Topic Review — Introduction to Differential Calculus

← Back to Topic›Maths Methods

1. Fluency

   Q1 — Average rate of change

   For f(x) = x² + 3x, find the average rate of change over each interval.

   (a) from x = 1 to x = 3   (b) from x = −2 to x = 0   (c) from x = 2 to x = 2 + h (leave in terms of h)

2. Fluency

   Q2 — Evaluating limits

   Evaluate each limit.

   (a) lim_x→3 (x² − 9)/(x − 3)   (b) lim_h→0 (3h² + 5h)/h   (c) lim_h→0 (h² + 4h)/h

3. Fluency

   Q3 — First principles for a constant and linear function

   Use the definition f′(x) = lim_h→0 [f(x + h) − f(x)]/h to differentiate:

   (a) f(x) = 7   (b) f(x) = 4x − 1   (c) f(x) = 3x

4. Fluency

   Q4 — First principles for x²

   Using first principles, prove that d/dx[x²] = 2x. Show all steps clearly.

5. Fluency

   Q5 — Power rule: differentiating monomials

   Differentiate each using the power rule d/dx[xⁿ] = nxⁿ⁻¹.

   (a) y = x⁵   (b) y = x⁸   (c) y = x^1/2   (d) y = x⁻³   (e) y = 6x⁴

6. Fluency

   Q6 — Differentiating polynomials

   Find f′(x) for each polynomial.

   (a) f(x) = x³ + 4x² − 7x + 2   (b) f(x) = 5x⁴ − 3x² + 1   (c) f(x) = 2x³ − x + 9

7. Understanding

   Q7 — First principles for x³

   Using first principles, show that d/dx[x³] = 3x². Expand (x + h)³ = x³ + 3x²h + 3xh² + h³ fully.

8. Understanding

   Q8 — Evaluating the derivative at a point

   Find the gradient of each curve at the specified x-value.

   (a) f(x) = x³ − 2x at x = 2   (b) f(x) = 4x² + 1 at x = −1   (c) f(x) = x⁴ − x at x = 1

9. Understanding

   Q9 — Interpreting the derivative as gradient

   For the curve y = 3x² − 12x + 5:

   (a) Find dy/dx.   (b) Find the gradient at x = 1 and at x = 3.   (c) What do your answers tell you about the curve at those points?

10. Understanding

    Q10 — Finding where the derivative equals a given value

    Find the x-value(s) where the gradient of the curve equals the given value.

    (a) y = x² − 4x + 1, gradient = 0   (b) y = 2x³ − 3x, gradient = 6   (c) y = x³ − 12x, gradient = 0

11. Understanding

    Q11 — Derivative notation

    Let f(x) = 2x⁴ − 5x² + 3x − 1.

    (a) Find f′(x).   (b) Evaluate f′(0) and f′(2).   (c) Write the derivative using dy/dx notation if y = f(x).

12. Understanding

    Q12 — First principles from the definition

    Use first principles to differentiate f(x) = x² − 5x.

    Show that your result matches the power rule applied term by term.

13. Problem Solving

    Q13 — The derivative function

    For the function f(x) = x³ − 3x:

    (a) Find f′(x).   (b) Sketch a sign diagram for f′(x) to identify where f is increasing and decreasing.   (c) State the x-coordinates of any turning points of f.

14. Problem Solving

    Q14 — Average rate from first principles limit

    The height (in metres) of a ball thrown upward is h(t) = −5t² + 20t + 1, where t is time in seconds.

    (a) Find the average rate of change of height from t = 1 to t = 3.   (b) Using first principles, find h′(t).   (c) Find the instantaneous velocity at t = 1 and t = 3.   (d) When is the ball at its highest point?

15. Problem Solving

    Q15 — Extended problem: connecting concepts

    Two curves are defined by f(x) = x³ − 6x and g(x) = −x² + 2.

    (a) Find f′(x) and g′(x).   (b) At what x-value do the two curves have the same gradient?   (c) At that x-value, are both curves increasing, decreasing, or one of each?   (d) Find the equation of the tangent to f at this x-value.