Test of Mathematics for University Admission (TMUA) Calculus Syllabus

Calculus syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Calculus in Test of Mathematics for University Admission (TMUA), not a summary of it.

1. 1

   Differentiation

   4 topics

   • Differentiation from First Principles Concept
     • Gradient of a curve as a limit
   • Differentiating Powers of x
     • Polynomial and fractional powers
     • Sums and constant multiples
   • Tangents and Normals
     • Equation of a tangent line
     • Equation of a normal line
   • Stationary Points
     • Locating maxima and minima
     • Second derivative test
     • Increasing and decreasing functions
2. 2

   Integration

   3 topics

   • Indefinite Integration
     • Reverse of differentiating powers
     • Constant of integration
   • Definite Integration
     • Evaluating with limits
     • Properties of definite integrals
   • Area Under a Curve
     • Area between a curve and the x-axis
     • Area between two curves
     • Handling regions below the axis
3. 3

   Applications of Calculus

   3 topics

   • Optimisation Problems
     • Forming an expression to optimise
     • Maximising and minimising in context
   • Rates of Change
     • Interpreting the derivative in context
   • Connecting Graphs and Calculus
     • Relating f, f' and f'' graphically

Calculus flashcards for Test of Mathematics for University Admission (TMUA)

25 of 50 cards from the Calculus deck — real questions with worked answers.

1. State the formal definition of the derivative $f'(x)$ from first principles.

   $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ provided this limit exists.

2. In differentiation from first principles, what does the quotient $\dfrac{f(x+h)-f(x)}{h}$ represent geometrically before the limit is taken?

   The gradient of the chord (secant line) joining the points $(x, f(x))$ and $(x+h, f(x+h))$ on the curve.

3. What does the derivative $f'(x)$ represent geometrically as $h \to 0$?

   The gradient of the tangent to the curve $y = f(x)$ at the point $(x, f(x))$ — the limiting position of the chord as $h \to 0$.

4. Use first principles to differentiate $f(x) = x^{2}$.

   $f'(x) = \lim_{h\to 0}\dfrac{(x+h)^{2}-x^{2}}{h} = \lim_{h\to 0}\dfrac{2xh + h^{2}}{h} = \lim_{h\to 0}(2x + h) = 2x.$

5. Use first principles to differentiate $f(x) = x^{3}$.

   $f'(x) = \lim_{h\to 0}\dfrac{(x+h)^{3}-x^{3}}{h} = \lim_{h\to 0}\dfrac{3x^{2}h + 3xh^{2} + h^{3}}{h} = \lim_{h\to 0}(3x^{2}+3xh+h^{2}) = 3x^{2}.$

6. List the three notations commonly used to denote the derivative of $y = f(x)$ with respect to $x$.

   $\dfrac{dy}{dx}$, $f'(x)$, and $\dfrac{d}{dx}\big(f(x)\big)$.

7. State the power rule for differentiating $y = x^{n}$.

   $$\frac{dy}{dx} = n x^{n-1}$$ valid for any rational constant $n$.

8. Differentiate $y = a x^{n}$ where $a$ is a constant.

   $\dfrac{dy}{dx} = a n x^{n-1}$ (the constant multiple is preserved).

9. What is the derivative of a constant function $y = c$, and why?

   $\dfrac{dy}{dx} = 0$, because a constant function has a horizontal graph with zero gradient everywhere.

10. State the sum/difference rule for derivatives.

    $\dfrac{d}{dx}\big(f(x) \pm g(x)\big) = f'(x) \pm g'(x)$ — differentiate term by term.

11. Differentiate $y = \sqrt{x}$ using the power rule.

    Write $y = x^{1/2}$, so $\dfrac{dy}{dx} = \dfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}.$

12. Differentiate $y = \dfrac{1}{x}$ using the power rule.

    Write $y = x^{-1}$, so $\dfrac{dy}{dx} = -x^{-2} = -\dfrac{1}{x^{2}}.$

13. Differentiate $y = \dfrac{1}{x^{2}}$.

    Write $y = x^{-2}$, so $\dfrac{dy}{dx} = -2x^{-3} = -\dfrac{2}{x^{3}}.$

14. Find $\dfrac{dy}{dx}$ for $y = 4x^{3} - 2x^{2} + 7x - 5$.

    $\dfrac{dy}{dx} = 12x^{2} - 4x + 7.$

15. Before differentiating an algebraic fraction such as $y = \dfrac{x^{2}+3x}{x}$, what must you do first?

    Simplify into a sum of powers of $x$ first: $y = x + 3$, then differentiate to get $\dfrac{dy}{dx} = 1.$

16. What is the second derivative, and what notations represent it?

    The derivative of the first derivative. Notations: $\dfrac{d^{2}y}{dx^{2}}$ and $f''(x)$.

17. How do you find the gradient of the curve $y = f(x)$ at a specific point $x = a$?

    Differentiate to get $f'(x)$, then substitute $x = a$: the gradient is $f'(a)$.

18. What is the gradient of the tangent to a curve at a point, in terms of the derivative?

    The gradient of the tangent at $x = a$ equals $f'(a)$ — the value of the derivative there.

19. State the equation of the tangent to the curve $y = f(x)$ at the point $(a, f(a))$.

    $$y - f(a) = f'(a)\,(x - a)$$ where $f'(a)$ is the gradient at that point.

20. What is the relationship between the gradient of a tangent and the gradient of the normal at the same point?

    They are perpendicular, so the product of their gradients is $-1$. If the tangent gradient is $m$, the normal gradient is $-\dfrac{1}{m}$.

21. State the equation of the normal to the curve $y = f(x)$ at the point $(a, f(a))$ (assuming $f'(a)\neq 0$).

    $$y - f(a) = -\frac{1}{f'(a)}\,(x - a).$$

22. Find the gradient of the normal to $y = x^{2}$ at the point $(1, 1)$.

    $\dfrac{dy}{dx} = 2x$, so the tangent gradient at $x=1$ is $2$. The normal gradient is $-\dfrac{1}{2}.$

23. What is a stationary point of a curve, defined in terms of the derivative?

    A point where $\dfrac{dy}{dx} = 0$, i.e. the tangent is horizontal (gradient zero).

24. List the three types of stationary point.

    Local maximum, local minimum, and point of inflection (horizontal/stationary inflection).

25. How do you find the coordinates of the stationary points of $y = f(x)$?

    Solve $f'(x) = 0$ for $x$, then substitute each solution back into $y = f(x)$ to find the corresponding $y$-values.

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