Applications of Derivatives Free Online FlashCards

What is a related-rates problem?

A related-rates problem connects quantities changing with respect to a common variable, usually time, through an equation.

When should values be substituted in related rates?

Differentiate the relationship before substituting numerical values. Substituting first can incorrectly treat a changing quantity as constant.

How fast does a circle’s area grow when r = 5 and dr/dt = 2?

For a circle, dA/dt = 2πr(dr/dt). If r = 5 cm and dr/dt = 2 cm/s, then dA/dt = 20π cm²/s.

What is the formula for linearization at x = a?

The linearization of f at x = a is L(x) = f(a) + f′(a)(x − a). It approximates f(x) near a.

Use linearization to estimate √4.1.

Using f(x) = √x at a = 4, L(x) = 2 + ¼(x − 4), so √4.1 ≈ L(4.1) = 2.025.

How is the differential dy defined?

If y = f(x), then dy = f′(x)dx. For a small change, the actual change Δy is approximately dy.

How does radius error propagate to a sphere’s volume?

For V = (4/3)πr³, dV/V ≈ 3dr/r. Thus a 0.2% radius error produces about a 0.6% volume error.

How are position, velocity, acceleration, and speed related?

For position s(t), velocity is v(t) = s′(t), acceleration is a(t) = s″(t), and speed is |v(t)|.

When does an object change direction?

An object changes direction when velocity is zero or undefined and its sign changes across that time.

What does the Extreme Value Theorem guarantee?

If f is continuous on a closed interval [a,b], it attains both an absolute maximum and an absolute minimum there.

What is the closed-interval method?

For a continuous function on [a,b], evaluate the function at every interior critical number and at both endpoints, then compare the values.

State the Mean Value Theorem.

If f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) satisfies f′(c) = [f(b) − f(a)]/(b − a).