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 / x²), (−1, 1)

검증된 단계별 안내

First, identify the function given: \( y = \frac{1}{x^2} \). 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 \( y = x^{-2} \) and differentiate: \( \frac{dy}{dx} = -2x^{-3} \).

Evaluate the derivative at the given point \((-1, 1)\) to find the slope of the tangent line. Substitute \( x = -1 \) into the derivative: \( \frac{dy}{dx} = -2(-1)^{-3} \).

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 \((-1, 1)\).

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 \( y = mx + b \).

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.

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/x²), the derivative can be found using the power rule, which helps determine how the function changes with respect to x. Calculating the derivative is essential for finding the equation of the tangent line.

Point-Slope Form

The point-slope form of a line equation is used to write the equation of a tangent line once the slope is known. It is expressed as y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. This form is particularly useful for constructing the tangent line equation using the slope from the derivative and the given point.

Sketching Curves and Tangent Lines

Sketching involves visually representing the function and its tangent line on a graph. Understanding the behavior of the function y = (1/x²) and its derivative helps in accurately plotting the curve and the tangent line at the point (-1, 1). This visualization aids in comprehending the relationship between the function and its tangent line.

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 / x²), (−1, 1)