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, not a summary of it.

1. 1

   Limits and Continuity

   7 topics

   • Understanding Limits
   • Evaluating Limits Algebraically
   • One-Sided Limits
   • Limits Involving Infinity
   • Continuity and Discontinuity
   • The Squeeze Theorem
   • Intermediate Value Theorem
2. 2

   Derivatives

   11 topics

   • Definition of the Derivative
   • Basic Differentiation Rules
   • Product and Quotient Rules
   • Chain Rule
   • Implicit Differentiation
   • Derivatives of Trigonometric Functions
   • Derivatives of Exponential and Logarithmic Functions
   • Higher-Order Derivatives
   • Applications of Derivatives
   • Related Rates
   • L'Hopital's Rule
3. 3

   Applications of Derivatives

   7 topics

   • Critical Points and Extrema
   • Concavity and Points of Inflection
   • Curve Sketching
   • Optimization Problems
   • Linear Approximations and Differentials
   • Newton's Method
   • Mean Value Theorem
4. 4

   Integrals

   9 topics

   • Antiderivatives and Indefinite Integrals
   • The Definite Integral
   • The Fundamental Theorem of Calculus
   • Techniques of Integration
   • Integration by Substitution
   • Integration by Parts
   • Trigonometric Integrals
   • Partial Fraction Decomposition
   • Improper Integrals
5. 5

   Applications of Integrals

   8 topics

   • Area Between Curves
   • Volumes of Solids of Revolution
   • Arc Length and Surface Area
   • Work and Fluid Forces
   • Center of Mass and Moments
   • Differential Equations
   • Separable Differential Equations
   • Modeling with Differential Equations
6. 6

   Sequences and Series

   9 topics

   • Sequences
   • Series and Convergence
   • The Integral Test and Estimates of Sums
   • Comparison Tests
   • Alternating Series
   • Ratio and Root Tests
   • Power Series
   • Taylor and Maclaurin Series
   • Applications of Taylor Series
7. 7

   Multivariable Calculus

   10 topics

   • Functions of Several Variables
   • Partial Derivatives
   • Multiple Integrals
   • Vector Calculus
   • Gradient, Divergence, and Curl
   • Line Integrals
   • Surface Integrals
   • Green's Theorem
   • Stokes' Theorem
   • Divergence Theorem

Calculus flashcards

The complete Calculus deck: 50 questions with worked answers.

1. What is the intuitive (informal) definition of the limit $\lim_{x \to a} f(x) = L$?

   It means that the values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ (but not equal to $a$). The value of $f$ at $a$ itself is irrelevant.

2. State the formal $\varepsilon$-$\delta$ definition of a limit.

   $\lim_{x \to a} f(x) = L$ means: for every $\varepsilon > 0$ there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \varepsilon$.

3. Under what condition does the two-sided limit $\lim_{x \to a} f(x)$ exist, in terms of one-sided limits?

   It exists if and only if both one-sided limits exist and are equal: $\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x)$.

4. What is the notation and meaning of a left-hand and right-hand limit?

   $\lim_{x \to a^{-}} f(x)$ is the left-hand limit (as $x$ approaches $a$ from values less than $a$); $\lim_{x \to a^{+}} f(x)$ is the right-hand limit (from values greater than $a$).

5. State the basic limit laws for sums, products, and quotients (assuming both limits exist).

   $\lim (f \pm g) = \lim f \pm \lim g$; $\lim (f \cdot g) = \lim f \cdot \lim g$; $\lim \frac{f}{g} = \frac{\lim f}{\lim g}$ provided $\lim g \neq 0$. Also $\lim (c \cdot f) = c \lim f$.

6. What is the standard algebraic technique to evaluate a limit that gives the indeterminate form $\frac{0}{0}$ for a rational function?

   Factor the numerator and denominator, cancel the common factor causing the zero, then substitute. E.g. $\lim_{x \to 2} \frac{x^{2}-4}{x-2} = \lim_{x \to 2}(x+2) = 4$.

7. How do you evaluate a limit involving a square root that yields $\frac{0}{0}$, such as $\lim_{x \to 0} \frac{\sqrt{x+1}-1}{x}$?

   Multiply numerator and denominator by the conjugate. $\frac{\sqrt{x+1}-1}{x} \cdot \frac{\sqrt{x+1}+1}{\sqrt{x+1}+1} = \frac{x}{x(\sqrt{x+1}+1)} = \frac{1}{\sqrt{x+1}+1} \to \frac{1}{2}$.

8. What is the value of the special trigonometric limit $\lim_{x \to 0} \frac{\sin x}{x}$?

   $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (with $x$ in radians).

9. What is the value of $\lim_{x \to 0} \frac{1 - \cos x}{x}$?

   $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$. (Related: $\lim_{x \to 0} \frac{1-\cos x}{x^{2}} = \frac{1}{2}$.)

10. How are limits at infinity of a rational function determined by comparing the degrees of numerator and denominator?

    If degree of numerator < denominator, the limit is $0$. If degrees are equal, the limit is the ratio of leading coefficients. If numerator's degree > denominator's, the limit is $\pm\infty$.

11. Define a vertical asymptote in terms of limits.

    The line $x = a$ is a vertical asymptote of $f$ if at least one of the one-sided limits is infinite: $\lim_{x \to a^{\pm}} f(x) = \pm\infty$.

12. Define a horizontal asymptote in terms of limits.

    The line $y = L$ is a horizontal asymptote of $f$ if $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$.

13. What is the value of $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x}$?

    $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x} = e$.

14. State the three conditions for a function $f$ to be continuous at a point $x = a$.

    1) $f(a)$ is defined; 2) $\lim_{x \to a} f(x)$ exists; 3) $\lim_{x \to a} f(x) = f(a)$.

15. What is a removable discontinuity, and how can it be 'removed'?

    A removable discontinuity occurs when $\lim_{x \to a} f(x)$ exists but does not equal $f(a)$ (or $f(a)$ is undefined). It is a 'hole' that can be removed by redefining $f(a)$ to equal the limit.

16. What is a jump discontinuity?

    A jump discontinuity occurs when both one-sided limits exist but are not equal: $\lim_{x \to a^{-}} f(x) \neq \lim_{x \to a^{+}} f(x)$, so the graph 'jumps'.

17. What is an infinite (essential) discontinuity?

    An infinite discontinuity occurs when at least one one-sided limit is $\pm\infty$, i.e., the function has a vertical asymptote at that point, such as $f(x) = \frac{1}{x}$ at $x = 0$.

18. State the Squeeze Theorem.

    If $g(x) \leq f(x) \leq h(x)$ for all $x$ near $a$ (except possibly at $a$), and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

19. Use the Squeeze Theorem to find $\lim_{x \to 0} x^{2} \sin\!\left(\frac{1}{x}\right)$.

    Since $-1 \leq \sin\!\left(\frac{1}{x}\right) \leq 1$, we have $-x^{2} \leq x^{2}\sin\!\left(\frac{1}{x}\right) \leq x^{2}$. Both bounds $\to 0$, so the limit is $0$.

20. State the Intermediate Value Theorem (IVT).

    If $f$ is continuous on $[a, b]$ and $N$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c \in (a, b)$ such that $f(c) = N$.

Calculus syllabus FAQ