Question

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

Accepted Answer

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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Key Concepts

Velocity and Displacement

Riemann Sums

Subintervals and Partitioning