Lesson Overview
Chapter
9 — AoD
Days
6 Days · 9 Periods
Board Wt.
~15 marks ISC
Dates
24 Apr – 2 May 2026
Sections Covered
Days 1–2 · Tangents & Normals
Slope · equations at given points · parametric curves · tangents from external point · parallel/perpendicular · through origin · curve intersections
Days 3–4 · Rate of Change
Chain rule · direct rates · geometric rates (circles/ladders/shadows) · conical & spherical containers · connected variables
Days 5–6 · Monotonicity
Strictly increasing/decreasing · First Derivative Test · intervals · trigonometric functions · proof-based questions
How to Use This Lesson
📝 Workspace Tab
Ruled workspace with pen/eraser draw tools. Click ↕ Expand for full-screen working space.
💡 Step Solution Tab
Manual: Click Next Step one at a time. Auto Play: Steps reveal at chosen speed. Edit steps inline with the step editor.
⚡ Surprise Test
Teacher login required. Choose mode (Quick 3Q / Day 5Q / Chapter 10Q / Custom), pick from days, swap questions, then launch fullscreen timed test.
DAY 1 · THU 24 APR 2026
Day 1 — Tangents & Normals: Fundamentals
✎Concept Board — Day 1Click to open · Draw diagrams & explain concepts▸ Open
1.1 Key Concepts — Tangents & Normals
SLOPESlope of tangent at P(x₁,y₁) on y=f(x): \(m = \left.\dfrac{dy}{dx}\right|_{(x_1,y_1)}\)
TANGENTEquation: \(y - y_1 = m(x - x_1)\)
NORMALSlope of normal = \(-\dfrac{1}{m}\). Equation: \(y - y_1 = -\dfrac{{1}}{{m}}(x - x_1)\)
PARALLELTangent ∥ x-axis: \(\dfrac{{dy}}{{dx}} = 0\) | Tangent ⊥ x-axis: \(\dfrac{{dx}}{{dy}} = 0\)
ORTHOGONALTwo curves cut at right angles: \(m_1 \cdot m_2 = -1\)
Type 1 — Slope of Tangent / Normal
Ex 1(i)
Find the slope of the tangent to \(y = x^3 - x + 1\) at \(x = 2\).
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Ex 1(ii)
Find the slope of the tangent to \(y = 2x^2 + 3\sin x\) at \(x = 0\).
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Type 2 — Equation of Tangent and Normal
Ex 2(i)
Find the equation of the tangent to \(y = 2\sin x + \sin 2x\) at \(x = \dfrac{{\pi}}{{3}}\).
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Ex 2(ii)
Find the equation of the normal to \(y = 2\sin^2 3x\) at \(x = \dfrac{{\pi}}{{6}}\).
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Ex 3
Find the equation of tangent to \(\dfrac{{x^2}}{{a^2}} + \dfrac{{y^2}}{{b^2}} = 1\) at \((x_1, y_1)\).
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Practice Questions
Q 1(i)
Find the slope of the tangent to \(y = \dfrac{{4}}{{x}}\) at \((2, 2)\).
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Q 1(iii)
Find the slope of the tangent to \(y = 2x - x^2\) at \(x = 1\).
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Q 2
Find the slope of normal to \(y = 3x^2\) at \(x = 2\).
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Q 3(i)
Find the tangent and normal to \(y = 2x^2 - 3x - 1\) at \((1, -2)\).
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📚 Homework
Q 3(iii): Tangent and normal to \(y = x^2 + 4x + 1\) at abscissa 3. | Q 4: Tangent and normal to \(y^2 = 4ax\) at \((at^2, 2at)\).📎 Assignments — Day 1 (Optional · Viewable by students)
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Day 1 — Video Resources
Exit Ticket
1State the formula for slope of tangent to y=f(x) at a point.
2Write the equation of normal at P(x₁,y₁) with slope m.
Key Formulas — T&N
•m_tan = dy/dx at point
•Normal slope = −1/m
•Tangent: y−y₁=m(x−x₁)
•∥ x-axis: dy/dx=0
DAY 2 · FRI 25 APR 2026
Day 2 — Tangents & Normals: Advanced
✎Concept Board — Day 2Click to open · Draw diagrams & explain concepts▸ Open
Type 3 — Parametric Curves
Ex 4
For \(ay^2 = x^3\), find the equation of the normal at \((am^2, am^3)\).
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Ex 5
Find the tangent and normal to \(x = a\sin^3\theta,\ y = a\cos^3\theta\) at \(\theta = \dfrac{{\pi}}{{4}}\).
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Type 4 — Tangent from External Point
Ex 7
Find normal to \(x^2 = 4y\) passing through \((1, 2)\). Also find the tangent.
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Type 5 — Parallel / Perpendicular Conditions
Ex 8
Find the point on \(y = x^2 - 4x + 3\) where the normal has slope \(\dfrac{{1}}{{2}}\).
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Ex 12
Find tangent to \(y = \sqrt{{5x-3}} - 2\) parallel to \(4x - 2y + 3 = 0\).
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Type 6 & 7 — Through Origin / Equally Inclined
Ex 9
For \(y = 4x^3 - 2x^5\), find all points where the tangent passes through the origin.
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Ex 11
Find the point on \(\sqrt{{x}} + \sqrt{{y}} = 4\) where the tangent is equally inclined to the axes.
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Practice Questions
Q 6
Find the point on \(y = 4x^3 - 3x + 5\) where the tangent is parallel to the x-axis.
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Q 8
Find points on \(x^2 + y^2 = 25\) where tangents are (i) ∥ x-axis (ii) ∥ y-axis.
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Q 9
Find point on \(y^2 = 4x\) where tangent is parallel to \(y = 2x + 4\).
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Q 15
Tangent at \((2, 3)\) on \(y^2 = ax^3 + b\) is \(y = 4x - 5\). Find \(a\) and \(b\).
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📚 Homework
Show that \(2x = y^2\) and \(2xy = k\) cut at right angles if \(k^2 = 8\). | Prove \(xy = 4\) and \(x^2 + y^2 = 8\) touch each other.📎 Assignments — Day 2 (Optional · Viewable by students)
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Exit Ticket
1For parametric curve x=f(t), y=g(t), what is dy/dx?
2Two curves are orthogonal when?
Key Formulas — Parametric
•dy/dx = (dy/dt)÷(dx/dt)
•Orthogonal: m₁·m₂=−1
•Touch: m₁=m₂ at common point
DAY 3 · MON 28 APR 2026
Day 3 — Rate of Change I: Direct & Geometric
✎Concept Board — Day 3Click to open · Draw diagrams & explain concepts▸ Open
2.1 Key Concepts — Rate of Change
CHAIN RULE\(\dfrac{{dy}}{{dt}} = \dfrac{{dy}}{{dx}} \cdot \dfrac{{dx}}{{dt}}\) — the fundamental tool for all rate problems
+/−Positive rate → increasing; Negative rate → decreasing
SPHERE\(V = \dfrac{{4}}{{3}}\pi r^3\) → \(\dfrac{{dV}}{{dt}} = 4\pi r^2 \cdot \dfrac{{dr}}{{dt}}\) | \(S = 4\pi r^2\) → \(\dfrac{{dS}}{{dt}} = 8\pi r \cdot \dfrac{{dr}}{{dt}}\)
CIRCLE\(A = \pi r^2\) → \(\dfrac{{dA}}{{dt}} = 2\pi r \cdot \dfrac{{dr}}{{dt}}\)
CUBE\(V = a^3\) → \(\dfrac{{dV}}{{dt}} = 3a^2 \cdot \dfrac{{da}}{{dt}}\)
Type 1 — Direct Rate
Ex 27
\(V = 500 - 3t - t^2\). Find the rate at which water is running out at \(t = 10\).
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Ex 34
Cube edge increasing at 10 cm/s. Find the rate of increase of volume when edge = 5 cm.
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Ex 36
Surface area \(A = 4\pi r^2\) with \(r = 3\) cm increasing at 2 cm/s. Find rate of increase of \(A\).
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Type 2 — Geometric Rates
Ex 30
Circular blot expanding at 5 cm/min. Find rate of increase of area when \(r = 5\) cm.
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Ex 37
Rectangle: length decreasing at 5 cm/min, width increasing at 4 cm/min; \(x = 8\), \(y = 6\). Find rate of (i) perimeter (ii) area.
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Ex 46
13 m ladder; bottom pulled at 2 m/s. Find rate of decrease of height when foot is 5 m from wall.
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Practice Questions
Q 1
\(y^2 = 4x\). Find rate of change of \(y\) w.r.t. \(x\) when \(x = 4\).
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Q 4
\(pV = 1000\); \(V\) increasing at 50 cm³/min. Find rate of change of \(p\) when \(V = 40\).
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Q 11
Circle area increases at 5 cm²/min. Find radius rate when circumference = 40 cm.
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Q 12
Kite 112 m above ground, string 130 m, moving at 8 m/s. Rate of string let out.
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📚 Homework
Q 7: \(y = x^3 + 10\); find \(x\) when \(y\) increases 27 times as fast as \(x\). | Q 15(ii): Equilateral triangle sides increasing at 2 cm/sec; area rate when side = 10 cm.📎 Assignments — Day 3 (Optional · Viewable by students)
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Day 3 — Video Resources
Exit Ticket
1State the chain rule for dy/dt in terms of dy/dx and dx/dt.
2Volume of sphere = ? Surface area = ?
Key Formulas — Rates
•dV/dt = 4πr²·(dr/dt) [sphere]
•dA/dt = 2πr·(dr/dt) [circle]
•dV/dt = 3a²·(da/dt) [cube]
DAY 4 · TUE 29 APR 2026
Day 4 — Rate of Change II: Conical & Connected Variables
✎Concept Board — Day 4Click to open · Draw diagrams & explain concepts▸ Open
2.2 Conical & Spherical Containers
CONE\(V = \dfrac{{1}}{{3}}\pi r^2 h\). Use similar triangles to relate \(r\) and \(h\), then differentiate w.r.t. \(t\).
SIMILAR △If semi-vertical angle \(\alpha\): \(\dfrac{{r}}{{h}} = \tan\alpha\). Substitute to get \(V\) in terms of \(h\) only.
SPHERE\(V = \dfrac{{4}}{{3}}\pi r^3\), \(S = 4\pi r^2\). \(\dfrac{{dV}}{{dt}} = 4\pi r^2 \cdot \dfrac{{dr}}{{dt}}\)
SETUPIdentify given rate, find formula, substitute similar triangles, differentiate w.r.t. \(t\).
Type 3 — Conical Containers
Ex 39
Cone (semi-vertical 45°, vertex down); water enters at 2 cm³/min. Find depth rate at \(h = 4\) cm.
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Ex 42
Conical vessel \(h = 10\) m, \(r = 5\) m; water enters at \(\dfrac{{3}}{{2}}\) m³/min. Rate of water level when level = 4 m.
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Ex 43
Conical funnel \(r = 10\) cm, \(h = 20\) cm; water out at 5 cm³/s. Rate of water level drop when 5 cm from top.
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Type 4 — Connected Variables
Ex 32
Gas escapes from sphere at 900 cm³/s. Rate of surface area shrinking when \(r = 360\) cm.
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Ex 44
Conical funnel semi-vertical angle \(\dfrac{{\pi}}{{4}}\); water drips at 2 cm³/s. Rate of decrease of slant height when slant \(h = 4\) cm.
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Practice Questions
Q 9
Inverted cone (semi-vertical 30°); depth increases at 1 cm/s. Volume rate at depth 24 cm.
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Q 14
Conical funnel drips at 2 cm³/s; slant \(h = 4\) cm, semi-vertical angle 60°. Rate of decrease of slant height.
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Q 15(ii)
Equilateral triangle sides increasing at 2 cm/sec. Area rate when side = 10 cm.
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📚 Homework
Man 180 cm moves from lamp post (4.5 m) at 1.2 m/s. Find rate of (i) shadow lengthening, (ii) tip of shadow moving.📎 Assignments — Day 4 (Optional · Viewable by students)
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Exit Ticket
1For an inverted cone with semi-vertical angle α, r/h = ?
2How do you eliminate r from V=(1/3)πr²h for a cone?
Cone Strategy
•Similar △: r/h = tan(α)
•Sub r=h·tan(α) into V formula
•Differentiate V w.r.t. t using chain rule
DAY 5 · WED 30 APR 2026
Day 5 — Monotonicity I: Definitions & Basic Proofs
✎Concept Board — Day 5Click to open · Draw diagrams & explain concepts▸ Open
3.1 Definitions & First Derivative Test
INCREASING\(f\) strictly increasing on \(I\): \(x_1 < x_2 \Rightarrow f(x_1) < f(x_2)\) for all \(x_1, x_2 \in I\)
DECREASING\(f\) strictly decreasing on \(I\): \(x_1 < x_2 \Rightarrow f(x_1) > f(x_2)\) for all \(x_1, x_2 \in I\)
FDT +\(f'(x) > 0\) for all \(x \in (a,b) \Rightarrow f\) is strictly increasing on \([a, b]\)
FDT −\(f'(x) < 0\) for all \(x \in (a,b) \Rightarrow f\) is strictly decreasing on \([a, b]\)
METHODFind \(f'(x)\) → set \(f'(x)=0\) → find critical points → test sign of \(f'\) in each interval
Type 1 — Without Derivative
Ex 47
Show \(f(x) = 3x + 1\) is increasing for all \(x \in \mathbb{{R}}\) (without using derivative).
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Type 2 — Proving Monotonicity Using Derivative
Ex 50(i)
Show \(f(x) = x^3 - 3x^2 + 3x - 100\) is increasing on \(\mathbb{{R}}\).
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Ex 51
Show \(y = \log(1+x) - \dfrac{{2x}}{{2+x}},\ x > -1\) is increasing throughout its domain.
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Type 3 — Determining Intervals
Ex 52
Determine intervals for \(f(x) = -2x^2 - 8x\).
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Ex 56
Determine intervals for \(f(x) = \dfrac{{x}}{{x^2+1}}\). Also find where tangent \(\parallel\) x-axis.
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Practice Questions
Q 2
Show \(x + \dfrac{{1}}{{x}}\) is increasing for \(x \geq 1\).
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Q 4
Prove \(f(x) = x^3 - 6x^2 + 12x - 18\) is increasing on \(\mathbb{{R}}\).
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Q 5
Find intervals for \(f(x) = \dfrac{{x-2}}{{x+1}},\ x \neq -1\).
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Q 10
Find intervals for \(f(x) = 2x^3 - 9x^2 + 12x + 15\).
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📚 Homework
Q 11: \(f(x) = 2x^3 - 15x^2 + 36x + 1\). Find intervals. | Q 17: Intervals for \(f(x) = x^2 - 6x + 9\); also find point where normal \(\parallel\ y = x + 5\).📎 Assignments — Day 5 (Optional · Viewable by students)
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Exit Ticket
1State FDT: if f'(x)>0 on (a,b), then f is ___?
2Find critical points of f(x)=x³−3x.
FDT Summary
•f'(x)>0 → strictly increasing
•f'(x)<0 → strictly decreasing
•f'(x)=0 → constant (only at points, not intervals)
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DAY 6 · FRI 2 MAY 2026
Day 6 — Monotonicity II: Trig & Advanced Proofs
✎Concept Board — Day 6Click to open · Draw diagrams & explain concepts▸ Open
Type 4 — Trigonometric Functions
Ex 57(i)
Show \(f(x) = \sin x\) is increasing in \(\left(0, \dfrac{{\pi}}{{2}}\right)\) and decreasing in \(\left(\dfrac{{\pi}}{{2}}, \pi\right)\).
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Ex 59
Prove \(y = \dfrac{{4\sin\theta}}{{2+\cos\theta}} - \theta\) is increasing in \(\left[0, \dfrac{{\pi}}{{2}}\right]\).
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Ex 60
Find intervals for \(f(x) = \sin x - \cos x\), \(0 \leq x \leq 2\pi\).
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Type 5 — Advanced / Proof-Based
Ex 63
Find values of \(x\) for which \(f(x) = [x(x-2)]^2\) is increasing.
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Ex 62(ii)
Find intervals where \(f(x) = x^4 - 2x^2\) is increasing/decreasing.
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Practice Questions
Q 3
State definition of increasing on \([a,b]\). Test \(f(x) = x^3 - 8\) on \([1, 2]\).
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Q 20
Find intervals for \(f(x) = \log(1+x) - \dfrac{{x}}{{1+x}}\).
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Q 11
Find intervals for \(f(x) = 2x^3 - 15x^2 + 36x + 1\).
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Q 17
Find intervals for \(f(x) = x^2 - 6x + 9\). Also find point where normal \(\parallel\ y = x + 5\).
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📚 Homework
Show \((x - \sin x)\) is increasing for all \(x\). | Find intervals for \(f(x) = e^x(x-2)^2\).📎 Assignments — Day 6 (Optional · Viewable by students)
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Exit Ticket
1For f(x)=sin x, f'(x) in (π/2,π) is negative/positive?
2Find intervals for f(x)=x⁴−2x².
Trig Monotonicity
•sin x: ↑ on (0,π/2), ↓ on (π/2,π)
•cos x: ↓ on (0,π)
•For trig: find f'(x)=0 in [0,2π]
📋 Multiple Choice Questions
Part A — Tangents & Normals
MCQ 1
The slope of the tangent to \(y = x^3 - x + 1\) at \(x = 2\) is:
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(a) 10
(b) 11
(c) 12
(d) 9
MCQ 2
The equation of the tangent to \(x^2 = 4y\) at \((2, 1)\) is:
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(a) \(x - y - 1 = 0\)
(b) \(x - y + 1 = 0\)
(c) \(x + y - 1 = 0\)
(d) \(x + y + 1 = 0\)
MCQ 3
The slope of the normal to \(y = 2x^2 + 3\) at \(x = 1\) is:
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(a) \(-\dfrac{{1}}{{4}}\)
(b) 4
(c) \(\dfrac{{1}}{{4}}\)
(d) \(-4\)
MCQ 4
The point on \(y = x^2\) at which tangent is parallel to \(y = 8x + 5\) is:
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(a) \((4,16)\)
(b) \((3,9)\)
(c) \((2,4)\)
(d) \((1,1)\)
MCQ 5
Tangent to \(x^2 + y^2 = 2\) at \((1,1)\) meets x-axis at:
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(a) \((2, 0)\)
(b) \((-2, 0)\)
(c) \((1, 0)\)
(d) \((0, 2)\)
Part B — Rate of Change
MCQ 6
Sphere radius increases at 0.2 cm/s. Rate of volume increase at \(r = 5\) cm:
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(a) \(20\pi\) cm³/s
(b) \(2\pi\) cm³/s
(c) \(200\pi\) cm³/s
(d) \(10\pi\) cm³/s
MCQ 7
For \(y = x^2\), rate of change of \(y\) when \(x\) changes at 2 units/s at \(x = 3\):
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(a) 6 units/s
(b) 12 units/s
(c) 9 units/s
(d) 4 units/s
MCQ 8
Radius of circle increases at 5 cm/min. Rate of increase of circumference:
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(a) \(10\pi\) cm/min
(b) \(5\pi\) cm/min
(c) \(25\pi\) cm/min
(d) \(2\pi\) cm/min
MCQ 9
Water fills conical vessel (semi-vertical 45°) at \(\pi\) cm³/min. Rate at \(h = 2\) cm:
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(a) \(\dfrac{{1}}{{4}}\) cm/min
(b) \(\dfrac{{1}}{{2}}\) cm/min
(c) \(1\) cm/min
(d) \(\dfrac{{1}}{{8}}\) cm/min
Part C — Monotonicity
MCQ 10
\(f(x) = x^3 - 3x\) is increasing in:
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(a) \((-\infty,-1)\cup(1,\infty)\)
(b) \((-1,1)\)
(c) \((0,\infty)\)
(d) \((-\infty,0)\)
MCQ 11
\(f(x) = 2x^3 - 9x^2 + 12x + 15\) is decreasing in:
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(a) \((1, 2)\)
(b) \((2, 3)\)
(c) \((-\infty, 1)\)
(d) \((3, \infty)\)
MCQ 12
\(\sin x - \cos x\) is increasing on:
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(a) \(\left(\dfrac{{3\pi}}{{4}}, \dfrac{{7\pi}}{{4}}\right)\)
(b) \((0, \pi)\)
(c) \(\left(0, \dfrac{{\pi}}{{2}}\right)\)
(d) \(\left(\dfrac{{\pi}}{{2}}, \pi\right)\)
MCQ 13
\(kx - \sin x\) strictly increasing for all \(x\) when:
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(a) \(k > 1\)
(b) \(k < 1\)
(c) \(k > 0\)
(d) \(k \geq 1\)
MCQ 14
\(f(x) = [x(x-2)]^2\) is increasing for:
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(a) \((0,1)\cup(2,\infty)\)
(b) \((-\infty,0)\cup(1,2)\)
(c) \((1,2)\)
(d) \((0,2)\)
📖 Case Studies
CS 1Spherical Balloon — Against Child Labour▼ Expand
A brave child inflates a spherical balloon bearing the slogan 'Against Child Labour' at 900 cm³/s.
\(V = \dfrac{4}{3}\pi r^3\), \(S = 4\pi r^2\), \(\dfrac{dV}{dt} = 900\) cm³/s
\(V = \dfrac{4}{3}\pi r^3\), \(S = 4\pi r^2\), \(\dfrac{dV}{dt} = 900\) cm³/s
CS1 (i)
Find the rate at which the radius is increasing when \(r = 15\) cm.
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CS1 (ii)
Find the rate at which the surface area is increasing when \(r = 15\) cm.
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CS1 (iii)
At what radius is the rate of increase of surface area exactly twice the rate of increase of radius?
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CS 2Plant Growth Model — Sunlight Exposure▼ Expand
Height of a plant (\(y\) cm) vs sunlight days (\(x\)): \(y = 4x - \dfrac{1}{2}x^2\), \(0 \leq x \leq 8\)
CS2 (i)
\(\dfrac{{dy}}{{dx}}\) (rate of growth) equals:
(a) \(4x - \frac{{1}}{{2}}x^2\)
(b) \(4 - x\)
(c) \(x - 4\)
(d) \(4 + x\)
CS2 (ii)
Days for maximum height:
(a) 4
(b) 6
(c) 7
(d) 8
CS2 (iii)
Maximum height of the plant:
(a) 8 cm
(b) 16 cm
(c) 10 cm
(d) 6 cm
CS2 (iv)
Height after 2 days. Verify it is still increasing at that point.
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CS2 (v)
Days of sunlight if height = \(\dfrac{{7}}{{2}}\) cm?
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CS 3Water Tank — Inverted Cone▼ Expand
Inverted conical tank, semi-vertical angle 45°. Water flows in at \(2\pi\) cm³/min. Since \(\tan 45° = 1\), we have \(r = h\).
CS3 (i)
Rate at which depth increases when depth = 4 cm.
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CS3 (ii)
Rate of increase of the water surface (circular top) area when depth = 4 cm.
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CS3 (iii)
At what depth is the rate of volume increase numerically equal to \(16\pi\) cm³/min?
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