Modelling Real-World Situations?

📌 CONCEPT: To find the total distance travelled by the particle in the first two seconds, we need to calculate the definite integral of the square root of the sum of the squares of the derivatives of the curve's x and y components with respect to time, denoted as √((dx/dt)^2 + (dy/dt)^2), over the given time interval.

📐 RULE / FORMULA: The total distance travelled is given by the definite integral of the square root of the sum of the squares of the derivatives of the curve's x and y components with respect to time, i.e., ∫[√((dx/dt)^2 + (dy/dt)^2)]dt, evaluated over the given time interval.

💡 WORKED EXAMPLE: Given the curve y = 3x^2 - 4x + 5, we need to find the total distance travelled in the first two seconds. First, we find the derivatives of x and y with respect to time, denoted as dx/dt and dy/dt. Then, we evaluate the definite integral ∫[√((dx/dt)^2 + (dy/dt)^2)]dt from t = 0 to t = 2.

⚠️ COMMON MISTAKE: Students often forget to consider the absolute value of the square root when calculating the total distance travelled, which can result in an incorrect answer.