Dynamic Calculus
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The derived function (Part 4)
The applet below shows the graph of a polynomial function \(y = f(x)\).
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- Reposition the point \(P\) by moving the 'slider' left and right. What is the significance of the red point?
- If \(y = f(x)\), find \(f'(x)\). Check your answer by clicking the 'checkbox'.
- Where does the graph of \(y = f'(x)\) cross the \(x\)-axis? How is this related to the gradient of the curve \(y = f(x)\) at these points?
- For what values of \(x\) is the graph of \(y = f'(x)\) positive? For what values of \(x\) is the graph of \(y = f'(x)\) negative?
Describe the graph of \(y = f(x)\) in each of these domains.
- What is the degree of the function \(y = f(x)\)? What is the degree of the derived function \(y = f'(x)\)? What generalisation can we make about the degree of a polynomial function and the degree of its derived function?
- Write down the equation of a function whose derivative is \(f'(x) = 4{x^3}\). In your book draw a sketch of the function and its derived function.