M1_VOL2_CALCULUS.pdf

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1. Limits and Continuity

1.1 Introduction

• Concept: Calculus builds on real numbers, sets, functions, and operations. It introduces limits and continuity to understand behavior at points and infinity.
• Example: In school, you learned about real numbers and functions like $ f(x) = x^2 $.

1.2 What is a Function?

• Definition: A function $ f: A \to B $ assigns exactly one output in $ B $ for every input in $ A $.
• Domain: Input set $ A $.
• Codomain: Output set $ B $.
• Range: Actual output values $ {f(a) \mid a \in A} $.
• Example: $ f(x) = x^2 $ is a function from $ \mathbb{R} $ to $ \mathbb{R} $.
• Counterexample: $ R = {(1,a), (2,b), (3,a), (1,b)} $ is not a function because 1 maps to both $ a $ and $ b $.

Graph of Functions

• Definition: Graph of $ f $ is $ {(x, f(x)) \mid x \in domain} $.
• Example: For $ f(x) = 7x+2 $, graph is all points $ (x, 7x+2) $.

Types of Functions

• Linear: $ f(x) = ax + b $
• Quadratic: $ f(x) = ax^2 + bx + c $
• Polynomial: $ f(x) = a_nx^n + ··· + a_0 $
• Exponential: $ f(x) = a^x $
• Logarithmic: $ f(x) = \log_a x $
• Trigonometric: $ \sin x, \cos x, \tan x $
• Step functions: Floor $ \lfloor x \rfloor $, Ceiling $ \lceil x \rceil $, Absolute $ |x| $
• Examples:
  • Floor in $[-1,2]$: $ \lfloor x \rfloor = -1 $ for $-1 \leq x < 0$, $0$ for $0 \leq x < 1$, $1$ for $1 \leq x < 2$.
  • Absolute: $ |x| = x $ if $ x \geq 0 $, $-x$ if $ x < 0 $.

Bounded Function

• Definition: $ f $ is bounded if $ m \leq f(x) \leq M $ for all $ x $.
• Example: $ f(x) = \frac{1}{x^2+1} $ is bounded ($0 \leq f(x) \leq 1$).
• Counterexample: $ f(x) = \frac{1}{x} $ on $ (0, \infty) $ is unbounded.

Monotonicity

• Increasing: If $ x \leq y \implies f(x) \leq f(y) $.
• Decreasing: If $ x \leq y \implies f(x) \geq f(y) $.
• Example: $ f(x) = x^2 $ is increasing on $[0, \infty)$.
• Example: $ f(x) = 7-4x $ is decreasing on $ \mathbb{R} $.
• Example: $ f(x) = |x| $ is neither increasing nor decreasing on $ \mathbb{R} $.

Arithmetic Operations on Functions

• Sum: $ (f+g)(x) = f(x) + g(x) $
• Difference: $ (f-g)(x) = f(x) - g(x) $
• Product: $ (fg)(x) = f(x)g(x) $
• Quotient: $ (f/g)(x) = f(x)/g(x) $ (if $ g(x) \neq 0 $)
• Example: If $ f(x) = x^3 + 5x + 1 $, $ g(x) = 3x^2 + 2x + 5 $, then $ (f-g)(x) = x^3 - 3x^2 + 3x - 4 $.

Composition of Functions

• Definition: $ (g \circ f)(x) = g(f(x)) $.
• Example: If $ f(x) = x^2 $, $ g(x) = 3x^3 + 2x $, then $ (g \circ f)(x) = 3x^6 + 2x^2 $.
• Question: If $ f(x) = \frac{x}{x+a} $, $ f(f(x)) = \frac{x}{3x+4} $, find $ a $.
• Answer: $ a = 2 $.

1.3 Curve and Tangent

• Curve: Path of a moving point.
• Tangent: Line touching curve at a point, representing instantaneous direction.
• Example: Graph of $ f(x) = x^2 $ is a curve. At point $ (a, a^2) $, tangent is unique.
• Example: Floor function $ f(x) = \lfloor x \rfloor $ is not a curve (has jumps).
• Question: Is tangent possible for $ f(x) = \lfloor x \rfloor $ at $ x=2 $ and $ x=3.5 $?
• Answer: No tangent at $ x=2 $ (jump), tangent is $ y=3 $ at $ x=3.5 $.

1.4 Sequence and Limit of Sequence

• Sequence: Function $ f: \mathbb{N} \to \mathbb{R} $, denoted $ {a_n} $.
• Limit of Sequence: $ \lim_{n \to \infty} a_n = L $ if $ a_n $ gets arbitrarily close to $ L $ as $ n $ increases.
• Example: $ a_n = 1 - \frac{1}{n^2} $ converges to 1.
• Example: $ a_n = n $ diverges.
• Example: $ a_n = (-1)^n $ diverges (oscillates).
• Example: $ a_n = \frac{n+1}{n} $ converges to 1.

Subsequence

• Definition: A sequence formed by selecting terms from another sequence in order.
• Example: For $ a_n = 5n^2 + 1 $, subsequence $ b_n = a_{2n} = 5(2n)^2 + 1 $.

Tools for Limits

• Sum/Difference: $ \lim (a_n \pm b_n) = \lim a_n \pm \lim b_n $
• Product: $ \lim (a_n b_n) = \lim a_n \cdot \lim b_n $
• Quotient: $ \lim (a_n / b_n) = \lim a_n / \lim b_n $ (if $ \lim b_n \neq 0 $)
• Sandwich Principle: If $ a_n \leq c_n \leq b_n $ and $ \lim a_n = \lim b_n = L $, then $ \lim c_n = L $.
• Example: $ c_n = \frac{\sin n}{n} \rightarrow 0 $ (since $ -\frac{1}{n} \leq \frac{\sin n}{n} \leq \frac{1}{n} $).

Important Theorems

• If $ \lim a_n = L $, then $ \lim \frac{a_1 + ··· + a_n}{n} = L $.
• If $ \lim \frac{a_{n+1}}{a_n} = \ell $, then:
  • If $ |\ell| < 1 $, $ \lim a_n = 0 $.
  • If $ \ell > 1 $, $ \lim a_n = \infty $.

Exercises

• Q5: $ a_n = \frac{5+3\sqrt{n}}{\sqrt{n}} \rightarrow 3 $
• Q6: $ a_n = 5^{1/n} \rightarrow 1 $
• Q7: $ a_n = \left(\frac{1}{2}\right)^n \rightarrow 0 $
• Q8: $ a_n = \frac{(-1)^n}{2n} \rightarrow 0 $
• Q9: If $ b_n \rightarrow 1 $, $ c_n \rightarrow \infty $, then $ \frac{b_n}{c_n} \rightarrow 0 $

1.5 Limit of Function

• Definition: $ \lim_{x \to a} f(x) = L $ if $ f(x) $ gets close to $ L $ as $ x $ approaches $ a $.
• Left/Right Limits: $ \lim_{x \to a^-} f(x) $, $ \lim_{x \to a^+} f(x) $
• Example: $ \lim_{x \to 1} x^2 = 1 $
• Example: $ \lim_{x \to -1} \lfloor x \rfloor $ does not exist (left limit is -2, right limit is -1).
• Example: $ f(x) = 1 $ if $ x $ is rational, $ 0 $ otherwise. $ \lim_{x \to \sqrt{2}} f(x) $ does not exist.

Limit at Infinity

• Definition: $ \lim_{x \to \infty} f(x) = L $ if $ f(x) $ approaches $ L $ as $ x $ becomes very large.
• Example: $ \lim_{x \to \infty} \frac{1}{x} = 0 $

Algebra of Limits

• Sum/Difference: $ \lim (f \pm g) = \lim f \pm \lim g $
• Product: $ \lim (f \cdot g) = \lim f \cdot \lim g $
• Quotient: $ \lim (f/g) = \lim f / \lim g $ (if $ \lim g \neq 0 $)
• Example: $ \lim_{x \to 2} (5x+9) = 19 $
• Example: $ \lim_{x \to -3} x^4 = 81 $
• Example: $ \lim_{x \to 5} \frac{25}{x^2} = 1 $

Sandwich Theorem

• If $ f(x) \leq h(x) \leq g(x) $ and $ \lim_{x \to a} f(x) = \lim_{x \to a} g(x) = L $, then $ \lim_{x \to a} h(x) = L $.
• Example: $ \lim_{x \to 0} x^2 \sin(1/x) = 0 $
• Example: $ \lim_{x \to 0} \frac{\sin x}{x} = 1 $
• Example: $ \lim_{x \to 0} \frac{\tan x}{x} = 1 $
• Example: $ \lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2} $

1.6 Continuity

• Definition: $ f $ is continuous at $ a $ if $ \lim_{x \to a} f(x) = f(a) $.
• Example: $ f(x) = |x| $ is continuous at $ x=0 $.
• Example: $ f(x) = \lfloor x \rfloor $ is not continuous at integers.
• Piecewise Example: $ f(x) = $$ \begin{cases} x+1 & -4 \leq x < 2 \\ x^2-4 & 2 \leq x \leq 3 \end{cases} $$ $ is not continuous at $ x=2 $.

Theorems on Continuity

• Sum/Difference/Product/Quotient: If $ f $ and $ g $ are continuous at $ a $, so are $ f \pm g $, $ f \cdot g $, $ f/g $ (if $ g(a) \neq 0 $).
• Composition: If $ g $ is continuous at $ a $ and $ f $ is continuous at $ g(a) $, then $ f \circ g $ is continuous at $ a $.

Exercises

• Q11: $ f(x) = $$ \begin{cases} 1 & x \geq 0 \\ -1 & x < 0 \end{cases} $$ $
  • Right limit at 0: $ \lim_{x \to 0^+} f(x) = 1 $
  • Left limit at 0: $ \lim_{x \to 0^-} f(x) = -1 $
  • Limit at 0 does not exist.
• Q12:
  • $ \lim_{x \to \infty} \frac{1}{x} = 0 $ (Option 1)
  • $ \lim_{x \to \infty} \frac{x^2}{1+x} = \infty $
  • $ \lim_{x \to -\infty} \frac{1+x}{x^2} = 0 $ (Option 3)
  • $ \lim_{x \to \infty} \frac{1+x+x^2}{5x^2+1} = \frac{1}{5} $ (Option 4)
• Q14:
  • $ \lim_{x \to -1} \frac{x^2-6x-7}{x^2+3x+2} = \lim_{x \to -1} \frac{(x+1)(x-7)}{(x+1)(x+2)} = \lim_{x \to -1} \frac{x-7}{x+2} = -8 $ (Option 1)
  • $ \lim_{x \to 0} \frac{x^2-6x-7}{x^2+3x+2} = \frac{-7}{2} $
  • $ \lim_{x \to 3} \frac{x^2-6x+9}{x-3} = \lim_{x \to 3} (x-3) = 0 $

2. Differentiation

2.1 Differentiability and the Derivative

• Definition: $ f $ is differentiable at $ a $ if $ \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} $ exists.
• Example: $ f(x) = x $ is differentiable everywhere, derivative is 1.
• Example: $ f(x) = \sin x $ is differentiable at 0, derivative is 1.
• Example: $ f(x) = |x| $ is not differentiable at 0 (left and right derivatives differ).
• Example: $ f(x) = x^{1/3} $ is not differentiable at 0 (derivative tends to infinity).
• Example: $ f(x) = \lfloor x \rfloor $ is not differentiable at integers.

Relation to Continuity

• Theorem: If $ f $ is differentiable at $ a $, then $ f $ is continuous at $ a $.
• Example: $ f(x) = \lfloor x \rfloor $ is not continuous at integers, so not differentiable.

Derivative Rules

• Sum/Difference: $ (f \pm g)’ = f’ \pm g’ $
• Product: $ (fg)’ = f’g + fg’ $
• Quotient: $ (f/g)’ = \frac{f’g - fg’}{g^2} $
• Chain Rule: $ (f(g(x)))’ = f’(g(x))g’(x) $
• Example: $ f(x) = x^2 $, $ f’(x) = 2x $
• Example: $ f(x) = \sin x $, $ f’(x) = \cos x $
• Example: $ f(x) = e^x $, $ f’(x) = e^x $
• Example: $ f(x) = \ln x $, $ f’(x) = 1/x $

Exercises

• Q27: $ f(x) = 5x $, derivative at $ x=2 $ is 5.
• Q28:
  • $ f(x) = a $ (constant), derivative is 0.
  • $ f(x) = x - c $, derivative is 1.
  • $ f(x) = x^2 $, derivative at $ c $ is $ 2c $.
  • $ f(x) = e^x $, derivative at $ c $ is $ e^c $.
• Q29: Check graphs for continuity and differentiability.
• Q30:
  • If $ \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} $ exists, $ f $ is differentiable at $ a $.
  • If $ f $ is differentiable at $ a $, it is continuous at $ a $.
  • There exist continuous functions not differentiable at some points (e.g., $ |x| $ at 0).

2.2 Indeterminate Limits and L’Hôpital’s Rule

• Indeterminate Form: $ \frac{0}{0} $ or $ \frac{\infty}{\infty} $.
• L’Hôpital’s Rule: If $ \lim_{x \to a} \frac{f(x)}{g(x)} $ is indeterminate, and $ f’ $, $ g’ $ exist near $ a $, then $ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f’(x)}{g’(x)} $.
• Example: $ \lim_{x \to 0} \frac{\log(1+x)}{x} = 1 $
• Example: $ \lim_{x \to 0} \frac{\sin x}{x} = 1 $
• Example: $ \lim_{x \to \infty} \frac{a+be^x}{c+de^x} = \frac{b}{d} $
• Example: $ \lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2} $

Exercises

• Q40: $ f(x) = \sqrt{9-x^2} $, $ \lim_{x \to 1} \frac{f(x)-f(1)}{x-1} = -\frac{1}{2\sqrt{2}} $, $ \sqrt{8} \times $ this is $-1$.
• Q42: $ \lim_{x \to \infty} x e^{-x} = 0 $

2.3 Tangents and Linear Approximation

• Tangent Line: $ y = f’(a)(x-a) + f(a) $
• Linear Approximation: $ L(x) = f(a) + f’(a)(x-a) $
• Example: $ f(x) = \cos x $, tangent at $ x=\pi/3 $: $ y = -\frac{\sqrt{3}}{2}(x-\pi/3) + \frac{1}{2} $
• Example: $ f(x) = x^3 $, linear approximation at 1: $ L(x) = 3x-2 $

Exercises

• Q44: $ f(x) = 4x^2 $, tangent at $ x=2 $: $ y = 16x - 16 $
• Q45: $ f(x) = 2x+5 $, linear approximation at 0: $ L(x) = 2x+5 $
• Q46: Tangent at $ (1,0) $, passes through $ (5,8) $, slope $ f’(1) = 2 $
• Q47: $ f(x) = x^3 + 3x $, slopes at $ x=-1,0,1 $: $ m_1 + m_2 + m_3 = 15 $
• Q48: Same as Q46, $ f’(1) = 2 $
• Q49: Tangent at $ (1, f(1)) $ is $ y=3x+2 $, so $ f(1) = 5 $

2.4 Finding Critical Points: Applications

• Critical Point: $ f’(a) = 0 $ or $ f $ not differentiable at $ a $.
• Local Max/Min: Use second derivative test:
  • $ f’’(a) > 0 $: local min
  • $ f’’(a) < 0 $: local max
  • $ f’’(a) = 0 $: test fails (saddle or inflection)
• Example: $ f(x) = x^3 - 12x $, critical points at $ x=2 $ (local min), $ x=-2 $ (local max)
• Example: $ f(x) = \cos x $, critical points at $ x=k\pi $, local max at even $ k $, local min at odd $ k $
• Example: $ f(x) = x^3 + x^2 - x + 5 $, critical points at $ x=-1 $ (local max), $ x=1/3 $ (local min)

Global Max/Min

• Definition: Maximum/minimum value of $ f $ over an interval.
• Example: $ f(x) = x^2 $ on $[-1,1]$, global min at $ x=0 $, global max at $ x=-1 $ and $ x=1 $.

Exercises

• Q51: $ f(x) = \frac{1}{3}x^3 - x^2 + x $, only one critical point at $ x=1 $, second derivative test inconclusive (saddle point).
• Q52: $ f(x) = $$ \begin{cases} -x^2 + 2x + 3 & 0 \leq x \leq 50 \\ x^3 + 3 & -50 \leq x < 0 \end{cases} $$ $
  • $ x=1 $ is local max.
  • $ x=-50 $ is global min.
  • $ x=50 $ is not global min.
• Q53: At local min $ x=2 $, slope $ f’(2) = 0 $. At local max $ x=5 $, slope $ f’(5) = 0 $.
• Q56: Minimum of $ (x-\alpha)(x-\beta) $ at $ x = \frac{\alpha+\beta}{2} $.
• Q57: Max of $ 2xy $ when $ x+y=50 $: $ 1250 $.

3. Integration

3.1 Introduction

• Concept: Integration is used to compute areas under curves, volumes, and more.
• Example: Area of rectangle is $ lb $.

3.2 Computing Areas

• Area of Parallelogram: $ bh $
• Area of Triangle: $ \frac{1}{2}bh $
• Area of Trapezium: $ \frac{1}{2}(a+b)h $
• Area of Circle: $ \pi r^2 $ (using limits or integration)

Exercises

• Q65: Area of trapezium $ ACDB $: $ 6 $ sq units
• Q66: Sequence of circles, radius $ r_n = \frac{2n-1}{2n+2} $, area of biggest circle $ \leq \pi $, smallest circle $ \frac{\pi}{16} $

3.3 Riemann Sums and the Integral

• Partition: Divide interval $[a,b]$ into subintervals.
• Riemann Sum: $ S(P) = \sum_{i=1}^n f(x_i^*) \Delta x_i $
• Definite Integral: $ \int_a^b f(x) dx = \lim_{||P|| \to 0} S(P) $
• Example: $ \int_1^2 (2x-1) dx = 2 $

Exercises

• Q70: For $ f(x) = x $ on $²$, Riemann sum with $ x_i^* = x_i $: $ \frac{25(n+1)}{2n} $
• Q71: $ \int_0^2 (3x+1) dx = 8 $

3.5 Anti-derivatives (Indefinite Integrals)

• Definition: $ F $ is anti-derivative of $ f $ if $ F’(x) = f(x) $.
• Fundamental Theorem of Calculus: $ \int_a^b f(x) dx = F(b) - F(a) $
• Integration Rules:
  • $ \int x^n dx = \frac{x^{n+1}}{n+1} + C $ ($ n \neq -1 $)
  • $ \int \sin x dx = -\cos x + C $
  • $ \int e^x dx = e^x + C $
  • $ \int \frac{1}{x} dx = \ln|x| + C $

Integration by Parts

• Formula: $ \int f(x)g(x) dx = f(x) \int g(x) dx - \int f’(x) (\int g(x) dx) dx $
• Example: $ \int x^2 2^x dx = \frac{x^2 2^x}{\ln 2} - \frac{x 2^{x+1}}{(\ln 2)^2} + \frac{2^{x+1}}{(\ln 2)^3} + C $

Integration by Substitution

• Formula: $ \int f(g(x))g’(x) dx = \int f(u) du $ where $ u = g(x) $
• Example: $ \int \sin(5x) dx = -\frac{1}{5} \cos(5x) + C $

Basic Properties of Definite Integrals

• Linearity: $ \int (cf + dg) = c \int f + d \int g $
• Additivity: $ \int_a^b f = \int_a^c f + \int_c^b f $
• Improper Integrals: $ \int_a^\infty f(x) dx = \lim_{b \to \infty} \int_a^b f(x) dx $
• Example: $ \int_1^\infty e^{-x} dx = \frac{1}{e} $

Piecewise Defined Functions

• Example: $ f(x) = $$ \begin{cases} x & 0 \leq x \leq 1 \\ 3-x & 1 < x \leq 2 \end{cases} $$ $, $ \int_0^2 f(x) dx = 2 $

Exercises

• Q75: $ \int_2^3 x^2 dx = \frac{19}{3} $
• Q76: $ \int_1^2 (3x^2 + \frac{1}{x}) dx = 7 + \ln 2 $
• Q77:
  • $ \int_2^3 x^2 dx = \frac{19}{3} $
  • $ \int_1^2 \frac{1}{x} dx = \ln 2 $
  • $ \int_0^{\pi/3} \tan x \sec x dx = 1 $
  • $ \int_0^2 \frac{1}{\sqrt{4-x^2}} dx = \frac{\pi}{2} $
• Q78:
  • $ \int_1^\infty e^{-x} dx = \frac{1}{e} $
  • $ \int_1^\infty \frac{1}{x} dx $ does not exist
• Q81: Area between $ 3x^2 $ and $ 4-x^2 $: $ 3A = 16 $

Summary Table

This structured approach covers all major concepts from the PDF with definitions, examples, questions, and answers for clarity and practice¹.