Worksheet Chapter 2: Differentiation

1. Find the coordinates of the turning points for the curve y = x³ - 3x + 2.

2. A curve has equation y = x³ - 6x² + 9x + 5. Determine the nature of the turning point at x = 1 using the second derivative test.

3. Which rule should be used to differentiate y = x² + 12x - 3?

4. The volume V cm³ of a sphere of radius r cm is given by V = 43 π r³. If the radius increases at a constant rate of 0.5 cm s⁻¹, find the rate of change of the volume when the radius is 4 cm.

5. State whether the following statement is True or False: 'At a stationary point of a curve y = f(x), the gradient dy/dx is always equal to zero.'

6. Fill in the blank: If two perpendicular lines are the tangent and normal to a curve at a point, and the gradient of the tangent is m, then the gradient of the normal is ________.

7. Find the gradient of the tangent to the curve y = 2x³ - 9x² + 12x - 4 at the point where x = 3.

8. Differentiate y = (5x² - 3)⁴ with respect to x using the chain rule.

9. A open rectangular box with a square base of side x cm and height h cm has a total surface area of 108 cm². (a) Express h in terms of x. (b) Show that the volume V cm³ is given by V = 27x - 14 x³. (c) Find the maximum volume of the box.

10. Given y = 16x³, use differentiation to approximate the value of 162.02³.