--- title: "Introduction to Derivatives — A Car on the Road" subtitle: "Preparatory Lesson: What Does «Rate of Change» Mean?" output: html_document: toc: true toc_float: true theme: flatly highlight: tango code_folding: show --- ```{r setup, include=FALSE} knitr::opts_chunk$set( echo = TRUE, # Show the R code message = FALSE, warning = FALSE, fig.width = 8, fig.height = 4.5 ) ``` --- ## 🎯 Goal of the Lesson Before we study how populations grow, we need a basic idea from mathematics: the **derivative**. > **The derivative of a function tells us how fast it changes.** Instead of abstract formulas, we will start with something very familiar: **a car moving along a road.** --- ## 🚗 The Story of the Car Imagine a car that: - **Phase A (0–30 seconds):** Moves at a **constant speed** of 20 m/s. It does not accelerate, it does not brake — steady motion. - **Phase B (30–60 seconds):** Steps on the gas and **accelerates** smoothly. Its speed increases from 20 m/s to 50 m/s. The question: How are **distance**, **speed** and **acceleration** connected? --- ## ⚙️ Step 1 — Defining Time and Parameters ```{r parameters} # ------------------------------------------------------- # We define the time scale of the motion # seq() creates a sequence of values # from 0 to 60 seconds, with a step of 0.1 sec # ------------------------------------------------------- t <- seq(0, 60, by = 0.1) # time in seconds # Phase A parameters — constant motion v0 <- 20 # initial speed (m/s) t_switch <- 30 # moment when acceleration begins (sec) # Phase B parameter — acceleration # We want the speed to go from 20 → 50 m/s between t=30 and t=60 # So: a = Δv / Δt = (50 - 20) / (60 - 30) = 1 m/s² a <- 1 # acceleration (m/s²) ``` --- ## ⚙️ Step 2 — Computing the Speed The speed changes depending on the phase: $$ v(t) = \begin{cases} 20 & \text{if } t \leq 30 \\ 20 + 1 \cdot (t - 30) & \text{if } t > 30 \end{cases} $$ ```{r velocity} # ------------------------------------------------------- # ifelse() works like a choice: # if the condition is TRUE → first value # if the condition is FALSE → second value # # Here: if t <= 30, speed = v0 (constant) # if t > 30, speed increases linearly # ------------------------------------------------------- v <- ifelse( t <= t_switch, v0, # Phase A: constant speed v0 + a * (t - t_switch) # Phase B: v = v₀ + a·(t - 30) ) ``` --- ## ⚙️ Step 3 — Computing the Distance The distance is the **integral** of the speed (or equivalently: the speed is the **derivative** of the distance). $$ x(t) = \begin{cases} v_0 \cdot t & \text{if } t \leq 30 \\ v_0 \cdot 30 + v_0(t-30) + \frac{1}{2}a(t-30)^2 & \text{if } t > 30 \end{cases} $$ ```{r distance} # ------------------------------------------------------- # Phase A: x = v₀ · t (uniform straight-line motion) # Phase B: x = distance of Phase A at t=30 # + v₀·(t-30) + ½·a·(t-30)² # # First we compute the distance covered # up to the moment t=30 (end of Phase A) # ------------------------------------------------------- x_at_switch <- v0 * t_switch # = 20 × 30 = 600 meters x <- ifelse( t <= t_switch, v0 * t, # Phase A x_at_switch + v0 * (t - t_switch) + 0.5 * a * (t - t_switch)^2 # Phase B ) ``` --- ## ⚙️ Step 4 — Computing the Acceleration The acceleration is the **derivative** of the speed. ```{r acceleration} # ------------------------------------------------------- # Phase A: speed constant → does not change → acceleration = 0 # Phase B: speed increases by 1 m/s every second # → acceleration = 1 m/s² # ------------------------------------------------------- acc <- ifelse(t <= t_switch, 0, a) ``` --- ## 📊 Step 5 — The Plots Now we will see the three quantities visually. Notice how the **shape** of each plot changes at 30 seconds. ```{r plots, fig.height=11, fig.width=8} # ------------------------------------------------------- # par(mfrow = c(3, 1)) → splits the graphics window # into 3 rows and 1 column (the 3 plots one below # the other) # mar = c(bottom, left, top, right) → margins in lines # ------------------------------------------------------- par(mfrow = c(3, 1), mar = c(4.5, 5, 3, 2)) # --- Plot 1: Distance vs Time --- plot(t, x, type = "l", # "l" = line lwd = 2.5, # line width col = "#2C7BB6", # color (blue) xlab = "Time (sec)", ylab = "Distance x (m)", main = "Distance versus Time — x(t)", cex.main = 1.3, cex.lab = 1.1) # Vertical dashed line at t=30 (phase change) abline(v = t_switch, lty = 2, col = "gray50") text(31, max(x) * 0.15, "Start of\nacceleration", adj = 0, col = "gray40", cex = 0.9) # Label for each phase text(13, max(x) * 0.85, "Phase A\n(straight line)", col = "#2C7BB6", cex = 0.95) text(47, max(x) * 0.20, "Phase B\n(parabola)", col = "#2C7BB6", cex = 0.95) # ------------------------------------------------------- # Tangent line in the acceleration phase # We pick the moment t₀ = 45 sec # At this point: # - the distance is x(45) # - the slope of the tangent = derivative = v(45) # ------------------------------------------------------- t0 <- 45 # moment x_t0 <- x_at_switch + v0*(t0 - t_switch) + 0.5*a*(t0 - t_switch)^2 v_t0 <- v0 + a*(t0 - t_switch) # instantaneous speed = slope of tangent # Equation of the tangent line: y = x(t₀) + v(t₀) · (t - t₀) t_tan <- seq(t0 - 10, t0 + 10, length.out = 50) y_tan <- x_t0 + v_t0 * (t_tan - t0) # We draw the tangent lines(t_tan, y_tan, col = "#E66101", lwd = 2.5) # Point of contact (dot) points(t0, x_t0, pch = 19, col = "#E66101", cex = 1.6) # Annotation with arrow arrows(x0 = 53, y0 = x_t0 + 280, x1 = t0 + 0.5, y1 = x_t0 + 30, length = 0.1, col = "#E66101", lwd = 1.5) text(53.5, x_t0 + 320, paste0("slope of tangent\n= v(", t0, ") = ", v_t0, " m/s"), col = "#E66101", cex = 0.9, adj = 0) # --- Plot 2: Speed vs Time --- plot(t, v, type = "l", lwd = 2.5, col = "#1A9641", # green xlab = "Time (sec)", ylab = "Speed v (m/s)", main = "Speed versus Time — v(t) = dx/dt", cex.main = 1.3, cex.lab = 1.1) abline(v = t_switch, lty = 2, col = "gray50") text(31, max(v) * 0.25, "Start of\nacceleration", adj = 0, col = "gray40", cex = 0.9) # Arrow showing that the slope of the line = acceleration text(40, 32, "slope = a = 1 m/s²", col = "#1A9641", cex = 0.9) # --- Plot 3: Acceleration vs Time --- plot(t, acc, type = "l", lwd = 2.5, col = "#D7191C", # red xlab = "Time (sec)", ylab = "Acceleration a (m/s²)", main = "Acceleration versus Time — a(t) = dv/dt", ylim = c(-0.3, 1.5), cex.main = 1.3, cex.lab = 1.1) abline(v = t_switch, lty = 2, col = "gray50") abline(h = 0, lty = 3, col = "gray70") text(31, 1.3, "Start of\nacceleration", adj = 0, col = "gray40", cex = 0.9) ``` --- ## 🔎 The Tangent and the Instantaneous Speed In the first plot we just added a **tangent line** to the distance curve, at the point t = 45 sec. > 💡 **The central geometric idea:** > The **slope** of the tangent line at any point of the distance > curve gives us the **instantaneous speed** at exactly that moment > in time. The **steeper** (larger angle with the horizontal axis) the tangent is, the **greater** the instantaneous speed. Let us see it by drawing **multiple tangents** at different moments in time: ```{r multiple_tangents, fig.height=6} # ------------------------------------------------------- # We choose four moments in time to show # how the slope of the tangent changes along the curve # ------------------------------------------------------- t_points <- c(15, 35, 45, 55) # moments for the tangents colors_t <- c("#1A9641", "#FDAE61", "#E66101", "#D7191C") par(mar = c(4.5, 5, 3, 2)) # Distance curve plot(t, x, type = "l", lwd = 2.5, col = "#2C7BB6", xlab = "Time (sec)", ylab = "Distance x (m)", main = "The slope of the tangent = instantaneous speed", cex.main = 1.2) abline(v = t_switch, lty = 2, col = "gray60") # ------------------------------------------------------- # Loop: for each moment in time we compute # - the position x(ti) # - the instantaneous speed v(ti) (= slope of tangent) # - and we draw the tangent # ------------------------------------------------------- for (i in seq_along(t_points)) { ti <- t_points[i] # Position at the point of contact if (ti <= t_switch) { xi <- v0 * ti # Phase A vi <- v0 # constant speed } else { xi <- x_at_switch + v0*(ti - t_switch) + 0.5*a*(ti - t_switch)^2 vi <- v0 + a*(ti - t_switch) # accelerating } # Tangent: y = xi + vi · (t - ti) tt <- seq(ti - 6, ti + 6, length.out = 30) yy <- xi + vi * (tt - ti) lines(tt, yy, col = colors_t[i], lwd = 2) points(ti, xi, pch = 19, col = colors_t[i], cex = 1.4) # Label with the instantaneous speed text(ti, xi - 130, paste0("t=", ti, "s\nv=", vi, " m/s"), col = colors_t[i], cex = 0.85, font = 2) } legend("topleft", legend = "Distance curve x(t)", col = "#2C7BB6", lty = 1, lwd = 2.5, bty = "n", cex = 0.95) ``` ### What we observe | Moment | Slope of tangent | Instantaneous speed | |--------|---------------------|---------------------| | t = 15 s (Phase A) | Moderate — constant | 20 m/s | | t = 35 s (Phase B, early) | A bit steeper | 25 m/s | | t = 45 s (Phase B, middle) | Steeper | 35 m/s | | t = 55 s (Phase B, end) | Very steep | 45 m/s | **In Phase A** all the tangents have the **same slope** — because the curve is a straight line, the line is its own tangent. So the speed is constant (20 m/s). **In Phase B** each tangent is **steeper** than the previous one. This means: the speed **increases** — that is, we have acceleration. ### Conclusion $$ v(t_0) = \frac{dx}{dt}\bigg|_{t=t_0} = \text{slope of the tangent of } x(t) \text{ at } t_0 $$ > **This is exactly the geometric meaning of the derivative:** > the derivative at a point = slope of the tangent line at that point. --- ## 💡 Step 6 — What Do We See? The Central Idea Look at the three plots together and observe: | Plot | Phase A (0–30 sec) | Phase B (30–60 sec) | |---------|-------------------|---------------------| | **Distance** x(t) | Straight line | Curve (parabola) | | **Speed** v(t) | Horizontal line | Rising line | | **Acceleration** a(t) | = 0 | = 1 m/s² (constant) | ### The connection: $$\boxed{v(t) = \frac{dx}{dt}} \qquad \text{and} \qquad \boxed{a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}}$$ > **In simple terms:** > - The **speed** tells us how fast the **position** changes. > - The **acceleration** tells us how fast the **speed** changes. > - This is exactly the relationship that the **derivative** expresses. --- ## 🔍 Step 7 — The Derivative Numerically (Optional) We don't need formulas to compute (approximately) the derivative. It is enough to see **how much the function changes over a very small step in time**. ```{r numerical_derivative} # ------------------------------------------------------- # diff() computes the differences between consecutive values # E.g. diff(c(1, 3, 6)) → c(2, 3) # # The numerical derivative: dx/dt ≈ Δx / Δt # Δt = 0.1 sec (the step of t we defined at the start) # ------------------------------------------------------- dt <- 0.1 # Numerical derivative of distance → should resemble v v_numerical <- diff(x) / dt # Numerical derivative of speed → should resemble acc a_numerical <- diff(v) / dt # The time for the derivatives has one element less t_deriv <- t[-1] # we remove the first element # Comparison: numerical vs analytical speed cat("Maximum deviation v_numerical vs v_analytical:", round(max(abs(v_numerical - v[-1])), 4), "m/s\n") ``` ```{r comparison_plot, fig.height=5} # ------------------------------------------------------- # Visual comparison: analytical speed (line) # vs numerical derivative (dashed) # ------------------------------------------------------- par(mar = c(4.5, 5, 3, 2)) plot(t, v, type = "l", lwd = 2.5, col = "#1A9641", xlab = "Time (sec)", ylab = "Speed (m/s)", main = "Speed: Analytical vs Numerical Derivative", cex.main = 1.2) lines(t_deriv, v_numerical, lty = 2, lwd = 2, col = "#D7191C") legend("topleft", legend = c("Analytical v(t)", "Numerical dx/dt"), col = c("#1A9641", "#D7191C"), lty = c(1, 2), lwd = 2, bty = "n", cex = 1.0) ``` > The two curves **coincide almost perfectly** — this confirms that > the derivative of the distance is indeed the speed! --- ## 🌿 Connection to Biology Why do we learn these things in a biology course? In biology we are often interested in **the rate** at which something changes: | Biological Quantity | «Distance» | «Speed» (derivative) | |---------------------|------------|--------------------------| | Population size N(t) | N | dN/dt = growth rate | | Drug concentration C(t) | C | dC/dt = excretion rate | | Biomass B(t) | B | dB/dt = production rate | > In the next lessons we will see exactly this: how the **rate of change of > the population** (i.e. the derivative dN/dt) determines whether a population grows, > shrinks or stabilizes. --- ## ✏️ Comprehension Questions 1. In Phase A, the distance-time graph is a **straight line**. What does this mean for its derivative (= speed)? 2. In Phase B, the distance follows a **parabola**. What shape does the speed have? And the acceleration? 3. If a population of bacteria grows exponentially, how do you expect the graph of the **growth rate** dN/dt versus time to look? --- *In the next lesson: Exponential Population Growth — N(t) = N₀ · e^(r·t)*