Dynamic Calculus
Home > The gradient function
The derived function (Part 1)
The applet below shows the graph of the linear function \(f(x) = x + 2\).
[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 \(f(x) = x + 2\), find \(f'(x)\). Check your answer by clicking the checkbox.
- What is the gradient of the line \(y = x + 2\)? How does this relate to your answer from question (2)?
- What is the range of \(y = f(x)\)? What is the range of \(y = f'(x)\)?
- Can you find a linear function \(y = f(x)\) for which the range of \(y = f'(x)\) would be a negative value?
- What is the degree of \(f(x) = x + 2\)? What is the degree of \(y = f'(x)\)?
- What generalisation can you make about the degree of the derivative of a linear function? Explain the significance of this with regard to the graph of \(y = f'(x)\).
- What would be the effect on the graph of \(y = f'(x)\) of moving \(y = f(x)\) 'up the \(y\)-axis'? What would be the effect on the graph of \(y = f'(x)\) of moving \(y = f(x)\) 'down the \(y\)-axis'?
- Give an example of a function whose derivative is \(f'(x) = -2\). In your book draw a sketch of the function and its derived function.