Calculus AB · BCUnit 4Summary sheet

Derivatives in context: theorems and formulas

Rates with units, motion, related rates, tangent line approximation and L'Hospital’s Rule. Print it, or keep it open beside a tool.

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The derivative in context Topics 4.1, 4.3

Meaning
f′(a) is the instantaneous rate of change of f with respect to x at x = a.
Units
The units of f′(x) are the units of f divided by the units of x. Example: V in liters and t in minutes: V′(t) is in liters per minute, and V″(t) in liters per minute per minute.
Sign
f′(a) > 0: f is increasing at a. f′(a) < 0: decreasing.
Estimate
From a table, the average rate of change over the closest interval: f′(a) ≈ f(b) − f(c)b − c
Net rate
rate in minus rate out: A′(t) = Rin(t) − Rout(t)

A complete sentence names the time, the quantity, increasing or decreasing, the rate and the units: “At t = 3 hours, the amount of water in the tank is decreasing at a rate of 12 liters per hour.”

Practice in Rates in Context

Straight-line motion Topic 4.2

Position
x(t)
Velocity
v(t) = x′(t)
Acceleration
a(t) = v′(t) = x″(t)
Speed
|v(t)|
Average velocity
x(b) − x(a)b − a on [a, b]
Average acceleration
v(b) − v(a)b − a on [a, b]

What the signs say

Moving right (or up)
v(t) > 0
Moving left (or down)
v(t) < 0
At rest
v(t) = 0
Changes direction
v(t) changes sign (being 0 is not enough)
Speeding up
v(t) and a(t) have the same sign
Slowing down
v(t) and a(t) have opposite signs
Toward the origin
x(t) and v(t) have opposite signs

Distance traveled: split the time interval where the particle turns around, and add the sizes of the moves: |x(t1) − x(a)| + |x(b) − x(t1)| for one turn at t1.

Practice in Particle Motion Lab

Tangent line approximation Topic 4.6

Tangent line
L(x) = f(a) + f′(a)(x − a)
Approximation
f(x) ≈ L(x) for x near a (local linearity: zoomed in, the curve looks like its tangent line)
From a rate equation
Given dydx in terms of x and y, find the slope at the known point, then use the tangent line.

Overestimate or underestimate

Concave up
f″ > 0 near a: the tangent line lies below the curve, so L(x) is an underestimate.
Concave down
f″ < 0 near a: the tangent line lies above the curve, so L(x) is an overestimate.

Example: √x at a = 4 gives L(x) = 2 + 14(x − 4), so √4.2 ≈ 2.05, an overestimate because √x is concave down.

Practice in Tangent Line Zoom

L'Hospital’s Rule Topic 4.7

L'Hospital’s Rule If lim f(x) = 0 and lim g(x) = 0, or lim f(x) = ±∞ and lim g(x) = ±∞, and lim f′(x)g′(x) exists or is ±∞, then lim f(x)g(x) = lim f′(x)g′(x)
  • It works as x → c, from one side, or as x → ±∞.
  • Differentiate the top and the bottom separately. It is not the quotient rule.
  • Show the form first: write that the top and the bottom both approach 0 (or both approach ±∞) before you use the rule.
  • If the new limit is still 00 or ∞∞, use the rule again.

Examples: limx → 0 ex − 1x = limx → 0 ex1 = 1, and limx → ∞ x2ex = limx → ∞ 2ex = 0 (twice).

Practice in L'Hospital Lens