Mathematics Admissions Test (MAT) 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 Mathematics Admissions Test (MAT), not a summary of it.

1. 1

   Differentiation

   4 topics

   • Differentiating Powers and Polynomials
     • The power rule for integer and fractional exponents
     • Differentiating sums of terms
   • Gradients, Tangents and Normals
     • Gradient of a curve at a point
     • Equations of tangents and normals
   • Stationary Points and Optimisation
     • Finding and classifying maxima, minima and points of inflection
     • Using the second derivative
     • Maximising or minimising a quantity in a worded problem
   • Increasing and Decreasing Functions
     • Determining intervals from the sign of the derivative
2. 2

   Integration

   3 topics

   • Indefinite Integration
     • Reversing the power rule and the constant of integration
     • Finding a curve from its gradient and a point
   • Definite Integration
     • Evaluating definite integrals using limits
     • Properties of definite integrals over split intervals
   • Area Under and Between Curves
     • Area between a curve and an axis, including signed area
     • Area enclosed between two curves
3. 3

   Calculus in Reasoning Problems

   3 topics

   • Linking Graphs and Derivatives
     • Matching a function to the graph of its derivative
     • Inferring shape from gradient information
   • Estimation and Bounds with Integrals
     • Comparing areas to bound an integral
   • Multi-step Optimisation
     • Combining geometry, algebra and calculus in a single problem

Calculus flashcards for Mathematics Admissions Test (MAT)

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

1. What is the power rule for differentiating $y = x^{n}$?

   $\dfrac{dy}{dx} = n x^{n-1}$ for any real constant $n$.

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

   $\dfrac{dy}{dx} = a n x^{n-1}$ (constant multiples carry through).

3. What is the derivative of a constant, $y = c$?

   $\dfrac{dy}{dx} = 0$.

4. How do you differentiate a polynomial such as $y = 3x^{4} - 5x^{2} + 7x - 2$?

   Differentiate term by term: $\dfrac{dy}{dx} = 12x^{3} - 10x + 7$. The sum/difference rule lets you treat each term separately.

5. Differentiate $y = \dfrac{1}{x^{3}}$ using the power rule.

   Write as $x^{-3}$: $\dfrac{dy}{dx} = -3x^{-4} = -\dfrac{3}{x^{4}}$.

6. Differentiate $y = \sqrt{x}$.

   Write as $x^{1/2}$: $\dfrac{dy}{dx} = \tfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}$.

7. What does the derivative $\dfrac{dy}{dx}$ represent geometrically?

   The gradient (slope) of the tangent to the curve at a given point; equivalently the instantaneous rate of change of $y$ with respect to $x$.

8. What is the gradient of the curve $y = f(x)$ at the point where $x = a$?

   $f'(a)$ — substitute $x = a$ into the derivative.

9. What is the equation of the tangent to $y = f(x)$ at the point $(a, f(a))$?

   $y - f(a) = f'(a)\,(x - a)$.

10. How is the gradient of the normal related to the gradient of the tangent at the same point?

    The normal is perpendicular to the tangent, so its gradient is $-\dfrac{1}{f'(a)}$ (the negative reciprocal), provided $f'(a) \neq 0$.

11. What is the equation of the normal to $y = f(x)$ at the point $(a, f(a))$?

    $y - f(a) = -\dfrac{1}{f'(a)}\,(x - a)$, provided $f'(a)\neq 0$.

12. What is the gradient of the tangent at a point where the tangent is horizontal?

    $f'(a) = 0$. (At such a point the normal is vertical.)

13. What condition defines a stationary point of $y = f(x)$?

    $f'(x) = 0$ — the gradient is zero there.

14. List the three types of stationary point.

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

15. Using the second derivative, how do you classify a stationary point at $x = a$ where $f'(a)=0$?

    If $f''(a) > 0$ it is a local minimum; if $f''(a) < 0$ it is a local maximum; if $f''(a) = 0$ the test is inconclusive.

16. What does the first-derivative (sign-change) test say about a local maximum?

    $f'(x)$ changes from positive (increasing) to negative (decreasing) as $x$ passes through the stationary point.

17. What does the first-derivative test indicate for a local minimum?

    $f'(x)$ changes from negative (decreasing) to positive (increasing) through the stationary point.

18. In the first-derivative test, what sign pattern indicates a stationary point of inflection?

    $f'(x)$ does not change sign across the point (e.g. positive–zero–positive or negative–zero–negative).

19. What is the general process for solving an optimisation problem with calculus?

    1. Express the quantity to optimise as a function of one variable (using any constraint to eliminate others). 2. Differentiate and set the derivative to zero. 3. Solve for the stationary point(s). 4. Classify to confirm it is a max/min and check endpoints/domain.

20. When optimising over a closed interval $[a,b]$, where can the maximum or minimum occur?

    At an interior stationary point (where $f'(x)=0$) or at an endpoint $x=a$ or $x=b$. All candidates must be compared.

21. For what values of $x$ is a function $f$ increasing?

    Where $f'(x) > 0$ — the curve rises as $x$ increases.

22. For what values of $x$ is a function $f$ decreasing?

    Where $f'(x) < 0$ — the curve falls as $x$ increases.

23. What does it mean for a function to be increasing on an interval in terms of values?

    For any $x_{1} < x_{2}$ in the interval, $f(x_{1}) \leq f(x_{2})$ (strictly increasing if $f(x_{1}) < f(x_{2})$).

24. How do you find the intervals where $f(x)=x^{3}-3x$ is increasing?

    $f'(x)=3x^{2}-3 = 3(x-1)(x+1)$. This is positive for $x<-1$ and $x>1$, so $f$ is increasing on $(-\infty,-1)$ and $(1,\infty)$.

25. What is the indefinite integral (antiderivative) power rule $\displaystyle\int x^{n}\,dx$?

    $\displaystyle\int x^{n}\,dx = \dfrac{x^{n+1}}{n+1} + C$, valid for $n \neq -1$.

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