Slope of a Curve at a Point

In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.

y=x³−3x²+4, P(2,0)

검증된 단계별 안내

First, find the derivative of the function y = x³ - 3x² + 4. The derivative, y', represents the slope of the curve at any point x. Use the power rule for differentiation: if y = xⁿ, then y' = n*xⁿ⁻¹.

Apply the power rule to each term in the function: the derivative of x³ is 3x², the derivative of -3x² is -6x, and the derivative of the constant 4 is 0. Therefore, the derivative of the function is y' = 3x² - 6x.

Evaluate the derivative at the given point P(2,0) to find the slope of the curve at this point. Substitute x = 2 into the derivative: y'(2) = 3(2)² - 6(2).

Now, use the point-slope form of the equation of a line to find the equation of the tangent line at P. The point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point P(2,0).

Substitute the slope found in step 3 and the coordinates of point P into the point-slope form to get the equation of the tangent line. This will give you the equation of the tangent line at the point P(2,0).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Derivative

The derivative of a function at a point 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 the context of a curve, the derivative at a specific point gives the slope of the tangent line to the curve at that point.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. The equation of the tangent line can be expressed in point-slope form, which utilizes the slope and the coordinates of the point of tangency.

Point-Slope Form

The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for writing the equation of a tangent line once the slope has been determined. By substituting the slope and the coordinates of the point of tangency into this formula, one can easily derive the equation of the tangent line.

Slope of a Curve at a PointIn Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.y=x³−3x²+4, P(2,0)