The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.
b. Find the velocity function for the marble.

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To find the velocity function, we need to determine the derivative of the position function s(t) with respect to time t. The position function is given as s(t) = \( \frac{100t}{t+1} \).

The function \( s(t) = \frac{100t}{t+1} \) is a rational function, which can be differentiated using the quotient rule. The quotient rule states that if you have a function \( \frac{u(t)}{v(t)} \), its derivative is \( \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \).

Identify the numerator and denominator of the function: \( u(t) = 100t \) and \( v(t) = t+1 \). Compute their derivatives: \( u'(t) = 100 \) and \( v'(t) = 1 \).

Apply the quotient rule: \( v(t)u'(t) - u(t)v'(t) = (t+1)(100) - (100t)(1) \). Simplify this expression to find the numerator of the derivative.

Divide the simplified numerator by the square of the denominator \( (t+1)^2 \) to obtain the velocity function v(t).

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주요 개념

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

Position Function

The position function describes the location of an object over time. In this case, the function s(t) = 100t / (t + 1) represents the position of the marble as a function of time t. Understanding this function is crucial for determining how the marble moves along the incline.

Velocity Function

The velocity function is the derivative of the position function with respect to time. It represents the rate of change of position, indicating how fast the marble is moving at any given moment. To find the velocity function, we differentiate the position function s(t) with respect to t.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative provides information about the function's rate of change and is essential for calculating the velocity from the position function. Mastery of differentiation techniques is necessary to solve problems involving motion.

The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.b. Find the velocity function for the marble.