Dynamic Calculus

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Finding the anti-derivative

The applet below shows the graph of \(f'(x) = ax + b\). What does the graph of \(y = f(x)\) look like?
[You can reset the applet at any time by clicking the 'circular arrows' icon in the top right corner]

 

1. Change the orientation and position of \(y = f'(x)\) by using the sliders 'a' and 'b'. In your book, draw the graph of \(y = f'(x)\) for your chosen values of a and b.
2. Given that \(f'(x) = ax + b\), use the process of anti-differentiation to find the equation of the primitive function. Check your answer by clicking the 'show anti-differentiation' checkbox.
3. The primitive function represents a family of curves; each member of which corresponds to a unique value of the constant, \(c\). On the same set of axes as question (1), draw the graph of \(y = f(x)\) for one value of \(c\).
4. Click the 'show graph of y = f(x)' checkbox. Move the slider 'c' and observe other possible graphs of \(y = f(x)\).
5. The gradient function of a curve is \(4 - 2x\) and the curve passes through the point \((2, 6)\). Use the process of anti-differentiation to find the equation of the curve. Check your answer with the applet by adjusting the sliders a, b and c.

State of New South Wales, Department of Education and Training 2008, Created with GeoGebra