Comprehensive Study Guide for Differential Calculus and Mean Value Theorems

Foundations of Differential Calculus

  • Differential calculus addresses the mathematical study of change, motion, growth, and decay.
  • It involves investigating the derivative, which represents the change of functions relative to varied inputs.

Successive Differentiation and Notation

  • If a function y=f(x)y = f(x) is differentiable, its first derivative provides the rate of change and is denoted as dydx\frac{dy}{dx}, y1y_1, y′(x)y'(x), or DyDy.
  • If the first derivative is itself differentiable, the resulting derivative is termed the second-order derivative, denoted as d2ydx2\frac{d^2y}{dx^2}, y2y_2, y′′(x)y''(x), or D2yD^2y.
  • This process continues for nth order derivatives, represented by symbols such as:
    • dnydxn\frac{d^ny}{dx^n}
    • DnyD^ny, where D=ddxD = \frac{d}{dx}
    • yn(x)y_n(x) or yny_n
    • f(n)(x)f^{(n)}(x)
  • The value of the nth differential coefficient evaluated at a specific point x=ax = a is represented as (yn)a(y_n)_a or f(n)(a)f^{(n)}(a).

nth Order Derivatives of Standard Functions

  • Power Functions of the form (ax+b)m(ax + b)^m:
    • The nth derivative is given by yn=m(m−1)(m−2)×...×(m−n+1)an(ax+b)m−ny_n = m(m - 1)(m - 2) \times \text{...} \times (m - n + 1) a^n (ax + b)^{m-n}.
    • In factorial notation: yn=m!(m−n)!an(ax+b)m−ny_n = \frac{m!}{(m-n)!} a^n (ax + b)^{m-n}.
  • Negative Power Functions (ax+b)−m(ax + b)^{-m}:
    • The nth derivative is yn=(−1)n(m+n−1)!(m−1)!an(ax+b)m+ny_n = \frac{(-1)^n (m + n - 1)!}{(m - 1)!} \frac{a^n}{(ax + b)^{m+n}}.
    • A specific case for m=1m = 1: dndxn(1ax+b)=(−1)nn!an(ax+b)n+1\frac{d^n}{dx^n} (\frac{1}{ax + b}) = \frac{(-1)^n n! a^n}{(ax + b)^{n+1}}.
  • Logarithmic Functions log(ax+b)\text{log}(ax + b):
    • The first derivative is y1=aax+by_1 = \frac{a}{ax + b}.
    • The nth derivative follows the pattern: yn=(−1)n−1(n−1)!an(ax+b)ny_n = \frac{(-1)^{n-1} (n - 1)! a^n}{(ax + b)^n}.
  • Exponential Functions:
    • For y=eaxy = e^{ax}, the nth derivative is yn=aneaxy_n = a^n e^{ax}.
    • For y=amxy = a^{mx}, the nth derivative is yn=mnamx(log(a))ny_n = m^n a^{mx} (\text{log}(a))^n.
  • Trigonometric Functions:
    • For y=sin(ax+b)y = \text{sin}(ax + b), the nth derivative is yn=ansin(ax+b+n×pi2)y_n = a^n \text{sin}(ax + b + \frac{n\times\text{pi}}{2}).
    • For y=cos(ax+b)y = \text{cos}(ax + b), the nth derivative is yn=ancos(ax+b+n×pi2)y_n = a^n \text{cos}(ax + b + \frac{n\times\text{pi}}{2}).
  • Exponential Product with Trigonometric Functions:
    • For y=eaxcos(bx+c)y = e^{ax} \text{cos}(bx + c), the derivative is yn=rneaxcos(bx+c+nθ)y_n = r^n e^{ax} \text{cos}(bx + c + n\theta).
    • For y=eaxsin(bx+c)y = e^{ax} \text{sin}(bx + c), the derivative is yn=rneaxsin(bx+c+nθ)y_n = r^n e^{ax} \text{sin}(bx + c + n\theta).
    • In these cases, constants are defined as r=sqrt(a2+b2)r = \text{sqrt}(a^2 + b^2) and θ=tan−1(ba)\theta = \text{tan}^{-1}(\frac{b}{a}).

Leibnitz's Theorem for Product Differentiation

  • This theorem provides a method to find the nth derivative of a product of two functions, uu and vv.
  • Statement: If y=uvy = uv, where uu and vv are functions of xx, then:
    • yn=unv+(n1)un−1v1+(n2)un−2v2+...+(nr)un−rvr+...+uvny_n = u_n v + \binom{n}{1} u_{n-1} v_1 + \binom{n}{2} u_{n-2} v_2 + \text{...} + \binom{n}{r} u_{n-r} v_r + \text{...} + u v_n
  • The proof is established via mathematical induction, demonstrating validity for n=1,2n = 1, 2 and then assuming truth for n=mn = m to prove it for n=m+1n = m + 1.

Continuity and Differentiability

  • Continuity Analysis: A function f(x)f(x) is continuous at x=ax = a if the limit as xx approaches aa from both sides equals the finite value f(a)f(a).
  • Differentiability Analysis: A function is differentiable at x=ax = a if the Right-Hand Derivative (RHDRHD) equals the Left-Hand Derivative (LHDLHD).
    • RHD=limh→0f(a+h)−f(a)hRHD = \text{lim}_{h \to 0} \frac{f(a + h) - f(a)}{h}
    • LHD=limh→0f(a−h)−f(a)−hLHD = \text{lim}_{h \to 0} \frac{f(a - h) - f(a)}{-h}
  • Key Principles:
    • All differentiable functions are continuous, though the reverse is not always true.
    • Common algebraic, trigonometric, logarithmic, and exponential functions are typically continuous and differentiable within their domains.

Rolle's Theorem

  • Conditions:
    1. f(x)f(x) is continuous in the closed interval [a,b][a, b].
    2. f(x)f(x) is differentiable in the open interval (a,b)(a, b).
    3. f(a)=f(b)f(a) = f(b).
  • Conclusion: There exists at least one point cc where a<c<ba < c < b such that f′(c)=0f'(c) = 0.
  • Geometrical Interpretation: If the endpoints of a continuous, smooth curve have the same y-coordinates, there is at least one point between them where the tangent is parallel to the x-axis.
  • Algebraic Interpretation: Between any two roots of the equation f(x)=0f(x) = 0, there exists at least one root of the derivative equation f′(x)=0f'(x) = 0.

Lagrange's Mean Value Theorem (LMVT)

  • Conditions:
    1. f(x)f(x) is continuous in [a,b][a, b].
    2. f(x)f(x) is differentiable in (a,b)(a, b).
  • Conclusion: There exists at least one point cc where a<c<ba < c < b such that f(b)−f(a)b−a=f′(c)\frac{f(b) - f(a)}{b - a} = f'(c).
  • Geometrical Interpretation: There is a point on the curve where the tangent is parallel to the chord joining the points (a,f(a))(a, f(a)) and (b,f(b))(b, f(b)).
  • Function Trends:
    • If f′(x)>0f'(x) > 0 in the interval, the function is strictly increasing.
    • If f′(x)<0f'(x) < 0 in the interval, the function is strictly decreasing.

Cauchy's Mean Value Theorem (CMVT)

  • Conditions: Functions f(x)f(x) and g(x)g(x) are continuous in [a,b][a, b] and differentiable in (a,b)(a, b), with g′(x)≠0g'(x) \neq 0 in the interval.
  • Conclusion: There exists at least one point cc in (a,b)(a, b) such that:
    • f(b)−f(a)g(b)−g(a)=f′(c)g′(c)\frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}.

Taylor's and Maclaurin's Series

  • Taylor's Series: Expands a function around a point x=ax = a:
    • f(x)=f(a)+(x−a)f′(a)+(x−a)22!f′′(a)+...+(x−a)nn!f(n)(a)+...f(x) = f(a) + (x - a) f'(a) + \frac{(x - a)^2}{2!} f''(a) + \text{...} + \frac{(x - a)^n}{n!} f^{(n)}(a) + \text{...}
  • Maclaurin's Series: A Taylor series centered at zero (a=0a = 0):
    • f(x)=f(0)+xf′(0)+x22!f′′(0)+x33!f′′′(0)+...f(x) = f(0) + x f'(0) + \frac{x^2}{2!} f''(0) + \frac{x^3}{3!} f'''(0) + \text{...}
  • Standard Maclaurin Expansions:
    • ex=1+x+x22!+x33!+...e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \text{...}
    • sin(x)=x−x33!+x55!−...\text{sin}(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \text{...}
    • cos(x)=1−x22!+x44!−...\text{cos}(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \text{...}
    • log(1+x)=x−x22+x33−x44+...\text{log}(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \text{...}
    • tan(x)=x+x33+2×x515+...\text{tan}(x) = x + \frac{x^3}{3} + \frac{2 \times x^5}{15} + \text{...}

Questions & Discussion

  • Exercise 2.1 Focus: Finding nth derivatives for complex rational functions, often involving partial fraction expansion before applying standard formulae. For example, for y=x(x−1)(x−2)(x−3)y = \frac{x}{(x-1)(x-2)(x-3)}, the strategy is to split the term into simpler fractions to compute the nth derivative.
  • Example 17 (Taylor Series): Proving that log(sin(x+h))=log(sin(x))+h×cot(x)−h22!×cosec2(x)+...\text{log}(\text{sin}(x + h)) = \text{log}(\text{sin}(x)) + h \times \text{cot}(x) - \frac{h^2}{2!} \times \text{cosec}^2(x) + \text{...} involves calculating successive derivatives of log(sin(x))\text{log}(\text{sin}(x)): y1=cot(x)y_1 = \text{cot}(x), y2=−cosec2(x)y_2 = -\text{cosec}^2(x), and y3=2×cosec2(x)×cot(x)y_3 = 2 \times \text{cosec}^2(x) \times \text{cot}(x).
  • Numerical Applications: Using series to approximate values like sqrt(1.02)\text{sqrt}(1.02). By setting f(x)=sqrt(x)f(x) = \text{sqrt}(x), a=1a = 1, and h=0.02h = 0.02 in the Taylor series, one obtains 1+0.01−0.00005...×approx.=1.00991 + 0.01 - 0.00005 \text{...} \times \text{approx.} = 1.0099.