Lesson 12 of 12 · 11 min
Chapter review
Watch a class
The whole chapter on YouTube
Loading the full lesson
Must-know facts
10 facts
- 01dy/dx is the rate of change of y with x; with time, dy/dt = (dy/dx)(dx/dt).
- 02Related rates: write the relation, differentiate in t, then substitute.
- 03Marginal cost = dC/dx, marginal revenue = dR/dx.
- 04f′ > 0 on (a, b) ⇒ increasing on [a, b]; f′ < 0 ⇒ decreasing; f′ = 0 ⇒ constant.
- 05Intervals of monotonicity come from the sign of f′ between its zeros.
- 06Critical point: f′(c) = 0 or f′(c) undefined.
- 07First test: + to − is a local max, − to + a local min, no change means inflection.
- 08Second test: f′(c) = 0 with f″(c) < 0 is a max, f″(c) > 0 a min, f″(c) = 0 no verdict.
- 09Closed interval: compare f at the critical points and at both ends.
- 10Tangent slope f′(x₀); normal slope −1/f′(x₀) (JEE).
Common traps
Where marks are lost
Formulas
7 to know
Rate chain
dy/dt = (dy/dx)·(dx/dt)
Also dy/dx = (dy/dt)/(dx/dt) where dx/dt ≠ 0.
Circle and sphere
dA/dt = 2πr dr/dt, dV/dt = 4πr² dr/dt
From A = πr² and V = 4πr³/3.
Marginal cost, revenue
MC = dC/dx, MR = dR/dx
Profit peaks where MR = MC and P″ < 0.
Monotonicity
f′ > 0 ⇒ increasing, f′ < 0 ⇒ decreasing
f continuous on [a, b], differentiable on (a, b).
First derivative test
f′: + → − max, − → + min
No sign change: point of inflection.
Second derivative test
f′(c) = 0: f″(c) < 0 max, f″(c) > 0 min
f″(c) = 0 gives no verdict.
Tangent and normal (JEE)
y − y₀ = m(x − x₀), m = f′(x₀); normal slope −1/m
Needs m ≠ 0 for the normal slope.
Key terms
6 terms
- Marginal cost
- The rate of change of total cost with output, dC/dx.
- Strictly increasing
- x₁ < x₂ always gives f(x₁) < f(x₂) on the interval.
- Critical point
- A point of the domain where f′ is zero or does not exist.
- Local maximum
- A value not exceeded by f anywhere in some open interval around the point.
- Point of inflection
- A point where f′ = 0 but f′ keeps its sign, so f has no extremum.
- Absolute maximum
- The largest value of f on the whole interval.