Find an equation of the line tangent to the given curve at a.
y = 2x² / (3x - 1); a = 1

Verified step by step guidance

1. Step 1: Identify the function and the point of tangency. The function given is \( y = \frac{2x^2}{3x - 1} \) and the point of tangency is at \( a = 1 \).
2. Step 2: Find the derivative of the function to determine the slope of the tangent line. Use the quotient rule: if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \). Here, \( u = 2x^2 \) and \( v = 3x - 1 \).
3. Step 3: Calculate \( u' \) and \( v' \). For \( u = 2x^2 \), \( u' = 4x \). For \( v = 3x - 1 \), \( v' = 3 \).
4. Step 4: Substitute \( u, u', v, \) and \( v' \) into the quotient rule formula to find \( y' \). Simplify the expression to get the derivative.
5. Step 5: Evaluate the derivative at \( x = 1 \) to find the slope of the tangent line at the point \( a = 1 \). Use the point-slope form of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point on the curve.

Key Concepts

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 represents the instantaneous rate of change of the function at that point, which can be found using the derivative.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is calculated as the limit of the average rate of change of the function over an interval as the interval approaches zero. For a function y = f(x), the derivative f'(x) gives the slope of the tangent line at any point x.

Point-Slope Form

The point-slope form of a linear equation is used to write the equation of a line when the slope and a point on the line are known. It is expressed as y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the point. This form is particularly useful for finding the equation of the tangent line once the slope has been determined.