Physical Interpretation of the Derivative
You are riding a bus on the federal highway from Puebla to Mexico City. The speedometer doesn't show the average speed of the whole trip: it shows your speed right at that instant. That is the idea Newton wanted to measure, and it is why he invented the derivative. How do you calculate that number "for one instant"?
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 meters 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 meters 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 meters 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 meters 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
Total distance traveled
If an object changes direction, the distance it travels is not $s(b)-s(a)$. That difference is the displacement: the outbound and return legs partly cancel. For the total distance:
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Calculate $v(t)$ and solve $v(t)=0$ to find where it changes direction within the interval.
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Split the interval into segments using those instants. In each segment $v$ does not change sign.
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In each segment calculate the absolute value of the difference in positions and add them all up.
A hot-air balloon rises such that its height in meters 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 meters is given by $h(t) = 15t - t^2$. At what time $t$ does its velocity become zero?
A toy rocket is launched vertically upward. Its height in meters 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 upward. Its height in meters 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.
A vehicle brakes sharply and the distance traveled in meters from the moment the brakes are applied is $d(t) = 24t - 3t^2$. How long does it take for the vehicle to come…
A vehicle brakes sharply and the distance traveled in meters from the moment the brakes are applied is $d(t) = 24t - 3t^2$. How long does it take for the vehicle to come to a complete stop? What total distance did it travel while braking?
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 ($v(t) < 0$).
An exploration robot moves along a straight rail. Its position in meters during the first $5$ seconds is given by $x(t) = t^3 - 6t^2 + 9t + 2$. Calculate the actual…
An exploration robot moves along a straight rail. Its position in meters during the first $5$ seconds is given by $x(t) = t^3 - 6t^2 + 9t + 2$. Calculate the actual total distance traveled by the robot in the interval $t \in [0, 5]$. Note that the total distance is not simply $x(5) - x(0)$, but you must analyze the changes in direction of motion using velocity.
Differentiation tools for other kinds of motion
Not all motions are polynomials. A spring oscillates with sines, a capacitor charges with exponentials, and the radius of an oil spill grows like a quotient. You need these rules, which you will see in more detail in the section on differentiation rules.
The position of a harmonic oscillator attached to a spring is $x(t) = 5 \sin(2t)$ in centimeters. Calculate its velocity and acceleration as functions of time. Show that…
The position of a harmonic oscillator attached to a spring is $x(t) = 5 \sin(2t)$ in centimeters. Calculate its velocity and acceleration as functions of time. Show that the acceleration is proportional to the position, that is, $a(t) = k \times x(t)$, and find the value of the constant $k$.
A conical tank empties such that the volume of water inside it is $V(t) = 1000 \left(1 - \frac{t}{20}\right)^3$ liters, for $0 \leq t \leq 20$ minutes. Calculate the…
A conical tank empties such that the volume of water inside it is $V(t) = 1000 \left(1 - \frac{t}{20}\right)^3$ liters, for $0 \leq t \leq 20$ minutes. Calculate the outflow rate of water (instantaneous rate of change of volume) at time $t = 10$ minutes.
An oil slick on the sea is circular in shape and its radius in meters grows according to the function $r(t) = \frac{20t}{t+2}$, where $t$ is the time in hours. Find the…
An oil slick on the sea is circular in shape and its radius in meters grows according to the function $r(t) = \frac{20t}{t+2}$, where $t$ is the time in hours. Find the rate at which the radius is growing at time $t = 3$ hours. Does it grow faster at the beginning or after a few hours? Justify your answer using the concept of acceleration or by analyzing the derivative.
The charge in a capacitor of an electric circuit varies with time according to the expression $q(t) = q_0 \left(1 - e^{-t/RC}\right)$, where $q_0$, $R$, and $C$ are…
The charge in a capacitor of an electric circuit varies with time according to the expression $q(t) = q_0 \left(1 - e^{-t/RC}\right)$, where $q_0$, $R$, and $C$ are positive constants. The current is $i(t) = q'(t)$. Show that the current decreases exponentially with time and calculate the limiting value of the charge and the current as $t \to +\infty$.
Other instantaneous rates of change
The derivative is not just velocity: it describes the instantaneous rate of change of any quantity with respect to another. The units and the context change, but the calculation is the same.
The volume of water in a tank changes according to the function $V(t) = 50 + 4t - t^3$, where $V$ is in liters and $t$ in minutes. Find the instantaneous rate of change…
The volume of water in a tank changes according to the function $V(t) = 50 + 4t - t^3$, where $V$ is in liters and $t$ in minutes. Find the instantaneous rate of change of volume at time $t = 1$ and state whether the tank is filling or emptying at that moment.
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 $C(x) = 30\,000 + 200x - 1x^2$ pesos. Marginal cost is defined as the derivative of the cost with respect…
The production cost of $x$ units of an electronic component is $C(x) = 30\,000 + 200x - 1x^2$ pesos. 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 meters 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 meters 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.
More complete problems
In the hard problems several ideas are combined: equating velocities, using initial conditions, finding the maximum of the acceleration, verifying an equation with an exponential, or requiring that a piecewise function have continuous velocity.
When is the acceleration maximum?
Acceleration is also a function of $t$, so it is optimized like any other: differentiate $a(t)$, set $a'(t)=0$, and check that it is a maximum (the derivative goes from positive to negative, or $a''<0$).
Newton's law of cooling and the natural logarithm
Sometimes, however, you are asked when a certain temperature is reached, and then $t$ ends up inside an exponent. To bring it out we use the natural logarithm $\ln$, which is the inverse function of $e^{x}$:
$$e^{u}=c \iff u=\ln c \qquad\text{and}\qquad \ln\left(e^{u}\right)=u$$Two race cars, A and B, travel along straight parallel tracks. Their positions relative to the starting line are given by $s_A(t) = t^3 - 6t^2 + 15t$ and…
Two race cars, A and B, travel along straight parallel tracks. Their positions relative to the starting line are given by $s_A(t) = t^3 - 6t^2 + 15t$ and $s_B(t) = 2t^2 + 3t$. Determine at what times $t$ both cars are travelling at the same velocity. For these times, calculate and compare the acceleration of each car.
A projectile is launched vertically from a platform. Its height in meters is given by $h(t) = at^2 + bt + c$. We know that the platform is $15$ m above the ground, that…
A projectile is launched vertically from a platform. Its height in meters is given by $h(t) = at^2 + bt + c$. We know that the platform is $15$ m above the ground, that the initial launch velocity is $20$ m/s upward, and that the acceleration due to gravity is constant and equal to $-9.8$ m/s$^2$. Using the relationship between position, velocity, and acceleration via derivatives, find the values of $a$, $b$, and $c$. Next, calculate the maximum height reached.
The velocity of a particle moving in a straight line is $v(t) = \frac{t^2 - 1}{t^2 + 1}$ m/s. Calculate the acceleration function $a(t)$. Determine at what time…
The velocity of a particle moving in a straight line is $v(t) = \frac{t^2 - 1}{t^2 + 1}$ m/s. Calculate the acceleration function $a(t)$. Determine at what time $t \geq 0$ the acceleration is maximum and calculate this maximum value.
Newton's law of cooling states that the rate of change of the temperature $T(t)$ of an object is proportional to the difference between its temperature and that of the…
Newton's law of cooling states that the rate of change of the temperature $T(t)$ of an object is proportional to the difference between its temperature and that of the surrounding environment $T_a$, that is, $T'(t) = -k(T(t) - T_a)$. Verify that the function $T(t) = T_a + (T_0 - T_a)e^{-kt}$ satisfies this differential equation. If $T_a = 20^\circ$C, $T_0 = 100^\circ$C and the cooling rate at the initial time $t=0$ is $-4^\circ$C/min, calculate the value of $k$.
The power consumed by an electronic device depends on the applied voltage according to the function $P(v) = \frac{v^3}{v^2 + 4}$ milliwatts. The rate of change of power…
The power consumed by an electronic device depends on the applied voltage according to the function $P(v) = \frac{v^3}{v^2 + 4}$ milliwatts. The rate of change of power with respect to voltage is called dynamic conductance $G(v) = P'(v)$. Determine for which voltage values $v$ the dynamic conductance is exactly $1$ milliwatt/volt.
A weather balloon rises vertically. Its height in meters is modeled by a piecewise function:…
A weather balloon rises vertically. Its height in meters is modeled by a piecewise function:
$h(t) = \begin{cases} 2t^2 & \text{if } 0 \leq t \leq 10 \\ at + b - \frac{100}{t} & \text{if } t > 10 \end{cases}$
Determine the values of the parameters $a$ and $b$ so that the position and velocity of the balloon are continuous at time $t = 10$. Once found, calculate the acceleration of the balloon for $t > 10$ and discuss what happens to the velocity in the long term (as $t \to +\infty$).
Back to the beginning: the speedometer shows $s'(t)$, the limit of the average velocities over shorter and shorter intervals.