Unit 4: Calculus

I. Orientation

Calculus studies continuous change, local behaviour, and accumulation. Its central idea is that a complicated function can be understood through limiting processes: the derivative measures instantaneous change, while the integral measures accumulated quantity. Limits provide the foundation for continuity and differentiability; derivatives support the analysis of maxima and minima and the Mean Value Theorem; integration reverses differentiation and calculates areas, volumes, and totals.

• Function framework: A function (f) assigns each input (x) in its domain a unique output (f(x)).
• Independent and dependent variables: (x) is usually the independent variable, while (y=f(x)) is the dependent variable.
• Local versus global behaviour: Limits, continuity, and derivatives describe behaviour near a point; maxima and minima may describe behaviour over an entire interval.
• Domain assumptions: Expressions such as (\frac{1}{x-a}), (\sqrt{x}), and (\ln x) require attention to where the function is defined.
• Notation convention: (f'(x)) denotes the derivative of (f), and (\int f(x)\,dx) denotes an indefinite integral.

II. Limits — Approaching a value

A limit describes the value that (f(x)) approaches as (x) approaches (a), whether or not (f(a)) is defined. It is the basic tool for expressing local behaviour.

A. Limits

The limit of (f(x)) as (x) approaches (a) is written (\lim_{x\to a}f(x)=L), where (L) is the approached value.

• Meaning: When (x) is sufficiently close to (a), (f(x)) is close to (L). The value (f(a)) may differ from (L) or may not exist.
• Two-sided limit: A finite limit exists only when the left-hand and right-hand limits agree.
  
  TEXT

    lim(x→a−) f(x) = lim(x→a+) f(x) = L

  
  Here, (a) is the approaching point and (L) is the common limit.
• Basic laws: For existing limits,
  
  TEXT

    lim(f + g) = lim f + lim g
    lim(fg) = (lim f)(lim g)
    lim(f/g) = (lim f)/(lim g), provided lim g ≠ 0

  
  The symbols (f) and (g) represent functions evaluated as (x\to a).
• Algebraic evaluation: Direct substitution works when the function is continuous at the point. For an indeterminate form such as (0/0), factorisation, rationalisation, or standard limits may be needed.
• Standard limits:
  
  TEXT

    lim(x→0) sin x/x = 1
    lim(x→0) (1 − cos x)/x = 0
    lim(x→0) (e^x − 1)/x = 1

  
  The angles in trigonometric limits are measured in radians.
• Worked example: For (f(x)=\frac{x^2-4}{x-2}), factor (x^2-4=(x-2)(x+2)). For (x\ne2), (f(x)=x+2), so
  
  TEXT

    lim(x→2) (x² − 4)/(x − 2) = lim(x→2)(x + 2) = 4.

  
  The function may be undefined at (x=2), but its limit exists.

III. Continuity — Unbroken functional behaviour

Continuity means that the function has no break, jump, or hole at the point under consideration. It connects the actual function value with its limiting value.

A. Continuity

A function (f) is continuous at (x=a) if all three conditions below hold.

• Existence of the function value: (f(a)) must be defined.
• Existence of the limit: (\lim_{x\to a}f(x)) must exist and be finite.
• Agreement condition:
  
  TEXT

    lim(x→a) f(x) = f(a)

  
  Here, (a) is the point of continuity and (f(a)) is the actual function value.
• Interval continuity: A function is continuous on an interval if it is continuous at every interior point. At a closed endpoint, one-sided continuity is used.
• Common continuous functions: Polynomials are continuous for all real (x); rational functions are continuous wherever their denominators are nonzero; (\sin x), (\cos x), (e^x), and (\ln x) are continuous on their natural domains.
• Types of discontinuity:
  • Removable: A hole occurs, as in (\frac{x^2-1}{x-1}) at (x=1).
  • Jump: Left and right limits exist but are unequal.
  • Infinite: The function becomes unbounded, as (\frac{1}{x}) near (x=0).
• Worked example: Define
  
  TEXT

    f(x) = (x² − 1)/(x − 1), x ≠ 1
           2,              x = 1.

  
  Since the quotient equals (x+1) for (x\ne1), (\lim_{x\to1}f(x)=2=f(1)). Therefore, (f) is continuous at (x=1).

IV. Differentiability — Instantaneous rate of change

Differentiability measures the instantaneous rate at which a function changes. A derivative exists when the graph has a well-defined tangent direction at the point.

A. Differentiability

The derivative of (f) at (x=a) is defined by the limiting difference quotient.

• Definition:
  
  TEXT

    f′(a) = lim(h→0) [f(a+h) − f(a)]/h

  
  Here, (h) is a small change in the input, and (f'(a)) is the instantaneous rate of change.
• Geometric meaning: (f'(a)) is the slope of the tangent to (y=f(x)) at ((a,f(a))). A positive derivative indicates increase; a negative derivative indicates decrease.
• Differentiability implies continuity: If (f) is differentiable at (a), then it is continuous at (a). The converse is false: (f(x)=|x|) is continuous at (0) but not differentiable there because its one-sided slopes are (-1) and (1).
• Basic rules:
  
  TEXT

    (x^n)′ = n x^(n−1)
    (cf)′ = c f′
    (f + g)′ = f′ + g′
    (fg)′ = f′g + fg′
    (f/g)′ = (g f′ − f g′)/g²

  
  Here, (c) is a constant and (f,g) are differentiable functions.
• Chain rule: If (y=f(g(x))), then
  
  TEXT

    dy/dx = f′(g(x)) · g′(x).

  
  This applies to composite functions.
• Worked example: For (y=(3x^2+1)^4), let (u=3x^2+1). Then
  
  TEXT

    dy/dx = 4u³ · 6x = 24x(3x²+1)³.

  
  The outer derivative is (4u^3), and the inner derivative is (6x).

V. Maxima and Minima — Extreme values of functions

Maxima and minima identify points where a function reaches locally or globally high and low values. Derivatives locate candidates, while interval and endpoint checks establish the conclusion.

A. Maxima and minima

A local maximum is greater than nearby function values, while a local minimum is smaller than nearby function values.

• Critical points: A point (x=c) is critical when (f'(c)=0) or (f'(c)) does not exist, provided (f(c)) is defined.
• First derivative test:
  • If (f') changes from positive to negative at (c), (f(c)) is a local maximum.
  • If (f') changes from negative to positive at (c), (f(c)) is a local minimum.
• Second derivative test:
  
  TEXT

    f′(c)=0 and f″(c)>0  → local minimum
    f′(c)=0 and f″(c)<0  → local maximum

  
  Here, (f''(c)) is the second derivative at the stationary point. If (f''(c)=0), the test is inconclusive.
• Absolute extrema on ([a,b]): For a continuous function, calculate (f) at every critical point inside ([a,b]) and at both endpoints (a) and (b), then compare the values.
• Worked example: Let (f(x)=x^3-3x). Then
  
  TEXT

    f′(x)=3x²−3=3(x−1)(x+1).

  
  Critical points are (x=-1,1). Since (f''(x)=6x), (f''(-1)=-6<0), giving a local maximum (f(-1)=2); (f''(1)=6>0), giving a local minimum (f(1)=-2).

VI. Mean Value Theorem — Relating average and instantaneous change

The Mean Value Theorem guarantees a point where instantaneous rate of change equals average rate of change. It converts endpoint information into a derivative statement.

A. Mean Value Theorem

If (f) is continuous on the closed interval ([a,b]) and differentiable on the open interval ((a,b)), then at least one (c\in(a,b)) satisfies the theorem.

• Formal statement:
  
  TEXT

    f′(c) = [f(b) − f(a)]/(b − a)

  
  Here, (a,b) are distinct endpoints and (c) is an interior point.
• Geometric meaning: The tangent at (x=c) is parallel to the secant line joining ((a,f(a))) and ((b,f(b))).
• Conditions: Continuity is required at both endpoints and throughout ([a,b]); differentiability is required only inside ((a,b)). A corner, cusp, or discontinuity can invalidate the theorem.
• Rolle’s Theorem: If (f(a)=f(b)) under the same regularity conditions, then some (c) satisfies (f'(c)=0). It is the special case of the Mean Value Theorem with equal endpoint values.
• Worked example: For (f(x)=x^2) on ([1,3]), the average rate is
  
  TEXT

    [f(3)−f(1)]/(3−1) = (9−1)/2 = 4.

  
  Since (f'(x)=2x), (2c=4), so (c=2), which lies in ((1,3)).

VII. Integration — Accumulation and inverse differentiation

Integration calculates accumulated quantities and reverses differentiation. It is represented by indefinite integrals, definite integrals, and the Fundamental Theorem of Calculus.

A. Integration

An indefinite integral gives a family of antiderivatives, while a definite integral gives a numerical accumulation over an interval.

• Indefinite integral:
  
  TEXT

    ∫ f(x) dx = F(x) + C, where F′(x)=f(x).

  
  Here, (x) is the variable, (F) is an antiderivative, and (C) is an arbitrary constant.
• Basic rules:
  
  TEXT

    ∫ x^n dx = x^(n+1)/(n+1) + C, n ≠ −1
    ∫ 1/x dx = ln|x| + C
    ∫ e^x dx = e^x + C
    ∫ cos x dx = sin x + C

  
  The power rule excludes (n=-1) because division by (n+1) would be undefined.
• Definite integral: The value
  
  TEXT

    ∫[a to b] f(x) dx

  
  represents signed accumulation from (x=a) to (x=b). Regions below the (x)-axis contribute negatively.
• Fundamental Theorem of Calculus: If (F'(x)=f(x)), then
  
  TEXT

    ∫[a to b] f(x) dx = F(b) − F(a).

  
  This links differentiation and integration.
• Methods: Substitution simplifies composite expressions by replacing (u=g(x)); integration by parts follows
  
  TEXT

    ∫u dv = uv − ∫v du,

  
  where (u) and (v) are differentiable functions or differentials.
• Worked example: Using the antiderivative (F(x)=x^3/3),
  
  TEXT

    ∫[0 to 2] x² dx = [x³/3]₀² = 8/3.

  
  This is the signed area under (y=x^2) between (x=0) and (x=2), measured in square units.