Derivatives as Rates of Change

Rates in every discipline

Motion along a line

Linear approximation

Rates from data

In economics, the derivative C prime of q of a cost function is called marginal cost. What is its practical interpretation?

A particle moving along a line has velocity v(t) = 3 - 2t. At what time does the particle change direction?

To approximate sqrt(4.1) by the tangent line at a = 4, which quantities enter the formula f(a) + f prime of a times (x - a)?

The volume V of a sphere is in cubic centimeters and the radius r is in centimeters. What are the units of the derivative dV/dr?

Linear approximation at a has an error bound involving |x - a| squared. How does the error behave as x approaches a?

A trip covers 120 miles in 2 hours. Which statement correctly distinguishes the two rates involved?

Near the surface of the earth, a falling object's position is s(t) = -4.9 t^2 + v0 t + s0 in meters. What is its acceleration in meters per second squared? Answer with a number.

The tangent line to f(x) = x^2 at a = 3 has equation f(3) + f prime of 3 (x - 3). What number does it give at x = 3.1? Answer with a number.

A cost function is C(q) = 100 + 4q + 0.1 q^2. What is the marginal cost C prime of 10, in dollars per unit? Answer with a number.

At an instant when a particle's velocity is zero, what is its speed, the magnitude of velocity? Answer with a number.