Mathematics Magazine for Grades 1-12

10/2004

## Theory:

Chain Rule

= f(g) g(x)

Proof of f(g(x)) = f(g)   g(x) from the definition

We can use the definition of the derivative:

 f(x) = lim     d-->0 f(x+d) - f(x)         d

Therefore, f(g(x)) can be written as such:

f(g(x)) = =

= =

## Solutions from the Previous Issue:

1.        A bicycle wheel of radius r = 1.5  m starts from rest and rolls 100 m without slipping in 30 s. Calculate a) the number of revolutions the wheel makes, b) the number of radians through which it turns, c)The average angular velocity.

Solution:

a) If there is no slipping, the arc length through which a point of the rim moves is equal to the distance traveled, so that the number of revolutions is:

n  = = 10.6.

b)

c) Average angular velocity:

2.        Assuming that the angular acceleration of the wheel given above was constant, calculate: a) The angular acceleration, b) the final angular velocity c) the tangential velocity and tangential acceleration of a point on the rim after one revolution.

Solution:

a) For constant angular acceleration:

Using  and solving for α gives:

b)

c) After one revolution, Δθ = 2π. Using  we get . The tangential velocity and acceleration are:

and

## Proposed Exercises: