Financial analysts use graphical models to predict stock values for a new stock. It is important for stock brokers to know about instantaneous changes of a stock price. To create a stock pricing model, analysts look at past behavior and future expectations for that stock.

A stock broker has developed the following model for the price of his favourite stock:

2^2 + + , < 0

() = { a, = 0

3( + 1)^4 + , > 0

a)- Why is it important for the stock broker to know about the instantaneous changes of a stock price?

b)- What should be true about the function f(t) in order for the stock broker to know how fast the stock price is changing in the present instant (t=0).

c) Determine the values of a, b and c that make the function f(t) differentiable at t=0

d) Determine algebraically the instantaneous rate of change at this present instant t=0

e) Make a sketch of f(t) on graph