Probability
Calculations". The boundary values can be dragged.
The results box table can be copied out to Excel or Word.
2. Here, Type 1 and 2 errors from
2 normal distributions can be explored visually. The
on-screen text boxes get data from the status bar,
and the Autograph font provides the symbols.
