Mathematics 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, not a summary of it.

1. 1

   Limits

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

2. 2

   Derivatives

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

3. 3

   Applications of Derivatives

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

4. 4

   Integrals

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

5. 5

   Applications of Integrals

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

6. 6

   Differential Equations

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

7. 7

   Sequences and Series

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

8. 8

   Multivariable Calculus

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

9. 9

   Vector Calculus

   overview

   Examined as a single unit within Calculus — no further topic split in the official outline.

Calculus flashcards for Mathematics

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

1. What is the formal (epsilon-delta) definition of the limit $\lim_{x \to a} f(x) = L$?

   For every $\varepsilon > 0$ there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \varepsilon$.

2. State the limit definition of the derivative of $f$ at $x$.

   $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

3. What does it mean for a function $f$ to be continuous at a point $a$?

   $f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$; this requires $f(a)$ to be defined, the limit to exist, and the two to be equal.

4. State the Power Rule for differentiation.

   For any real number $n$, $\frac{d}{dx} x^{n} = n x^{n-1}$.

5. State the Product Rule for derivatives.

   $$\frac{d}{dx}\big[f(x)g(x)\big] = f'(x)g(x) + f(x)g'(x)$$

6. State the Quotient Rule for derivatives.

   $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^{2}}$$

7. State the Chain Rule for the derivative of a composite function $f(g(x))$.

   $$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$$

8. What is the derivative of $\sin x$ and of $\cos x$?

   $\frac{d}{dx}\sin x = \cos x$ and $\frac{d}{dx}\cos x = -\sin x$.

9. What is the derivative of $\tan x$?

   $\frac{d}{dx}\tan x = \sec^{2} x$.

10. What is the derivative of $e^{x}$ and of $a^{x}$?

    $\frac{d}{dx} e^{x} = e^{x}$ and $\frac{d}{dx} a^{x} = a^{x} \ln a$.

11. What is the derivative of $\ln x$ and of $\log_a x$?

    $\frac{d}{dx}\ln x = \frac{1}{x}$ and $\frac{d}{dx}\log_a x = \frac{1}{x \ln a}$.

12. What is the derivative of $\arcsin x$?

    $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^{2}}}$.

13. What is the derivative of $\arctan x$?

    $\frac{d}{dx}\arctan x = \frac{1}{1 + x^{2}}$.

14. State the Power Rule for integration (the reverse power rule).

    $$\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$

15. What is $\int \frac{1}{x}\,dx$?

    $\int \frac{1}{x}\,dx = \ln|x| + C$.

16. State the formula for integration by parts.

    $$\int u\,dv = uv - \int v\,du$$

17. State the First Fundamental Theorem of Calculus.

    If $F(x) = \int_a^x f(t)\,dt$ and $f$ is continuous, then $F'(x) = f(x)$.

18. State the Second Fundamental Theorem of Calculus (evaluation form).

    If $F$ is an antiderivative of a continuous $f$ on $[a,b]$, then $\int_a^b f(x)\,dx = F(b) - F(a)$.

19. What is the geometric interpretation of the definite integral $\int_a^b f(x)\,dx$?

    It is the net signed area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$.

20. What is the limit definition of the definite integral (Riemann sum)?

    $$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^{*})\,\Delta x, \quad \Delta x = \frac{b-a}{n}$$

21. State L'Hopital's Rule and the indeterminate forms it applies to.

    If $\lim \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$, provided the latter limit exists.

22. What is the geometric meaning of the derivative $f'(a)$?

    It is the slope of the tangent line to the graph of $f$ at the point $(a, f(a))$; equivalently the instantaneous rate of change of $f$ at $a$.

23. State the Mean Value Theorem (MVT).

    If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

24. State Rolle's Theorem.

    If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists $c \in (a,b)$ with $f'(c) = 0$.

25. How do you find critical points of a function $f$?

    Critical points are values $x$ in the domain where $f'(x) = 0$ or $f'(x)$ does not exist.

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