Physical interpretation of the derivative
A car's speedometer does not measure the average speed over the whole journey: it shows the speed at that very instant. This is precisely the physical idea behind the derivative, and it is the historical reason why Newton invented it.
Instantaneous velocity: the derivative of position
The position of a free-falling object is given by the function $s(t) = 100 - 4,9t^2$, where $s$ is measured in metres and $t$ in seconds. Determine the function…
The position of a free-falling object is given by the function $s(t) = 100 - 4,9t^2$, where $s$ is measured in metres and $t$ in seconds. Determine the function describing its velocity $v(t)$ and its acceleration $a(t)$.
A train travels along a straight track with an equation of motion $e(t) = 2t^3 - 5t^2 + 8t$, with $e$ in metres and $t$ in seconds. Calculate the velocity of the train…
A train travels along a straight track with an equation of motion $e(t) = 2t^3 - 5t^2 + 8t$, with $e$ in metres and $t$ in seconds. Calculate the velocity of the train at time $t = 2$ s.
Acceleration: the derivative of velocity
The position of a particle as a function of time is $x(t) = t^4 - 3t^2 + 5$. Determine the acceleration of the particle at time $t = 1$.
The position of a particle as a function of time is $x(t) = t^4 - 3t^2 + 5$. Determine the acceleration of the particle at time $t = 1$.
A test car has a velocity given by the function $v(t) = 3t^2 - 12t + 4$ in m/s. Calculate its acceleration at time $t = 3$ s.
A test car has a velocity given by the function $v(t) = 3t^2 - 12t + 4$ in m/s. Calculate its acceleration at time $t = 3$ s.
The sign of $v(t)$ and $a(t)$: direction of motion
A hot-air balloon rises such that its height in metres is given by $h(t) = 15t - t^2$. At what time $t$ does its velocity become zero?
A hot-air balloon rises such that its height in metres is given by $h(t) = 15t - t^2$. At what time $t$ does its velocity become zero?
A toy rocket is launched vertically upwards. Its height in metres after $t$ seconds is $h(t) = 30t - 4,9t^2$. Determine the time it takes to reach its maximum height and…
A toy rocket is launched vertically upwards. Its height in metres after $t$ seconds is $h(t) = 30t - 4,9t^2$. Determine the time it takes to reach its maximum height and calculate the exact value of this height.
The position of a particle is given by $s(t) = \frac{t^3}{3} - 3t^2 + 8t$. Determine the time intervals in which the particle moves to the right ($v(t) > 0$) and in…
The position of a particle is given by $s(t) = \frac{t^3}{3} - 3t^2 + 8t$. Determine the time intervals in which the particle moves to the right ($v(t) > 0$) and in which it moves back to the left ($v(t) < 0$).
Other instantaneous rates of change
The derivative is not just velocity: it is used to describe the instantaneous rate of change of any quantity with respect to another. Outside kinematics, the quantities and units change, but the calculation is identical.
The electric charge passing through a cross-section of a wire as a function of time is $q(t) = 2t^2 + 3t$ coulombs. Given that the electric current is the derivative of…
The electric charge passing through a cross-section of a wire as a function of time is $q(t) = 2t^2 + 3t$ coulombs. Given that the electric current is the derivative of charge with respect to time, calculate the current at $t = 4$ s.
The production cost of $x$ units of an electronic component is $C(x) = 1500 + 10x - 0,05x^2$ euros. Marginal cost is defined as the derivative of the cost with respect…
The production cost of $x$ units of an electronic component is $C(x) = 1500 + 10x - 0,05x^2$ euros. Marginal cost is defined as the derivative of the cost with respect to the number of units. Calculate the marginal cost for $x = 50$ units and explain what this value represents in the context of the factory.
Average velocity versus instantaneous velocity
The height of a projectile in metres is $y(t) = 40t - 5t^2$. Calculate the average velocity in the interval $[1, 3]$ and compare it with the instantaneous velocity at…
The height of a projectile in metres is $y(t) = 40t - 5t^2$. Calculate the average velocity in the interval $[1, 3]$ and compare it with the instantaneous velocity at $t = 2$.
The temperature of an industrial oven during a cooling process is given by $T(t) = 200 - 10t + 0,5t^2$ in degrees Celsius, with $t$ in minutes. Calculate the cooling…
The temperature of an industrial oven during a cooling process is given by $T(t) = 200 - 10t + 0,5t^2$ in degrees Celsius, with $t$ in minutes. Calculate the cooling rate (instantaneous rate of change of temperature) at $t = 5$ min and $t = 15$ min. Interpret the physical meaning of the signs obtained.