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Chapter 01 · Part I · Mechanics
Revised on ____________________
Chapter 01 · The Calculus Toolkit

The mathematics, before the physics needs it

2 sheets
2 diagrams
33 results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

I

Differentiation

A DERIVATIVE IS A SLOPE
I.Definition  dfdx = limh→0 f(x+h) − f(x)h
II.Power  ddx xn = n xn−1
III.Product  (uv)′ = u′v + uv′
IV.Quotient  ( uv )′ = u′v − uv′v²
V.Chain  dydx = dydu · dudx
VI.Trigonometric  (sin x)′ = cos x , (cos x)′ = − sin x x in radians, always
VII.More trig  (tan x)′ = sec²x , (sin ax)′ = a cos ax
VIII.Exponential & log  (eax)′ = a eax , (ln x)′ = 1x
IX.Parametric  dydx = dy/dtdx/dt
X.Second derivative  d²ydx² = ddx ( dydx ) curvature, not slope
The secant becomes the tangent in the limit. Everything else on this sheet is bookkeeping around that one idea.
xyPQtangentsecants
II

Reading a curve

WHERE THE MAXIMUM ACTUALLY IS
I.Stationary point  f′(x) = 0
II.Maximum  f′ = 0 and f″ < 0
III.Minimum  f′ = 0 and f″ > 0
IV.Small change  Δf ≈ f′(x) Δx
V.Binomial, |x| ≪ 1  (1 + x)n ≈ 1 + n x
VI.Small angle  sin θ ≈ tan θ ≈ θ , cos θ ≈ 1 − θ²2
III

Integration

AN INTEGRAL IS AN AREA
I.Power (n ≠ −1)  ∫ xn dx = xn+1n + 1 + C
II.The exception  ∫ dxx = ln|x| + C
III.Exponential  ∫ eax dx = eaxa + C
IV.Trigonometric  ∫ sin ax dx = − cos axa + C
V.Substitution  ∫ f(u) dudx dx = ∫ f(u) du change the limits with the variable
VI.Definite integral  ∫ab f dx = F(b) − F(a)
VII.Signed area  area below the axis counts negative
VIII.Average value  ⟨f⟩ = 1b − a ∫ab f dx
IX.Symmetry  odd f over (−a, a) integrates to 0 worth two minutes in an exam
A definite integral is the limit of Σ f·Δx as Δx → 0. If you can see the area, you can often skip the antiderivative entirely.
tvn → ∞ ⇒ exact area
IV

The bridge into physics

EVERY MECHANICS RELATION IS ON THIS LADDER
I.Velocity  v = dxdt
II.Acceleration  a = dvdt = d²xdt²
III.When a depends on x  a = v dvdx no time anywhere in the problem
IV.Displacement  Δx = ∫ v dt area under v–t
V.Change in velocity  Δv = ∫ a dt area under a–t
VI.Work  W = ∫ Fx dx
VII.Impulse  J = ∫ F dt
VIII.Charge from current  q = ∫ i dt the same ladder, other Parts
Where marks are lost in this chapter
1Differentiating sin x with x in degrees. The rule (sin x)′ = cos x holds only in radians.
2Forgetting + C on an indefinite integral, then losing the initial condition that fixes it.
3Using a = v dv/dx when acceleration is given as a function of time — that route needs a = dv/dt.
4Reading area under a v–t graph as distance. It is displacement; distance needs |v|.