A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e^-t cos t), for t ≥ 0.
Determine her velocity at t = 1 and t = 3.

검증된 단계별 안내

To find the velocity of the woman at a given time, we need to determine the derivative of her height function y(t) with respect to time t. The derivative, y'(t), represents the velocity.

The height function is given as y(t) = 15(1 + e^(-t) * cos(t)). We will apply the product rule and chain rule to differentiate this function.

First, identify the components of the function: u(t) = e^(-t) and v(t) = cos(t). The product rule states that the derivative of u(t) * v(t) is u'(t) * v(t) + u(t) * v'(t).

Calculate the derivatives: u'(t) = -e^(-t) (using the chain rule for e^(-t)) and v'(t) = -sin(t) (derivative of cos(t)).

Substitute these derivatives into the product rule formula: y'(t) = 15 * [(-e^(-t) * cos(t)) + (e^(-t) * (-sin(t)))]. Evaluate y'(t) at t = 1 and t = 3 to find the velocity at these times.

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주요 개념

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

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable, which in this case is time. To determine the velocity of the woman at specific times, we need to differentiate her height function y(t) with respect to t, yielding the velocity function v(t).

Exponential Functions

Exponential functions are mathematical functions of the form f(t) = a * e^(bt), where e is Euler's number. In the given height function y(t), the term e^(-t) indicates that the height changes exponentially over time. Understanding how exponential decay affects the height is crucial for accurately calculating the velocity at different time points.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In the height function y(t), the cosine term introduces oscillatory behavior to the height over time. Recognizing how the cosine function influences the overall height is important for evaluating the velocity at specific moments during the jump.

A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e-t cos t), for t ≥ 0.Determine her velocity at t = 1 and t = 3.