Application of Derivatives
1. Rate of Change - How Fast?
Derivative dy/dx is rate of change of y with respect to x. If y is distance, dy/dt is speed. If volume V of sphere V=4/3 π r³ then dV/dt = 4π r² dr/dt rate of volume change with radius. Example: radius increasing at 2 cm/s, when r=10 cm dV/dt=4π×100×2=800π cm³/s. If two quantities related, differentiate with respect to time t. Related rates: find dr/dt from dV/dt. Used in balloon inflating, water tank filling.
Rate of change dy/dx instantaneous rate. If y distance dy/dt speed. V=4/3 π r³ dV/dt=4π r² dr/dt. If radius ↑ at 2 cm/s r=10 dV/dt=800π. Related rates differentiate w.r.t t. dr/dt from dV/dt.
• dy/dx = rate of change of y w.r.t x
• Distance→ speed dy/dt, Volume→ dV/dt
• V=4/3 π r³ → dV/dt=4π r² dr/dt
• Example r=10 dr/dt=2 → dV/dt=800π cm³/s
• Related rates: differentiate w.r.t time t
• Use: Balloon, tank filling, growing circle
2. Increasing and Decreasing Functions - Going Up or Down
Function increasing if as x increases y increases, graph goes up. Condition f'(x)≥0 for increasing, f'(x)≤0 for decreasing, strictly increasing if f'(x)>0, strictly decreasing if f'(x)<0. Example f(x)=x² f'(x)=2x >0 for x>0 so increasing on (0,∞), <0 for x<0 decreasing on (-∞,0). At x=0 f'=0 stationary. Find intervals: find f', solve f'=0 critical points, test sign in intervals. Increasing slope positive, decreasing slope negative.
Increasing if x1
• Increasing: As x↑ y↑ graph up, f'(x)≥0
• Decreasing: As x↑ y↓ graph down, f'(x)≤0
• Strictly: f'>0 inc, f'<0 dec
• Example x² f'=2x inc on (0,∞) dec on (-∞,0)
• Method: f'=0 critical points, test sign in intervals
• Stationary point f'=0 flat
3. Tangents and Normals - Touching the Curve
Tangent touches curve at one point, slope = derivative at that point m=f'(x0). Equation: y-y0=m(x-x0). Normal is perpendicular to tangent, slope m_n=-1/m = -1/f'(x0). Equation y-y0=(-1/m)(x-x0). If tangent parallel to x-axis f'=0 horizontal, parallel to y-axis f' undefined vertical. Example y=x² at (1,1) f'=2x=2 tangent y-1=2(x-1) → y=2x-1 normal y-1=(-1/2)(x-1).
Slope of tangent m=f'(x0). Tangent equation y-y0=m(x-x0). Normal slope -1/m perpendicular. Normal equation y-y0=(-1/m)(x-x0). If f'=0 tangent ∥ x-axis, if f'→∞ normal ∥ x-axis. Example x² at (1,1) m=2 tangent y=2x-1 normal 2y+x=3.
• Tangent slope m = f'(x0) at point (x0,y0)
• Tangent: y-y0=m(x-x0)
• Normal: ⊥ to tangent slope -1/m, y-y0=(-1/m)(x-x0)
• If f'=0 tangent ∥ x-axis horizontal
• If f' undefined vertical tangent
• Example x² at (1,1) m=2 tangent y=2x-1 normal 2y+x=3
4. Approximations - Using Differentials Δy≈dy
Small change in x dx causes small change in y dy = f'(x) dx approximates Δy = f(x+dx)-f(x). So f(x+dx)≈f(x)+f'(x)dx. Example approximate √25.2 using f(x)=√x at x=25 dx=0.2 f'=1/(2√x)=1/10=0.1 dy=0.1×0.2=0.02 so √25.2≈5+0.02=5.02 actual 5.0199 close. Used to estimate cube root, small errors. Smaller dx better approximation.
Δy=f(x+dx)-f(x) ≈ dy=f'(x)dx. So f(x+dx)≈f(x)+f'(x)dx. Example √25.2: f=√x x=25 dx=0.2 f'=1/(2√25)=0.1 dy=0.02 √25.2≈5.02. Error small if dx small.
• Δy = f(x+dx)-f(x) exact change
• dy = f'(x)dx approx change tangent line
• Approximation: f(x+dx)≈f(x)+f'(x)dx
• Example √25.2: 5+0.1×0.2=5.02 actual 5.0199
• Use: Cube roots, small errors, quick estimate
• Smaller dx better accuracy
5. Maxima and Minima - Peaks and Valleys
Maxima is peak highest in neighbourhood, minima valley lowest. Local maxima f(c)≥f(x) near c, local minima f(c)≤f(x). Absolute maxima highest in whole domain. Critical point where f'(c)=0 or f' not defined. Example f(x)=x³-3x f'=3x²-3=0 at x=±1 critical. To find max/min, find critical points, test. Global max on [a,b] check critical points and endpoints a,b choose largest value.
Local max peak f(c)≥f(x) nearby, Local min valley f(c)≤f(x). Critical where f'=0 or undefined. Absolute max/min highest/lowest in [a,b]. To find global on [a,b]: find critical in (a,b), evaluate f at critical and endpoints, largest=max smallest=min.
• Local max peak highest nearby, local min valley lowest nearby
• Critical point: f'(c)=0 or f' undefined
• Absolute max highest in full domain
• Example x³-3x f'=3x²-3=0 at x=±1 critical
• Global on [a,b]: evaluate f at critical + endpoints a,b
• Largest value = absolute max, smallest = absolute min
6. First Derivative Test - Sign Change
First derivative test: At critical point c, check sign of f' left and right. If f' changes + to - then c is local maxima (going up then down peak). If changes - to + then local minima (down then up valley). If same sign both sides (+ to + or - to -) then neither max nor min inflection. Example f(x)=x³-3x critical -1,1: left of -1 f'>0, between -1 and1 f'<0 so + to - at -1 ⇒ maxima at -1, at 1 - to + ⇒ minima at 1. Simple sign chart.
First derivative test at critical c: f' left + right - ⇒ local max, left - right + ⇒ local min, same sign ⇒ neither. Example x³-3x critical -1 maxima +→-, 1 minima -→+. Make sign chart.
• At critical c check f' left and right sign
• + → - : Local maxima peak
• - → + : Local minima valley
• + → + or - → - : Neither inflection
• Example x³-3x: -1 +→- maxima, 1 -→+ minima
• Method: Number line sign chart of f'
7. Second Derivative Test - Concavity
Second derivative test: Find critical where f'=0, then find f'' at that point. If f''(c)<0 concave down cap shape ⇒ local maxima. If f''(c)>0 concave up cup shape ⇒ local minima. If f''=0 test fails use first derivative test. Example f(x)=x³-3x f''=6x at x=-1 f''=-6<0 maxima, at x=1 f''=6>0 minima. Faster than first test when second derivative easy. f'' tells concavity.
Second derivative test at critical f'=0: f''(c)<0 ⇒ local max concave down, f''(c)>0 ⇒ local min concave up, f''(c)=0 test fails use first test. Example x³-3x f''=6x -1→-6<0 max, 1→6>0 min.
• Find critical f'=0, then find f'' at c
• f''(c)<0 concave down cap → local maxima
• f''(c)>0 concave up cup → local minima
• f''(c)=0 test fails → use first derivative test
• Example x³-3x f''=6x -1 maxima, 1 minima
• Faster when f'' easy
8. Real Life Applications - Maximum Profit Minimum Cost
Real life: Want maximum area with fixed perimeter fencing, maximum volume box from sheet, minimum cost material, maximum profit, minimum distance, maximum height projectile. Steps: Make function of one variable using constraint, find critical f'=0, test max/min. Example fence 100m to make rectangular field with river one side no fence, area A=x(100-2x) maximize A' =100-4x=0 x=25 max area 1250. Engineering: beam strength maximize, medicine dosage effective.
Real: Max area fixed perimeter, max volume box, min cost material, max profit, min distance, max height. Steps: Constraint→ function one variable→ f'=0 critical→ test. Example 100m fence river one side A=x(100-2x) max at x=25 A=1250. Beam strength, dosage.
• Max area with fixed perimeter fencing
• Max volume box from sheet cutting squares
• Min cost material, max profit revenue
• Min distance point to curve
• Steps: Constraint → single variable function → f'=0 → test
• Example 100m fence river side: A=x(100-2x) max at 25
• Engineering beam strength, medicine dosage, projectile max height
• Rate: dy/dx rate, V=4/3 π r³ dV/dt=4π r² dr/dt, related rates w.r.t t
• Inc/Dec: f'≥0 inc, f'≤0 dec, f'>0 strictly inc, f'<0 strictly dec, f'=0 stationary, sign chart
• Tangent: m=f'(x0), equation y-y0=m(x-x0), Normal slope -1/m y-y0=(-1/m)(x-x0), f'=0 ∥ x-axis
• Approximation: Δy≈dy=f'(x)dx, f(x+dx)≈f(x)+f'(x)dx, small dx better
• Maxima Minima: Local max peak f(c)≥nearby, min valley, critical f'=0 or undefined, global on [a,b] check endpoints
• First test: +→- max, -→+ min, same sign neither
• Second test: f'=0 then f''<0 max concave down, f''>0 min concave up, f''=0 fails
• Real: Max area perimeter, max volume box, min cost, max profit, min distance, beam strength