A bug is moving along the right side of the parabola y=x² at a rate such that its distance from the origin is increasing at 1 cm/min.
b. Use the equation y=x² to find an equation relating dy/dt to dx/dt.

검증된 단계별 안내

Start by understanding the problem: We have a bug moving along the parabola y = x², and its distance from the origin is increasing at a rate of 1 cm/min. We need to find a relationship between dy/dt and dx/dt.

Recall that the distance from the origin to a point (x, y) on the parabola is given by the distance formula: d = √(x² + y²). Since y = x², substitute y in the distance formula to get d = √(x² + (x²)²) = √(x² + x⁴).

Differentiate the distance formula with respect to time t to find the rate of change of distance: d(d)/dt = (1/2) * (1/√(x² + x⁴)) * (2x * dx/dt + 4x³ * dx/dt). Set this equal to 1 cm/min, as given in the problem.

Now, differentiate the equation y = x² with respect to time t to find dy/dt: dy/dt = 2x * dx/dt.

Combine the results from the differentiation of the distance formula and the parabola equation to express dy/dt in terms of dx/dt. Use the fact that d(d)/dt = 1 cm/min to solve for dy/dt in terms of dx/dt.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to relate the rates of change of the bug's position along the parabola (dx/dt and dy/dt) to its distance from the origin. This concept is essential for setting up the equations that will allow us to solve for the desired rates.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations that define y implicitly in terms of x. Since the bug's position is described by the equation y = x², we can differentiate both sides with respect to time t to find a relationship between dy/dt and dx/dt. This method is crucial for relating the rates of change in this context.

Distance Formula

The distance formula calculates the distance between two points in a coordinate system. In this scenario, the distance from the origin to the bug's position on the parabola can be expressed as D = √(x² + y²). Understanding this formula is vital for determining how the distance changes over time, which is central to solving the problem.

A bug is moving along the right side of the parabola y=x² at a rate such that its distance from the origin is increasing at 1 cm/min.b. Use the equation y=x² to find an equation relating dy/dt to dx/dt.