ICT for AS Core Mathematics
Autograph Version
Differentiation/Integration
1. Plot a function and find the gradient of the chord between two points
a. What happens to the gradient of the chord as the chord reduces in length?
b. How is this related to the gradient of the tangent?
c. Could you predict what the gradient of the chord would be if it was infinitely
short?
2. Explore the gradient at a point on a curve
a. Plot a curve; e.g. y = x 2
b. Find the tangent to the curve at a point:
i. Add a point on the curve
ii. Add the tangent to the curve at the point
iii. The gradient of the curve at a point can be defined as the gradient of
the tangent to the curve at a point
c. In a spreadsheet, record the value of the gradient for different x -values
d. Try changing the initial equation of the curve to a different function; e.g.
y = 3x 2 , y = x3 , y = x 4 , y = 5 x 2 , =
y x 2 + 3x
3. Gradient functions and stationary points
a. Plot a function and it’s gradient function
i. Plot a curve, e.g. y =x 3 − 3 x 2 − 2 x (a cubic function works well for this
task)
ii. Plot the gradient function: select the curve and press
b. Add a point to the original curve and display the tangent to the curve at this
point
c. Describe the gradient of the tangent in terms of the gradient function
d. When is the tangent horizontal?
e. When is the gradient of the tangent at a maximum/minimum?
4. Tangents and Normals
a. Draw the curve y = x 2
b. Add two points to the curve
c. Display the tangents to the curve at each point
d. Find the point of intersection of the two tangents: solve f( x) = g( x) with the
tangents selected
e. Measure the angle between two tangents
f. Add the line segment between two points on the curve and find where this
intersects the y -axis (by adding the line x = 0 )
g. Find positions of the two tangents so they are perpendicular (you may find it
helpful to edit the axes settings so that the pips on the x -axis are set at small
intervals such as 0.1)
i. Record the point of the intersection of segment between the two
points with the y -axis
ii. Record the coordinates of the point of intersection of the tangents
h. Repeat f. for different positions of the tangents that are perpendicular. What is
the relationship between the points of intersection?
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5. Stationary points
a. Find a cubic with two distinct stationary points and plot it, e.g.
y = x3 + 3x 2 − 5 x − 3
b. Find the turning stationary points: solve f '( x) = 0
c. Add points on the curve on top of the stationary points and find the mid-point
of these points.
d. Repeat this for different cubics and observe a relationship between the points
i. Prove this relationship using differentiation
6. Introduction to integration (finding areas under curves)
a. Draw a curve, e.g. y = x 2
b. Find the area under the curve:
i. Select the curve and choose the “Find Area” function
ii. Start with the area between 0 and 1 using 5 divisions
iii. Select the rectangles and use the “Animate Object” option (the
lightning bolt next to the constant controller) to increase the number of
divisions
iv. What is the area under the curve tending to?
v. Repeat this for the area between 0 and 2, 0 and 3, etc.
vi. Can you generalise this? What is the area under y = x 2 from 0 to a?
vii. Try some other curves and some straight lines
7. Integration (reverse of differentiation)
a. Plot =
y ax + b
b. Find a function that would have =
y ax + b as its gradient function:
i. Plot a second curve e.g. y = x 2
ii. Plot the gradient function for the second curve
iii. Edit the second function so that its gradient function is the same as
=
y ax + b
c. Repeat c. for different values of a and b
d. Change the first curve to y = ax 2 + bx + c
e. Find a function that would have y = ax 2 + bx + c as its gradient function
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8. Finding integral functions
a. Plot a function and its integral
i. Plot a curve, e.g. y = x 2
ii. Select the curve and choose the integral function
iii. Add a point for the integral function to pass through
b. What function would give the integral function? How does changing the
selected point change the curve given?
c. Try some different functions for f. How is the Area function related to the
original function?
9. Finding the general cases of cubics
a. Plot y = ax 3 + bx 2 + cx + d
b. Find values of a , b , c and d so that the curve has…
i. …2 stationary points
ii. …1 stationary point
iii. …0 stationary points
c. What are the conditions on a cubic that determine whether it has 0, 1, 2
stationary points?
10. Trapezium rule
a. Plot a function that you can’t integrate, e.g. y = sin x , y= x ⋅ cos x ,
=
y
1 − x2
b. Open a spreadsheet and use this to estimate the area under the curve
between 0 and 1 by the trapezium rule with a strip size of 0.2:
i. In cells A1:A6 enter 0, 0.2, 0.4, 0.6, 0.8, 1
ii. In cell B1 enter a function of A1, e.g. =SIN(A1) and fill down to B6
iii. In cells C1 and C6 enter 1, in cells C2:C6 enter 2
iv. In cell D1 enter =B1*C1 and fill down to D6
v. In cell D7 enter =SUM(D1:D6)
vi. In cell D8 enter 0.2 (the strip size)
vii. In cell D9 enter =.5*D8*D7
c. Compare the value of the area to area calculated by the trapezium rule on the
graph: select the curve, choose “Find Area” and select the Trapezium rule
with 5 strips
d. How could you find an improved trapezium rule estimate?
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