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 = 4 − x², (−1, 3)

검증된 단계별 안내

First, identify the function given: \( y = 4 - x^2 \). This is a quadratic function representing a parabola that opens downwards.

To find the equation of the tangent line, we need the derivative of the function, which gives us the slope of the tangent line at any point \( x \). Differentiate \( y = 4 - x^2 \) with respect to \( x \) to get \( \frac{dy}{dx} = -2x \).

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

Use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (-1, 3) \) and \( m = 2 \). Substitute these values into the equation to get \( y - 3 = 2(x + 1) \).

Simplify the equation \( y - 3 = 2(x + 1) \) to get the final equation of the tangent line. This will give you the equation in the form \( y = mx + b \), which can be used to sketch the tangent line along with the curve \( y = 4 - x^2 \).

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 = 4 - x², the derivative is found using basic differentiation rules, resulting in dy/dx = -2x. Evaluating this derivative at x = -1 gives the slope of the tangent line at the point (-1, 3).

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 point. Using the slope from the derivative and the point (-1, 3), you can find the equation of the tangent line.

Graphing Functions and Tangent Lines

Graphing involves plotting the curve of the function and the tangent line to visualize their relationship. For y = 4 - x², the graph is a downward-opening parabola. The tangent line at (-1, 3) will touch the curve at this point, illustrating the concept of tangency where the line just 'kisses' the curve without crossing it.

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 = 4 − x², (−1, 3)