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=7−x², P(2,3)

Verified step by step guidance

First, identify the function given: \( y = 7 - x^2 \). This is a quadratic function, and we need to find its derivative to determine the slope of the curve at point P.

To find the slope of the curve at point P, calculate the derivative of the function \( y = 7 - x^2 \) with respect to \( x \). The derivative, \( \frac{dy}{dx} \), represents the slope of the tangent line at any point \( x \).

The derivative of \( y = 7 - x^2 \) is \( \frac{dy}{dx} = -2x \). This derivative gives us the slope of the curve at any point \( x \).

Substitute \( x = 2 \) into the derivative \( \frac{dy}{dx} = -2x \) to find the slope at point P(2, 3). This will give you the slope of the tangent line at that specific point.

To find the equation of the tangent line at point P, use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope found in the previous step, and \( (x_1, y_1) \) is the point P(2, 3). Substitute the values to get the equation of the tangent line.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 and the point of tangency are known. It allows for a straightforward way to express the line based on its slope and a specific point.

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