





							COMPUTER ALGEBRA SYSTEMS - TECHNOLOGY IN MATHEMATICS EDUCATION

Solution:

Consider the cubic function

f (x) = x³

The gradient function is:

f´(x) = 3x². Choose P(1,1).

f´(1) = 3. Therefore the equation of the line tangent to the curve at P is: y = 3x - 2

The student should choose various other points and perform the same procedures as above.

The student can make the following conjectures based on the examples above:

• The x-coordinate of Q is –2 times the x-coordinate of P.

• The gradient at Q is 4 times the gradient at P.

• The y-intercept at Q is –8 times the y-intercept at P.

An Investigation in Cubics

Q is the point of intersection of the cubic and the tangent line through P. Hence, we need to solve:

x³ = 3x - 2 . Using either technology, or by hand. x = 1 or x = -2. So, Q is the point whose coordinates are

(-2,-8). The gradient of the curve at

Q is 12, hence the equation of the tangent at Q is y = 12x + 16.

> TOP

Q is the point of intersection of the

cubic and the tangent line at P.

Solving the equation: x³ = 3a²x - 2a³

for x, we get: Q(-2a, -8a³).

Hence, the x-coordinate of Q is

–2 times the x-coordinate of P.

• The gradient at P is 3a². The gradient at Q is 12a². Hence the gradient at Q is 4 times the gradient at P.

• The equation through Q is: y = 12a²x + 16a³. Hence the y-intercept at Q is –8 times the

y-intercept at P.

> TOP

1. Consider any point on the graph of y = x³ , e.g., P(1.1). The tangent line at P will intersect the curve again at some point Q.

Find:

• the gradient function of the above cubic.

• the gradient of the cubic at P

• the equation of the tangent at P in the form y = mx + c

• Q, the point of intersection of the cubic and the tangent line at P

• the gradient of the cubic at Q

• the equation of the tangent line at Q in the form y = mx + c

2. After repeating the above steps for several other points on the above cubic, formulate conjectures on the relationship of:

• the x-coordinates of P and Q

• the gradients of the curve at P and Q

• the y-intercepts of the tangent lines at P and Q

3. Confirm your conjectures symbolically.

Proofs:

• f (x) = x³; P(a, a3); gradient of f (x) at P is 3a². The equation of the tangent at P is:

y = 3a²x - 2a³.

White Box/Black Box Model

- Buchberger

CAS as a “white box”

- “PeCAS”

CAS as a “black box”

- Investigations

- Mathematical Modeling

- Scaffolding

- Picture mathematics