In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

y = (1 / x³), (−2, −1/8)

검증된 단계별 안내

First, identify the function given: \( y = \frac{1}{x^3} \). We need to find the derivative of this function to determine the slope of the tangent line at the given point.

To find the derivative, use the power rule. Rewrite the function as \( y = x^{-3} \) and differentiate: \( \frac{dy}{dx} = -3x^{-4} \).

Evaluate the derivative at the given point \( x = -2 \) to find the slope of the tangent line. Substitute \( x = -2 \) into the derivative: \( \frac{dy}{dx} = -3(-2)^{-4} \).

With the slope calculated, use the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point \((-2, -\frac{1}{8})\).

Substitute the slope and the point into the point-slope form to get the equation of the tangent line. Simplify the equation to express it in the form \( y = mx + b \).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

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

Derivative

The derivative of a function at a point provides the slope of the tangent line to the curve at that point. For the function y = (1 / x³), the derivative can be found using the power rule, which helps in determining the rate of change of the function with respect to x. This slope is crucial for writing the equation of the tangent line.

Point-Slope Form of a Line

The point-slope form is a method for writing the equation of a line when you know a point on the line and its slope. It is expressed as y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the given point. This form is used to find the equation of the tangent line once the slope is determined from the derivative.

Graphing Functions and Tangent Lines

Graphing involves plotting the function and its tangent line to visualize their relationship. For y = (1 / x³), sketching the curve helps in understanding its behavior, while the tangent line at the point (-2, -1/8) shows the instantaneous rate of change. This visual representation aids in comprehending how the tangent line approximates the curve locally.

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.y = (1 / x³), (−2, −1/8)