Polynomials

Exercises - guess the derivative function!

The constant function

Let \(a=b=c=d=0\). Find a formula for \(f'(x)\)! Explain!

First degree polynomials

Let \(a=b=c=0\), try different values of \(d\). Find an exact formula for \(f'(x)\)! Explain!

What happens with the graph of \(f'(x)\) when you change the slider e? Explain!

Higher degree polynomials

What kind of a function is \(f'(x)\) when \(f(x)\) is a second degree polynomial?

What kind of a function is \(f'(x)\) when \(f(x)\) is a third degree polynomial? Make a guess!

What kind of a function is \(f'(x)\) when \(f(x)\) is a fourth degree polynomial? Make a guess!

What happens with the graph of \(f'(x)\) when you change the slider e?

The derivative of a polynomial of degree n

Try to find an exact formula for \(f'(x)\) when \(f(x)=cx^2\) (let the sliders \(a, b, d, e\) be zero). How is the gradient of the line related to the number \(c\)? Try to figure it out by changing the slider c.

Having the formulae for \(f'(x)\) when \(f(x)=ax^0, f(x)=ax^1, f(x)=ax^2\); try to guess the formula for \(f'(x)\) when \(f(x)=ax^3\) and \(f(x)=ax^4\).

Antiderivative

If \(f'(x)=3\) what is \(f(x)\)? Could many functions have the same derivative functions?

If \(f'(x)=4x\) what is \(f(x)\)?

If \(f'(x)=6x^2\) what is \(f(x)\)?

If \(f'(x)=x^{-1}\) what is \(f(x)\)?