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, P(2,8)

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To find the slope of the curve at point P, we need to calculate the derivative of the function y = x³. The derivative, denoted as dy/dx or y', gives us the slope of the tangent line at any point on the curve.

Differentiate the function y = x³ with respect to x. Using the power rule, which states that the derivative of x^n is n*x^(n-1), we find that the derivative of x³ is 3x².

Now, substitute the x-coordinate of point P, which is x = 2, into the derivative to find the slope at that specific point. This gives us the slope m = 3*(2)².

With the slope m calculated, we can now find the equation of the tangent line at point P(2, 8). The equation of a line in point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is the point on the line.

Substitute the slope m and the coordinates of point P into the point-slope form equation: y - 8 = m(x - 2). Simplify this equation to get the equation of the tangent line in slope-intercept form.

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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. For the curve y = x³, the derivative will provide the slope of the tangent line at any point, including point P(2,8).

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The equation of the tangent line can be expressed in point-slope form, which utilizes the slope obtained from the derivative and the coordinates of the point of tangency. This line provides a linear approximation of the curve near that point.

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 the tangent line once the slope has been determined from the derivative. For the curve y = x³ at point P(2,8), this form will allow us to easily express the tangent line's equation.

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