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Calculus I (Differential & intro Integral) Unit 4: Derivatives in context

Derivatives describe real rates: how fast position, volume, cost or temperature is changing. This unit is about reading and using them in context.

Covers: Motion (velocity/acceleration), related rates, linearization, L'Hospital's rule

Study guide

Derivatives describe change in real situations. For motion along a line, position s(t) has velocity v(t) = s′(t) and acceleration a(t) = v′(t) = s″(t). Speed is |v(t)|. The object moves right (or up) when v > 0, left when v < 0, and changes direction where v changes sign. It speeds up when v and a have the same sign and slows down when they differ. Units matter: the derivative's units are output units per input unit.

Related rates problems connect quantities that change over time. Draw a picture, write an equation relating the quantities (often geometry such as the Pythagorean theorem or a volume formula), differentiate both sides with respect to t, and only then substitute the known values to solve for the unknown rate.

Linearization approximates a function near a point by its tangent line: L(x) = f(a) + f′(a)(x − a). It gives quick estimates, like √4.1 ≈ 2.025. If the graph is concave up near a, the tangent line lies below it, so the estimate is too small; concave down gives an overestimate.

L'Hospital's rule evaluates limits of indeterminate form 0/0 or ∞/∞: lim f(x)/g(x) = lim f′(x)/g′(x), differentiating the top and bottom separately (not the quotient rule). Check the form first, and repeat if needed.

To interpret f′(a) = k in context, say that at the input value a, the quantity is increasing (or decreasing, if k < 0) at a rate of |k| output units per input unit. For a second derivative, describe how the rate itself is changing. A quantity that is shrinking has a negative rate, so enter its rate with a minus sign. A negative answer means the quantity you solved for is decreasing.

When an angle changes, relate it to lengths with a trig ratio such as tan θ = opposite/adjacent, then differentiate with the chain rule (d/dt tan θ = sec²θ·dθ/dt). Keep angles in radians. L'Hospital's rule also works when functions are given only by tables or graphs. Confirm that the numerator and denominator both approach 0 (or both approach ±∞), then evaluate f′(a)/g′(a) using the given derivative values.

Key ideas

• Position s(t) → velocity v(t) = s′(t) → acceleration a(t) = v′(t).
• An object speeds up when velocity and acceleration have the same sign, and slows down when they differ.
• Related rates: draw it, write an equation linking the quantities, differentiate with respect to t, then plug in the numbers.
• Linearization uses the tangent line to estimate values near a known point.
• L'Hospital's rule handles 0/0 and ∞/∞ forms by taking the derivative of top and bottom separately.
• Units of f′ are units of f per unit of x.

Formulas and rules

Common mistakes

• Plugging in numbers before differentiating in a related-rates problem.
• Using the quotient rule with L'Hospital. Differentiate top and bottom separately.

Words to know

Velocity in terms of position

v(t) = s′(t)

Speeding up (motion)

Velocity and acceleration have the same sign.

Linearization of f at x = a

L(x) = f(a) + f′(a)(x − a)

L'Hospital's rule

For 0/0 or ∞/∞: lim f/g = lim f′/g′.

Related rates

Relate the changing quantities in an equation, differentiate in t, then plug in values.

Units of a derivative

Output units per input unit (e.g. liters per minute).

Acceleration

a(t) = v′(t) = s″(t)

Speed

|v(t)|, the magnitude of velocity.

Change of direction

Where velocity changes sign.

Linearization

L(x) = f(a) + f′(a)(x − a), the tangent-line approximation.

Tangent-line over/underestimate

Concave up → underestimate; concave down → overestimate.

Indeterminate forms

0/0 and ∞/∞ (and others rewritten into them).

Sample practice questions

37 more questions like these, from level 1 to 5, are in the practice set.