Exercises

1. For each of the following functions calculate \(\frac{df}{dx}\), \(\frac{d^2f}{dx^2}\) and \(\int f(x) dx\).

   1. \(f(x) = x\)

   2. \(f(x) = x ^{\frac{1}{3}}\)

   3. \(f(x) = 2 x (x - 3) (\sin(x) - 5)\)

   4. \(f(x) = 3 x ^ 3 + 6 \sqrt{x} + 3\)

2. Consider the function \(f(x)=2x+1\). By differentiating from first principles show that \(f'(x)=2\).

3. Consider the second derivative \(f''(x)=6x+4\) of some cubic function \(f(x)\).

   1. Find \(f'(x)\)

   2. You are given that \(f(0)=10\) and \(f(1)=13\), find \(f(x)\).

   3. Find all the stationary points of \(f(x)\) and determine their nature.

4. Consider the function \(f(x)=\frac{2}{3}x ^ 3 + b x ^ 2 + 2 x + 3\), where \(b\) is some undetermined coefficient.

   1. Find \(f'(x)\) and \(f''(x)\)

   2. You are given that \(f(x)\) has a stationary point at \(x=2\). Use this information to find \(b\).

   3. Find the coordinates of the other stationary point.

   4. Determine the nature of all stationary points.

5. Consider the functions \(f(x)=-x^2+4x+4\) and \(g(x)=3x^2-2x-2\).

   1. Create a variable turning_points which has value the turning points of \(f(x)\).

   2. Create variable intersection_points which has value of the points where \(f(x)\) and \(g(x)\) intersect.

   3. Using your answers to parts 2., calculate the area of the region between \(f\) and \(g\). Assign this value to a variable area_between.