91Ó°ÊÓ

The power rule is a fundamental concept in calculus, used to find the derivative of functions that are expressed as powers. Essentially, it makes taking derivatives simple for polynomial expressions. Whether you’re dealing with a term like \(x^3\) or any variable raised to a power, the power rule states that if you have a function \(f(x) = x^n\), its derivative \(f'(x)\) would be \(nx^{n-1}\). This involves:

• Multiplying the power 'n' by the coefficient of the term.
• Reducing the power by one to form the new exponent.

In our example, with the function \(f(x) = 2x^3\), applying the power rule means taking 3 (the exponent), multiplying it by 2 (the coefficient), and reducing the exponent by 1. Therefore, \(f'(x)\) becomes \(6x^2\). The power rule helps us easily compute derivatives, which are crucial in determining tangent lines and rates of change.

A tangent line is a straight line that touches a curve at a single point, matching the curve's slope at that point. Understanding tangent lines is important in many fields, both mathematical and applied, from analyzing graphs to modeling real-world data. When you have a function like \(f(x) = 2x^3\), the tangent line at any point \((a, f(a))\) gives you an approximation of the curve near that point. The slope of this tangent line is given by the derivative, \(f'(a)\).

• First, find the derivative of the function to obtain the slope.
• Then, evaluate this derivative at the desired point to get the specific slope at that point.
• Use this slope in conjunction with the point to determine the equation of the tangent line.

In our specific case, with \(a = 10\), the slope of the tangent line is found to be 600 through the derivative \(f'(a) = 6(10)^2 = 600\). Thus, the tangent line approximates \(f(x)\) near \((10, 2000)\). It provides a linear approach to understanding how the curve behaves around that point.

The point-slope form is a formula used to find the equation of a line when you know the slope and a point on the line. It’s especially useful when dealing with tangent lines in calculus, as it makes finding linear equations straightforward. The formula is:\[ y - y_1 = m(x - x_1) \]where \((x_1, y_1)\) are the coordinates of the point and \(m\) is the slope. To use this form:

• First, determine the slope of the line using the derivative of the function.
• Identify the specific point on the curve that intersects with the tangent line.
• Substitute both the point and the slope into the point-slope formula to get the tangent line's equation.

In the case where \(a = 10\), the slope \(m\) is 600, and the point on the line is \((10, 2000)\). Thus, using the point-slope formula, you find: \[ y - 2000 = 600(x - 10) \]This calculated equation represents the tangent line, providing a clear linear relationship between \(x\) and \(y\) close to the point on the original curve.