★ Topic Review — Applications of Differential Calculus

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Fluency

1. Fluency

   Find the equation of the tangent to y = x³ − 2x² + 1 at x = 2.

2. Fluency

   Find the equation of the normal to y = x² + 3x at x = 1.

3. Fluency

   A particle has displacement x(t) = t³ − 3t² + 4 (metres, t in seconds). Find the velocity and acceleration at t = 2, and determine the direction of motion.

4. Fluency

   Find and classify all stationary points of f(x) = x³ − 6x² + 9x + 1 using the first derivative test.

5. Fluency

   Determine the end behaviour of y = 3x⁴ − x² + 5 and state whether the function has a global minimum.

Understanding

6. Understanding

   Find the point(s) on y = x² + x where the tangent is parallel to the line y = 5x − 3.

7. Understanding

   A particle has displacement x = 2t³ − 12t² + 18t (cm, t ≥ 0 s).

   1. (a) Find when the particle is at rest.
   2. (b) Find the total distance travelled from t = 0 to t = 4.
8. Understanding

   For f(x) = 2x³ + 3x² − 12x − 7:

   1. (a) Find all stationary points and classify them.
   2. (b) State the intervals where f is increasing and decreasing.
9. Understanding

   Use the 5-step method to sketch y = x³ − 3x, labelling all intercepts and stationary points.

10. Understanding

    A 6 m length of wire is bent to form three sides of a rectangle (the fourth side lies along a wall). Find the dimensions that maximise the area of the rectangle.

Problem Solving

11. Problem Solving

    The curve y = ax³ + bx passes through (1, −2) and has a tangent gradient of 1 at that point. Find a and b, and find the equation of the tangent at (1, −2).

12. Problem Solving

    A particle moves along a line with displacement x = t³ − 6t² + 12t − 5 (m, t ≥ 0 s).

    1. (a) Show that the particle never reverses direction by showing v(t) ≥ 0 for all t ≥ 0.
    2. (b) Find the acceleration when v is at its minimum, and interpret the result.
13. Problem Solving

    A farmer wants to build a rectangular pen divided into 3 equal sections by internal fences parallel to the width. The total fencing available is 240 m. Find the overall dimensions that maximise the total enclosed area.

14. Problem Solving

    The profit from selling x items is P(x) = −2x³ + 18x² − 30x for x ≥ 0. Find the production level that maximises profit, and verify it is a maximum using the first derivative test.

15. Problem Solving

    A ladder of length L = 5 m leans against a vertical wall. The base of the ladder slides away from the wall at 0.5 m/s.

    Using Pythagoras: if the base is x metres from the wall, the height up the wall is h = √(25 − x²).
    1. (a) When the base is 3 m from the wall, what is the height of the top of the ladder?
    2. (b) Find h′(x) = dh⁄dx.
    3. (c) Using dx⁄dt = 0.5, find the rate at which the height is decreasing when x = 3 m (i.e. find dh⁄dt).