Lesson Overview
Chapter
7 — C&D
Days
6 Days · 9 Periods
Board Wt.
~15 marks ISC
Dates
24 Apr – 2 May 2026
Sections Covered
Days 1–2 · Continuity Basics
Slope · equations at given points · parametric curves · tangents from external point · parallel/perpendicular · through origin · curve intersections
Days 3–4 · Finding Constants
Chain rule · direct rates · geometric rates (circles/ladders/shadows) · conical & spherical containers · connected variables
Days 5–6 · Continuity & Differentiability
Continuous vs Differentiable · |x| counter-example · Finding constants · ISC board questions
How to Use This Lesson
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DAY 1 · CHAPTER 7
Day 1 — Continuity — Definition & Basic Examples
✎Concept Board — Day 1Click to open · Draw diagrams & explain concepts▸ Open
Definition of Continuity at x = a
DEF\(f\) continuous at \(a\) if: (i) \(f(a)\) defined; (ii) \(\lim_{x\to a}f(x)\) exists; (iii) \(\lim_{x\to a}f(x)=f(a)\)
KEYEquivalent: LHL = f(a) = RHL, i.e., \(\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)\)
DISCDiscontinuity if: \(f(a)\) undefined, or LHL≠RHL, or limit ≠ \(f(a)\)
Questions — Day 1
Ex 1
Examine the continuity of \(f(x)=2x^2-1\) at \(x=3\).
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Ex 2
Find the points of discontinuity of \(f(x)=\dfrac{x^2-1}{x-1}\).
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Ex 3
Show \(f(x)=x^2\) (\(x\neq1\)), \(f(1)=2\) is discontinuous at \(x=1\).
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Ex 4
Is \(f(x)=x\) if \(x\leq1\), \(5\) if \(x>1\) continuous at \(x=0,1,2\)?
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Ex 5
Discuss continuity of \(f(x)=x^3-3\) (\(x\leq2\)); \(x^2+1\) (\(x>2\)).
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Ex 6
Is \(f(x)=x^2-\sin x+5\) continuous at \(x=\pi\)?
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Q 1
\(f(x)=\dfrac{x^2-4x+3}{x^2-1}\) (\(x\neq1\)), \(f(1)=2\). Test continuity at \(x=1\).
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Q 2
Examine: \(f(x)=x+1\) (\(x\geq1\)); \(x^2+1\) (\(x<1\)).
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Q 3
Examine: \(f(x)=x^{10}-1\) (\(x\leq1\)); \(x^2\) (\(x>1\)).
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Q 4
Verify \(f(x)=x\sin(1/x)\) (\(x\neq0\)), \(f(0)=0\) is continuous at \(x=0\).
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Day 1 — Video Resources
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DAY 2 · CHAPTER 7
Day 2 — Discontinuities & Algebra of Continuous Functions
✎Concept Board — Day 2Click to open · Draw diagrams & explain concepts▸ Open
Types of Discontinuity & Algebra
RemovableLimit exists but ≠ \(f(a)\). Fix by redefining \(f(a)\).
JumpLHL ≠ RHL — limit DNE. Cannot be fixed by redefining.
Infinite\(f\to\pm\infty\) — vertical asymptote.
ALGEBRAIf f, g continuous at c: \(f\pm g,\ fg,\ f/g\) (g(c)≠0), \(f\circ g\) all continuous.
Questions — Day 2
Ex 7
Find all points of discontinuity: \(|x|+3\) (\(x\leq-3\)); \(-2x\) (\(-3
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Ex 8
Show \(f(x)=\dfrac{e^{1/x}-1}{e^{1/x}+1}\) (\(x\neq0\)), \(f(0)=0\) is discontinuous at \(x=0\).
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Ex 9
Show \(x=2\) is a removable discontinuity for \(f(x)=2x\) (\(x<2\)), \(2\) (\(x=2\)), \(x^2\) (\(x>2\)).
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Ex 10
Show \(f(x)=\cos(x^2)\) is continuous everywhere.
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Ex 11
Show \(\dfrac{\sin x}{x}+\cos x\) (\(x>0\)); \(2\) (\(x=0\)); \(\dfrac{\sqrt{1-x}-1}{-x}\) (\(x<0\)) is continuous at \(x=0\).
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Q 5
Find all points of discontinuity of \(f(x)=2x+3\) (\(x\leq2\)); \(2x-3\) (\(x>2\)).
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Q 6
Discuss discontinuity of \(f(x)=2x-1\) (\(x<0\)); \(2x+1\) (\(x\geq0\)) at \(x=0\).
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Q 7
Examine: \(f(x)=-2\) (\(x\leq-1\)); \(2x\) (\(-11\)).
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Q 8
\(f(x)=\dfrac{x^2-4}{x+2}\) (\(x\neq-2\)), \(f(-2)=0\). Discuss continuity at \(x=-2\).
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Day 2 — Video Resources
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DAY 3 · CHAPTER 7
Day 3 — Finding Constants for Continuity
✎Concept Board — Day 3Click to open · Draw diagrams & explain concepts▸ Open
Finding Constants for Continuity
STEP 1Compute LHL using left-piece formula.
STEP 2Compute RHL using right-piece formula.
STEP 3Set LHL = f(a) = RHL, solve for constant(s).
TRIGFor \(\frac{\cos x}{\pi-2x}\): sub \(x=\pi/2-h\) ⟹ \(\cos x=\sin h,\ \pi-2x=2h\).
Questions — Day 3
Ex 12
Find \(k\): \(f(x)=2x+1\) (\(x<2\)); \(k\) (\(x=2\)); \(3x-1\) (\(x>2\)) continuous.
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Ex 13
Find \(a,b\): \(f(x)=5\) (\(x\leq2\)); \(ax+b\) (\(2
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Ex 14
Find \(a\): \(\dfrac{1-\cos4x}{x^2}\) (\(x<0\)); \(a\) (\(x=0\)); \(\dfrac{x}{\sqrt{16+x}-4}\) (\(x>0\)) continuous.
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Ex 15
Find \(k\): \(f(x)=\dfrac{k\cos x}{\pi-2x}\) (\(x\neq\pi/2\)), \(f(\pi/2)=3\) is continuous.
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Ex 16
Find \(a\): \(f(x)=a\sin\dfrac{\pi}{2}(x+1)\) (\(x\leq0\)); \(\dfrac{\tan x-\sin x}{x^3}\) (\(x>0\)) continuous.
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Ex 17
\(f(x)=\dfrac{\log(1+ax)-\log(1-bx)}{x}\) undefined at \(x=0\). Define \(f(0)\) for continuity.
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Q 9
Find \(a\): \(f(x)=2x-1\) (\(x<2\)); \(a\) (\(x=2\)); \(x+1\) (\(x>2\)) continuous.
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Q 10
Find \(k\): \(f(x)=kx+1\) (\(x\leq\pi\)); \(\cos x\) (\(x>\pi\)) continuous at \(x=\pi\).
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Q 11
For what \(\lambda\) is \(\lambda(x^2-2x)\) (\(x\leq0\)); \(4x+1\) (\(x>0\)) continuous at \(x=0\)?
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Q 12
Find \(p,q\): \(\dfrac{1-\sin^3x}{3\cos^2x}\) (\(x<\pi/2\)); \(p\) (\(x=\pi/2\)); \(\dfrac{q(1-\sin x)}{(\pi-2x)^2}\) (\(x>\pi/2\)) continuous.
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Day 3 — Video Resources
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DAY 4 · CHAPTER 7
Day 4 — Differentiability — LHD, RHD & First Principles
✎Concept Board — Day 4Click to open · Draw diagrams & explain concepts▸ Open
Differentiability — LHD & RHD
LHD\(Lf'(a)=\lim_{h\to0^+}\dfrac{f(a-h)-f(a)}{-h}\)
RHD\(Rf'(a)=\lim_{h\to0^+}\dfrac{f(a+h)-f(a)}{h}\)
COND\(f\) differentiable at \(a\) iff LHD = RHD = \(f'(a)\)
GRAPHNot differentiable at corner points, cusps, or vertical tangents.
Questions — Day 4
Ex 18
Show \(f(x)=x^2\) is differentiable at \(x=1\) and find \(f'(1)\).
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Ex 19
Show \(f(x)=[x]\) (GIF) is not differentiable at \(x=1\) and \(x=2\).
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Ex 20
Show \(f(x)=2+x\) (\(x\geq0\)); \(2-x\) (\(x<0\)) is not derivable at \(x=0\).
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Ex 21
Show \(f(x)=x^2\sin(1/x)\) (\(x\neq0\)), \(f(0)=0\) is differentiable at \(x=0\).
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Ex 22
Discuss differentiability of \(f(x)=|x-1|+|x-2|\).
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Q 1
Show \(f(x)=|x-4|\) is continuous but not differentiable at \(x=4\).
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Q 2
Show \(f(x)=|x-5|\) is continuous at \(x=5\) but \(f'(5)\) does not exist.
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Q 3
Show \(f(x)=x^2\) (\(x\leq1\)); \(1/x\) (\(x>1\)) is continuous but not differentiable at \(x=1\).
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Q 4
Examine differentiability at \(x=2\): \(f(x)=1+x\) (\(x\leq2\)); \(5-x\) (\(x>2\)).
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Day 4 — Video Resources
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DAY 5 · CHAPTER 7
Day 5 — Continuity vs Differentiability — The Key Theorem
✎Concept Board — Day 5Click to open · Draw diagrams & explain concepts▸ Open
Continuity vs Differentiability
THMDifferentiable at \(a\) ⟹ Continuous at \(a\)
CONTRANot continuous ⟹ Not differentiable (contrapositive)
FALSEContinuous ⟹ Differentiable is FALSE. Counter-ex: \(|x|\) at \(x=0\)
ISC TIPAlways compute BOTH LHD and RHD and check equality.
Questions — Day 5
Ex 23
Show \(f(x)=|x|\) is continuous at \(x=0\) but not differentiable there.
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Ex 24
If \(f\) is everywhere differentiable, find \(a,b\): \(f(x)=x^2+3x+a\) (\(x\leq1\)); \(bx+2\) (\(x>1\)).
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Ex 25
Prove \(f(x)=|x-1|\) is continuous but not differentiable at \(x=1\).
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Ex 26
Show \(3x-2\) (\(02\)) is continuous but not differentiable at \(x=2\).
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Ex 27
If \(f\) is differentiable at \(x=a\), evaluate \(\displaystyle\lim_{x\to a}\dfrac{x^2f(a)-a^2f(x)}{x-a}\).
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Q 5
Find \(a,b\): \(f(x)=ax^2+b\) (\(x<1\)); \(2x+1\) (\(x\geq1\)) is differentiable at \(x=1\).
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Q 6
For what \(a,b\) is \(f(x)=x^2\) (\(x\leq c\)); \(ax+b\) (\(x>c\)) differentiable at \(x=c\)?
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Q 7
\(f(x+y)=f(x)f(y)\), \(f(x)\neq0\), \(f'(0)=2\). Prove \(f'(x)=2f(x)\).
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Q 8
Use IVT to show \(x^3-6x+2=0\) has a root in \((0,1)\).
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Day 5 — Video Resources
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DAY 6 · CHAPTER 7
Day 6 — ISC Board Review — Mixed Problems
✎Concept Board — Day 6Click to open · Draw diagrams & explain concepts▸ Open
ISC Board — Key Results
CContinuity: LHL = f(a) = RHL
DDifferentiability: LHD = RHD
LIMITS\(\frac{\sin x}{x}\to1\), \(\frac{1-\cos x}{x^2}\to\frac{1}{2}\), \(\frac{\tan x}{x}\to1\), \(\frac{\log(1+x)}{x}\to1\)
KEYD⟹C (not vice versa). Canonical: \(|x|\) continuous but not diff at 0.
Questions — Day 6
ISC 1
Find \(k\): \(\dfrac{k\cos x}{\pi-2x}\) (\(x\neq\pi/2\)), \(f(\pi/2)=3\) continuous.
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ISC 2
Define \(f(0)\) for continuity of \(f(x)=\dfrac{3x+2\sin x}{x}\), \(x\neq0\).
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ISC 3
Examine continuity: \(f(x)=\dfrac{\tan2x}{3x}\) (\(x\neq0\)), \(f(0)=\dfrac{2}{3}\) at \(x=0\).
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ISC 4
Find \(k\): \(\dfrac{\sqrt{1+kx}-\sqrt{1-kx}}{x}\) (\(-1\leq x<0\)); \(\dfrac{2x+1}{x-1}\) (\(0\leq x<1\)) continuous.
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ISC 5
Find \(a,b\): \(ax^2+b\) (\(x<1\)); \(2x+1\) (\(x\geq1\)) differentiable at \(x=1\).
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ISC 6
For what \(\lambda\): \(\lambda(x^2-2x)\) (\(x\leq0\)); \(4x+1\) (\(x>0\)) continuous at \(x=0\)?
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ISC 7
Evaluate \(\displaystyle\lim_{x\to a}\dfrac{x^2f(a)-a^2f(x)}{x-a}\) given \(f\) differentiable at \(x=a\).
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ISC 8
Find \(p,q\): \(\dfrac{1-\sin^3x}{3\cos^2x}\) (\(x<\pi/2\)); \(p\) (\(x=\pi/2\)); \(\dfrac{q(1-\sin x)}{(\pi-2x)^2}\) (\(x>\pi/2\)) continuous.
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ISC 9
Find \(a\): \(a\sin\dfrac{\pi}{2}(x+1)\) (\(x\leq0\)); \(\dfrac{\tan x-\sin x}{x^3}\) (\(x>0\)) continuous at \(x=0\).
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Day 6 — Video Resources
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