Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.
{Use of Tech} v = 4 √(t +1) (mi/hr) . for 0 ≤ t ≤ 15 ; n = 5

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Step 1: Understand the problem. The goal is to approximate the displacement of the object over the interval [0, 15] using the velocity function v(t) = 4√(t + 1). The interval is subdivided into n = 5 subintervals, and the left endpoint of each subinterval is used to compute the height of the rectangles.

Step 2: Determine the width of each subinterval. The interval [0, 15] is divided into 5 equal subintervals, so the width of each subinterval (Δt) is calculated as Δt = (15 - 0) / 5 = 3.

Step 3: Identify the left endpoints of the subintervals. The left endpoints are the starting points of each subinterval: t₀ = 0, t₁ = 3, t₂ = 6, t₃ = 9, and t₄ = 12.

Step 4: Compute the velocity at each left endpoint. Substitute each left endpoint into the velocity function v(t) = 4√(t + 1) to find the velocity values: v(t₀), v(t₁), v(t₂), v(t₃), and v(t₄). For example, v(t₀) = 4√(0 + 1), v(t₁) = 4√(3 + 1), and so on.

Step 5: Approximate the displacement. Multiply the velocity at each left endpoint by the width of the subinterval (Δt = 3) to find the area of each rectangle. Sum these areas to approximate the total displacement: Displacement ≈ Δt * [v(t₀) + v(t₁) + v(t₂) + v(t₃) + v(t₄)].

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Velocity and Displacement

Velocity is the rate of change of an object's position with respect to time, often expressed as a function of time. Displacement, on the other hand, is the total distance traveled in a specific direction over a given time interval. In this context, understanding how velocity relates to displacement is crucial for approximating the total displacement of the object over the specified interval.

Riemann Sums

Riemann sums are a method for approximating the integral of a function, which in this case represents the area under the velocity curve. By dividing the interval into subintervals and using the left endpoint to determine the height of rectangles, we can estimate the total area, which corresponds to the displacement. This technique is foundational in calculus for understanding how to approximate integrals.

Subintervals and Partitioning

Partitioning an interval into subintervals involves dividing the total time interval into smaller segments, allowing for a more manageable calculation of the area under the curve. In this problem, the interval from 0 to 15 is divided into 5 subintervals, which helps in applying the left endpoint method to compute the height of the rectangles. This concept is essential for implementing Riemann sums effectively.