Calculus

Limits

Formal Definition

Let $f$ be defined on an open interval containing $a$Except possibly at $a$ itself. We write $\displaystyle\lim_{x \to a} f(x) = L$ if for every $\varepsilon \gt 0$There exists a $\delta \gt 0$ such that:

$0 \lt |x - a| \lt \delta \implies |f(x) - L| \lt \varepsilon$

Limit Laws

If $\displaystyle\lim_{x \to a} f(x) = L$ and $\displaystyle\lim_{x \to a} g(x) = M$Then:

$\lim_{x \to a}\bigl[f(x) + g(x)\bigr] = L + M$

$\lim_{x \to a}\bigl[f(x) \cdot g(x)\bigr] = L \cdot M$

$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}, \quad M \ne 0$

$\lim_{x \to a} \bigl[f(x)\bigr]^n = L^n, \quad n \in \mathbb{Z}^{+}$

One-Sided Limits

A two-sided limit exists if and only if both one-sided limits exist and are equal:

$\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$

Indeterminate Forms

Expressions of the form \dfrac{0}{0}$$\dfrac{\infty}{\infty}$$0 \cdot \infty \infty - \infty$$1^\infty$$0^0And $\infty^0$ are indeterminate — the limit may or may Not exist, and algebraic manipulation or l’H^opital’s rule is required.

Common Limits

Continuity

Definition

A function $f$ is continuous at $a$ if and only if all three conditions hold:

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

Intermediate Value Theorem

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

This theorem underpins the bisection method for root-finding and guarantees that a continuous Function on a closed interval attains every intermediate value.

Types of Discontinuity

Differentiation

Definition from First Principles

The derivative of $f$ at $a$ is:

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

Provided this limit exists. When it does, $f$ is said to be differentiable at $a$.

Differentiability implies continuity, but continuity does not imply differentiability. The function $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$.

Differentiation Rules

Power rule. For $n \in \mathbb{R}$:

$\frac{d}{dx}\bigl[x^n\bigr] = nx^{n-1}$

Constant multiple and sum rules:

$\frac{d}{dx}\bigl[cf(x)\bigr] = c\,f'(x), \qquad \frac{d}{dx}\bigl[f(x) \pm g(x)\bigr] = f'(x) \pm g'(x)$

Product rule:

$\frac{d}{dx}\bigl[uv\bigr] = u'v + uv'$

Quotient rule:

$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}, \quad v \ne 0$

Chain rule. If $y = f(g(x))$Then:

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

Derivatives of Standard Functions

Implicit Differentiation

When a relation between $x$ and $y$ cannot be solved for $y$Differentiate both sides with Respect to $x$Treating $y$ as a function of $x$. Every occurrence of $y$ produces a factor of $\dfrac{dy}{dx}$ via the chain rule.

Example. Find $\dfrac{dy}{dx}$ for $x^2 + y^2 = 25$.

Differentiating implicitly: $2x + 2y\dfrac{dy}{dx} = 0$So $\dfrac{dy}{dx} = -\dfrac{x}{y}$.

Higher-Order Derivatives

The second derivative $f''(x) = \dfrac{d^2y}{dx^2}$ is the derivative of $f'(x)$. Higher-order Derivatives $f^{(n)}(x)$ are defined recursively. The second derivative governs concavity: $f''(x) \gt 0$ implies concave up; $f''(x) \lt 0$ implies concave down.

Applications of Differentiation

Stationary Points

A stationary point occurs where $f'(x) = 0$. The second derivative test classifies them:

• $f'(a) = 0$ and $f''(a) \gt 0$: local minimum
• $f'(a) = 0$ and $f''(a) \lt 0$: local maximum
• $f'(a) = 0$ and $f''(a) = 0$: test is inconclusive; use the first derivative test

Optimisation

To solve optimisation problems:

1. Identify the quantity to be maximised or minimised as a function of a single variable.
2. Determine the domain of the function from the problem constraints.
3. Find stationary points by setting the derivative to zero.
4. Verify the nature of each stationary point (second derivative test or sign analysis).
5. Check endpoints and boundary values.

Related Rates

When two or more quantities vary with time and are related by an equation, their rates of change are Related by implicit differentiation with respect to time $t$.

Example. A spherical balloon is inflated at $\dfrac{dV}{dt} = 100\;\mathrm{cm^3/s}$. Find $\dfrac{dr}{dt}$ when $r = 5\;\mathrm{cm}$.

Since $V = \dfrac{4}{3}\pi r^3$Differentiating: $\dfrac{dV}{dt} = 4\pi r^2 \dfrac{dr}{dt}$. Substituting: $100 = 4\pi(25)\dfrac{dr}{dt}$So $\dfrac{dr}{dt} = \dfrac{1}{\pi}\;\mathrm{cm/s}$.

Curve Sketching

Systematic curve sketching involves:

1. Domain and intercepts: where $f(x)$ is defined; $x$- and $y$-intercepts.
2. Symmetry: even ($f(-x) = f(x)$), odd ($f(-x) = -f(x)$), periodic.
3. Asymptotes: vertical (where denominator vanishes), horizontal (end behaviour), oblique.
4. First derivative: intervals of increase/decrease; stationary points.
5. Second derivative: concavity and points of inflection.

Tangents and Normals

The tangent to $y = f(x)$ at $x = a$ has equation:

$y - f(a) = f'(a)(x - a)$

The normal is perpendicular to the tangent, with gradient $-\dfrac{1}{f'(a)}$ (provided $f'(a) \ne 0$).

Integration

Indefinite Integrals

The antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$. The general Antiderivative includes an arbitrary constant:

$\int f(x)\,dx = F(x) + C$

Fundamental Theorem of Calculus

First part. If $f$ is continuous on $[a, b]$ and $F'(x) = f(x)$Then:

$\int_a^b f(x)\,dx = F(b) - F(a)$

Second part. If $f$ is continuous on $[a, b]$Then:

$\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$

Definite Integrals

Properties of definite integrals (assuming $f, g$ are integrable):

$\int_a^b \bigl[cf(x)\bigr]\,dx = c\int_a^b f(x)\,dx$

$\int_a^b \bigl[f(x) \pm g(x)\bigr]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx$

$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$

$\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$

Area Under a Curve

If $f(x) \ge 0$ on $[a, b]$The area between the curve and the $x$-axis is:

$A = \int_a^b f(x)\,dx$

If $f$ changes sign, compute the integral piecewise. The total enclosed area is:

$A = \int_a^b |f(x)|\,dx$

Area Between Curves

The area between $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$Where $f(x) \ge g(x)$:

$A = \int_a^b \bigl[f(x) - g(x)\bigr]\,dx$

Standard Integrals

| $f(x)$ | $\displaystyle\int f(x)\,dx$ | | :-------------------------- | :-------------------------------------- | --- | ---- | | $x^n$ | $\dfrac{x^{n+1}}{n+1} + C, \; n \ne -1$ | | | | $\dfrac{1}{x}$ | $\ln | x | + C$ | | $e^x$ | $e^x + C$ | | | | $\sin x$ | $-\cos x + C$ | | | | $\cos x$ | $\sin x + C$ | | | | $\sec^2 x$ | $\tan x + C$ | | | | $\dfrac{1}{\sqrt{1 - x^2}}$ | $\arcsin x + C$ | | | | $\dfrac{1}{1 + x^2}$ | $\arctan x + C$ | | |

Integration Techniques

Substitution

Theorem. Let $u = g(x)$ where $g'$ is continuous and $f$ is continuous on the range of $g$. Then:

$\int f(g(x))\,g'(x)\,dx = \int f(u)\,du$

Strategy. Identify an “inner function” and its derivative as a factor. Substitute $u = g(x)$ Replace $dx$ with $\dfrac{du}{g'(x)}$And integrate with respect to $u$.

Example. Evaluate $\displaystyle\int 2x\sqrt{x^2 + 1}\,dx$.

Let $u = x^2 + 1$So $du = 2x\,dx$. Then:

$\int 2x\sqrt{x^2 + 1}\,dx = \int \sqrt{u}\,du = \frac{2}{3}u^{3/2} + C = \frac{2}{3}(x^2 + 1)^{3/2} + C$

Integration by Parts

Theorem. If $u$ and $v$ are differentiable functions:

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

LIATE priority for choosing $u$: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Choose $u$ as the function that appears highest on this list.

Example. Evaluate $\displaystyle\int x e^x\,dx$.

Let u = x$$dv = e^x\,dx. Then du = dx$$v = e^x:

$\int x e^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C = (x - 1)e^x + C$

Repeated Integration by Parts

For integrals such as $\int x^n e^x\,dx$ or $\int x^n \sin x\,dx$Apply integration by parts Repeatedly. Tabular (DI) integration provides a systematic shortcut: differentiate the algebraic Factor until it reaches zero, integrate the other factor, and alternate signs.

Volumes of Revolution

About the $x$-axis. Rotating $y = f(x)$ about the $x$-axis from $x = a$ to $x = b$:

$V = \pi \int_a^b \bigl[f(x)\bigr]^2\,dx$

About the $y$-axis. Rotating $x = g(y)$ about the $y$-axis from $y = c$ to $y = d$:

$V = \pi \int_c^d \bigl[g(y)\bigr]^2\,dy$

Cylindrical shells (rotating about the $y$-axis):

$V = 2\pi \int_a^b x\,|f(x)|\,dx$

Improper Integrals

An improper integral involves an infinite limit of integration or an unbounded integrand. Evaluate As a limit:

$\int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx$

The integral converges if this limit exists (is finite); otherwise it diverges.

Common Pitfalls

1. Forgetting the chain rule. When differentiating composite functions, always identify the inner function and multiply by its derivative.

2. Sign errors in the quotient rule. Remember: $u'v - uv'$Not $uv' - u'v$. A mnemonic: “low d-high minus high d-low, over the square of what’s below.”

3. Dropping the $+C$. Every indefinite integral requires an arbitrary constant.

4. Integration by parts choice. Choosing the wrong factor for $u$ can lead to a more complicated integral. Follow the LIATE rule.

5. Incorrect limits after substitution. When using substitution in a definite integral, either transform the limits to the new variable or back-substitute before evaluating.

6. Volume formula confusion. Ensure the correct axis of rotation is used. The variable in the integrand corresponds to the axis perpendicular to the one of rotation.

7. Treating $\dfrac{dy}{dx}$ as a fraction in implicit differentiation. While the notation suggests a ratio, it is a limit. Manipulations such as cross-multiplying are justified by the chain rule, not by treating derivatives as fractions.

Practice Problems

Worked Examples

Worked Example: Evaluating a Limit Using L’Hopital’s Rule

Evaluate $\displaystyle\lim_{x \to 0} \frac{e^{3x} - 1 - 3x}{x^2}$.

Worked Example: Curve Sketching with Full Analysis

Sketch the curve $y = \dfrac{x^2 - 4}{x^2 - 1}$Identifying all asymptotes, stationary points, and Intervals of increase and decrease.

Worked Example: Integration by Parts with Definite Integral

Evaluate $\displaystyle\int_0^{\pi/2} x \cos x\,dx$.

Worked Example: Volume of Revolution with Shells

Find the volume generated when the region bounded by $y = x^2$, $y = 0$And $x = 2$ is rotated $360^\circ$ About the $y$-axis.

Worked Example: Implicit Differentiation and Second Derivative

Given $x^3 + y^3 = 6xy$Find $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ at the point $(3, 3)$.

Additional Common Pitfalls

• L’H^opital’s rule overuse. Only apply when the limit is genuinely indeterminate ($\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$). Applying it to determinate forms produces incorrect results.

• Second derivative test inconclusiveness. When $f''(a) = 0$The test fails. Use a sign chart of $f'$ around $a$ (the first derivative test) to classify the stationary point.

• Area vs. Signed integral. $\displaystyle\int_a^b f(x)\,dx$ gives the signed area. When the curve crosses the $x$-axis, split the integral at each root and take absolute values to find total enclosed area.

• Forgetting substitution limits. In $\displaystyle\int_a^b f(g(x))g'(x)\,dx$If you substitute $u = g(x)$Change the limits to $g(a)$ and $g(b)$. Failing to do so and using the original limits gives the wrong answer.

• Chain rule on trigonometric functions. $\dfrac{d}{dx}[\sin(f(x))] = f'(x)\cos(f(x))$Not $\cos(f(x))$. Students frequently omit the inner derivative factor.

• Power rule for negative exponents. $\dfrac{d}{dx}\bigl[x^{-3}\bigr] = -3x^{-4}$Not $-3x^{-2}$. Subtract one from the exponent, then multiply by the original exponent.

Exam-Style Problems

If You Get These Wrong, Revise:

• Differentiation rules and chain rule → Review Calculus (sections on Differentiation Rules and Implicit Differentiation)
• Integration techniques → Review Calculus (sections on Substitution and Integration by Parts)
• Trigonometric identities → Review the geometry and trigonometry topics in this subject
• Exponential and logarithmic functions → Review algebra and functions material
• Curve sketching fundamentals → Review Calculus (section on Curve Sketching)

Related Content at Other Levels

• A-Level Differentiation and Integration: Mathematics
• University Calculus: Multivariable Calculus

Summary

This topic covers the mathematical techniques and concepts related to calculus, including key theorems, methods, and problem-solving approaches.

Key concepts include:

• differentiation from first principles
• product, quotient, and chain rules
• integration techniques (by parts, substitution)
• differential equations
• applications to kinematics

Regular practice with a variety of question types is essential to build fluency and confidence in applying these mathematical techniques.