Dynamic Calculus
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Curve sketching (Part 3)
The applet below sketches the graphs of four mystery functions. Move the 'slider' for each function and test your curve-sketching ability by observing the values of the first and second derivatives.
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]
For each function, consider the following points as you move the slider left to right.
- Click the 'checkbox' for \(f(x)\). Approximately where does the curve cross the \(x\)-axis?
- Click the checkbox for \(f'(x)\). If \(f'(x)\) is positive at a point \(P\) on the curve, what does this indicate about \(y = f(x)\) at \(P\)? If \(f'(x)\) is negative at a point \(P\) on the curve, what does this indicate about \(y = f(x)\) at \(P\)? If \(f'(x)\) is zero at a point \(P\) on the curve, what does this indicate about the point \(P\)?
- What conditions must exist for a curve to have a maximum turning point? What conditions must exist for a curve to have a minimum turning point? Identify any turning points on the curve \(y = f(x)\).
- Click the checkbox for \(f''(x)\). If \(f''(x)\) is negative in a given domain, what does this indicate about \(y = f(x)\) in that domain? If \(f''(x)\) is positive in a given domain, what does this indicate about \(y = f(x)\) in that domain? If \(f''(x)\) is zero at a point \(P\) on the curve, what does this indicate about the point \(P\)?
- What conditions must exist for a curve to have a point of inflexion? What conditions must exist for a curve to have a horizontal point of inflexion? Identify any points of inflexion on the curve \(y = f(x)\).
When you have identified all critical points on the curve, sketch the function in your book. Check your answer by clicking the appropriate checkbox.