ICT for A2 Core Mathematics: e, natural logarithms and exponentials
TI-Nspire version
e
1. Definition of e
a. Add 2 sliders (labelled a and k) to the screen. Edit the settings of both
sliders so they have minimum 0, maximum 5, step-size 0.1. Set the value of
a to 2 initially.
b. Plot f1(x) = a^x.
c. Find the gradient of the curve at a point and use it to trace the gradient
function.
i. Add a point on the x-axis
ii. Find the perpendicular to the x-axis through the point
iii. Find the intersection of the perpendicular and the curve
iv. Find the derivative at this point (Analyse graph: dy/dx)
v. Use Measurement Transfer to transfer the derivative to the y-axis
vi. Find the perpendicular to the y-axis through this point
vii. Find the intersection of the two perpendicular lines
viii. Switch Geometry Trace on for this point and vary the original point on
the x-axis
d. Plot f2(x)=k*a^x
e. Vary k to verify that the gradient function is y=k*a^x for some value of k.
f. Find the value of k for different values of a.
g. Is there a value of a where the y=a^x is its own gradient function (i.e. k=1)?
2. Modelling growth functions in terms of e
a. Add 2 sliders (labelled a and b) to the screen. Edit the settings of both
sliders so they have minimum 0, maximum 5, step-size 0.1. Set the value of
a to 2 initially.
b. Plot f1(x) = a^x.
c. Add a point on the curve and display its coordinates.
d. Analyse the dy/dx at the point.
e. Plot f2(x) = e^(b*x).
f. Vary b so that that graph of f2(x) sits on top of f1(x).
g. What is the approximate relationship between the y-value of the point, dy/dx
and the value of b?
h. Move the point and verify that the relationship still holds.
i. Vary the value of a and verify that the relationship still holds.
3. Use a spreadsheet to investigate the following ways of generating the number e:
(change the Documents settings: Calculation mode to Approximate for these)
a. It is possible to conjecture that an infinite polynomial that would differentiate
x 2 x3 x 4
to itself is: f( x)  1  x     ... Using this as the definition of ex and
2! 3! 4!
1 1 1 1 1
substituting x  1 into it gives e x       ...
0! 1! 2! 3! 4!
 1
b. e  lim 1  
n 
 n
x
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n
Natural log
4. Inverse of ex (see prepared file)
The file shows the graphs of f1(x)=e^x, f2(x)=ln(x) and f3(x)=x
a. The gradient on y= e^x at the point (a, ea) is ea, this is displayed as a
gradient triangle (i.e. one with base 1).
b. The image of (a, ea) when reflected in y=x is (ea, a). The gradient at this
point is 1/ea, as displayed by the image of the gradient triangle under the
reflection.
c. The gradient on y=ln(x) is 1/(x-value at the point), i.e. the derivateive of ln(x)
is 1/x.
5. Area under 1/x
a. Plot f1(x)=1/x
b. Reset the window to XMin: –1, XMax: 7, YMin: –1.5, YMax: 1.5
c. Add two points on the x-axis.
d. Display the coordinates of the points and set the x-coordinate of the lefthand point to 1. Store the x-coordinate of the right-hand point as a.
e. Analyse the Integral of the curve between the two points.
f. What is the relationship between the integral and a?
Functions
6. Transformations of functions
a. f(x+a)+b
i. Add 2 sliders (labelled a and b) to the screen.
ii. Plot a function f1(x).
iii. Plot f2(x)=f1(x+a)+b
iv. Describe the relationship between the graphs of y=f(x) and
y= f(x+a)+b
v. Verify that this relationship works for different functions for f1(x).
b. cf(x) and f(dx)
i. Add a new Graphs page.
ii. Add 2 sliders (labelled c and d) to the screen.
iii. Plot a function f3(x).
iv. Plot f4(x)=c*f3(x)
v. Describe the relationship between the graphs of y=f(x) and y=cf(x)
vi. Plot f5(x)=f3(d*x)
vii. Describe the relationship between the graphs of y=f(x) and y=f(dx)
viii. Verify that these relationships work for different functions for f3(x).
c. f(–x) and –f(x)
i. Add a new Graphs page.
ii. Plot a function f6(x).
iii. Plot f7(x)= f6(–x)
iv. Describe the relationship between the graphs of y=f(x) and y=f(–x)
v. Plot f8(x)=–f6(x)
vi. Describe the relationship between the graphs of y=f(x) and y=f(–x)
vii. Verify that these relationships work for different functions for f6(x)
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7. Composite functions (see prepared file)
The file shows two functions and a point on the x-axis (representing the input.
a. Move the point on the x-axis. The value of f1(x) is displayed – this is
converted as an input to f2(x) by converting the y-value to an x-value. f2(x)
is calculated and this value is plotted as the y-coordinate with the xcoordinate being the original input.
b. Select Geometry Trace and select the new point. Drag the point on the xaxis and verify the curve traced is f2(f1(x)).
c. Verify that the relationship holds for other values of f1(x) and f2(x).
8. Inverse functions
a. Plot a function f1(x).
b. Add a Point On the curve and display its Coordinates.
c. Plot a point with the x and y coordinates reversed:
i. Use Measurement transfer to transfer the x-coordinate of the point to
the y-axis and the y-coordinate of the point to the x-axis.
ii. Add perpendiculars to the axes through these points on the axes.
iii. Add a Geometry trace to the point of intersection and drag the
original point on the curve.
d. What function would fit the curve traced by the point?
e. Is the curve traced by the point a function?
i. Is there a function which fits all the points traced?
ii. If there isn’t a function that fits all the points how would you need to
limit the domain of the original function so the inverse is a function.
NB to plot a function over a limited domain use “|” after the function to
define the domain (ctrl= on the handheld keypad): e.g. f1(x)=x2|x>0
9. Odd and even (see prepared file)
a. Plot a function f1(x).
b. Add a point on the curve, display its coordinates and store the x-coordinate
as a.
c. Plot the point on the curve with x-coordinate –a.
d. Drag the initial point and investigate f1(–a) for different functions:
i. Can you find functions where f1(–a)= f1(a) for all a? What
properties/symmetries do this curves display?
ii. Can you find functions where f1(–a)= –f1(a) for all a? What
properties/symmetries do this curves display?
10. Periodic
a. Add a function f1(x).
b. Add a slider a.
c. Add a function f2(x)=f1(x+a).
d. Investigate functions for f1(x) and a such that the functions are the same.
i. Try sin(x), cos(x), tan(x)
ii. Try sin(2x), sin (3x) …
Hint: you may find it easier to set the graphing angle to degrees
(Document Settings: Graphs & Geometry: Graphing Angle: degrees).
You may also need to edit the limits of the slider.
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