Revision sheet

Velocity

• Average: $\dfrac{s(a + h) - s(a)}{h}$, change in position ÷ change in time.
• Instantaneous: the limit of that as $h \to 0$.

Watch out: divide by the time; a change in position alone is not a velocity.

The derivative at a point

1. Write $f(a + h) - f(a)$ and simplify.
2. Divide by $h$.
3. Let $h \to 0$: the result is $f'(a)$.

Watch out: $f'(a)$ is a slope, not the value $f(a)$.

The tangent line

• Point: $(a, f(a))$. Slope: $f'(a)$.
• $y = f(a) + f'(a)(x - a)$.

Watch out: $y = f'(a)\,x$ has the right slope but usually misses the point.

The derivative function

• $f'(x)$ is the slope of $f$ at each $x$.
• $f$ rises: $f' > 0$. $f$ falls: $f' < 0$. Horizontal tangent: $f' = 0$.

Watch out: $f' = 0$ (flat) is not the same as $f = 0$ (on the axis).

Reading a derivative

• Units: output units per input unit.
• $f'(a) = r$: at $x = a$, $f$ changes by about $r$ per extra unit of $x$.

Watch out: $f'(a)$ is a rate at one moment, not a value and not a total change.

The second derivative

• $f'' > 0$: slopes grow, concave up. $f'' < 0$: slopes shrink, concave down.
• Inflection point: the concavity changes; there $f'$ has a maximum or minimum.
• Acceleration is the derivative of velocity.

Watch out: $f'$ says rising or falling; $f''$ says how it bends.

Differentiability

• Differentiable ⇒ continuous; not the other way round.
• No derivative at a corner, a cusp, a vertical tangent, or a break.
• To join two pieces smoothly, match the heights and the slopes.

Watch out: an unbroken graph can still have no tangent line.

Local linearization

• $L(x) = f(a) + f'(a)(x - a)$, the tangent line used as an estimate.
• Good near $a$; concave up underestimates, concave down overestimates.

Watch out: multiply the step $x - a$ by the slope $f'(a)$.