Dynamic Calculus
Home > Curve sketching and problems on maxima and minima
Curve sketching (Part 1)
The graph of \(f(x) = a{x^3} + b{x^2} + cx + d\) is shown below. Investigate the function of the 'sliders' by moving each left and right. What do the graphs of \(y = f'(x)\) and \(y = f''(x)\) look like?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
- Click the checkbox 'show trace point \(P\)'. Reposition the point \(P\) on the curve by moving the slider left and right. Draw a rough sketch of the function \(y = f(x)\) in your book.
- Where on your graph of \(y = f(x)\) do stationary points occur? Confirm your answer by clicking the checkbox 'show turning point(s)'. What is the value of \(f'(x)\) at these points?
- On the same set of axes, mark the position of \(y = f'(x)\), where \(f'(x) = 0\).
- Move the slider so that the point \(P\) is left then right of the stationary points. What does the sign of \(f'(x)\) indicate about the graphs of \(y = f(x)\) and \(y = f'(x)\) in these domains?
- When \(y = f(x)\) is of degree 3, what is the degree of \(y = f'(x)\)? What would this tell you about the shape of the graph of \(y = f'(x)\)?
- In your book complete the graph of \(y = f'(x)\). Check your graph by clicking the checkbox 'show y = f'(x)'.
- Where on your graph of \(y = f(x)\) does a point of inflexion occur? Confirm your answer by clicking the checkbox 'show point of inflexion'. Describe the features of the graphs of \(y = f(x)\) and \(y = f'(x)\) at this point.
- On the same set of axes, mark the position of \(y = f''(x)\), where \(f''(x) = 0\).
- Move the slider so that the point \(P\) is left then right of the point of inflexion. What does the sign of \(f''(x)\) indicate about the graphs of \(y = f(x)\), \(y = f'(x)\) and \(y = f''(x)\) in these domains?
- When \(y = f(x)\) is of degree 3, what is the degree of \(y = f''(x)\)? What would this tell you about the shape of the graph of \(y = f''(x)\)?
- In your book complete the graph of \(y = f''(x)\). Check your graph by clicking the checkbox 'show y = f"(x)'.
- Draw another graph by moving the sliders a, b, c and d and then repeat the steps above.