Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s \u003e 0 corresponding to positions right of the origin.Determine the velocity and acceleration of the object at t = 1. f(t) = t2 − 4t; 0 ≤ t ≤ 5

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

문제 15c

3장, 문제 15c

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
Determine the velocity and acceleration of the object at t = 1.
f(t) = t² − 4t; 0 ≤ t ≤ 5

검증된 단계별 안내

Step 1: Understand the problem. We are given a position function s = f(t) = t^2 - 4t, which describes the position of an object moving along a line. We need to find the velocity and acceleration of the object at t = 1.

Step 2: Find the velocity function. The velocity of an object is the derivative of the position function with respect to time. So, calculate the derivative of f(t) with respect to t: v(t) = f'(t) = d/dt (t^2 - 4t).

Step 3: Calculate the derivative. Use the power rule to differentiate each term: the derivative of t^2 is 2t, and the derivative of -4t is -4. Therefore, v(t) = 2t - 4.

Step 4: Evaluate the velocity at t = 1. Substitute t = 1 into the velocity function: v(1) = 2(1) - 4.

Step 5: Find the acceleration function. The acceleration is the derivative of the velocity function. Differentiate v(t) = 2t - 4 with respect to t to find a(t).

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주요 개념

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

Position Function

The position function, denoted as s = f(t), describes the location of an object at any given time t. In this case, the function f(t) = t² - 4t represents the position of the object moving along a horizontal line. Understanding this function is crucial for determining how the object's position changes over time.

Velocity

Velocity is the rate of change of position with respect to time, mathematically represented as the derivative of the position function. To find the velocity at a specific time, we compute f'(t), which gives us the instantaneous rate of change of the position at that moment. For this problem, we need to evaluate the derivative at t = 1.

Acceleration

Acceleration is the rate of change of velocity with respect to time, which can be found by taking the derivative of the velocity function. In this context, it is essential to compute the second derivative of the position function, f''(t), to determine how the velocity of the object changes over time. Evaluating this at t = 1 will provide the acceleration at that instant.