In Exercises 19–22, find the slope of the curve at the point indicated.

y = x³ − 2x + 7, x = −2

검증된 단계별 안내

To find the slope of the curve at a given point, we need to determine the derivative of the function. The function given is \( y = x^3 - 2x + 7 \).

Differentiate the function with respect to \( x \). The derivative of \( y = x^3 - 2x + 7 \) is \( \frac{dy}{dx} = 3x^2 - 2 \).

Now that we have the derivative, we can find the slope of the curve at any point \( x \) by substituting the \( x \)-value into the derivative.

Substitute \( x = -2 \) into the derivative \( \frac{dy}{dx} = 3x^2 - 2 \) to find the slope at this specific point.

Calculate \( 3(-2)^2 - 2 \) to determine the slope of the curve at \( x = -2 \).

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주요 개념

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

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides the slope of the tangent line to the curve at any given point.

Slope of a Curve

The slope of a curve at a specific point is determined by the derivative of the function evaluated at that point. It represents the instantaneous rate of change of the function's value with respect to its input. For the function y = x³ − 2x + 7, finding the slope at x = -2 involves calculating the derivative and substituting -2 into that derivative.

Evaluating Functions

Evaluating a function involves substituting a specific value for the variable in the function's expression to find the corresponding output. In this context, after finding the derivative of the function, we will evaluate it at x = -2 to determine the slope of the curve at that point. This process is essential for obtaining numerical results from algebraic expressions.

In Exercises 19–22, find the slope of the curve at the point indicated.y = x³ − 2x + 7, x = −2