Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = √3x; a= 12

Verified step by step guidance

1. Step 1: Identify the function and the point of tangency. The function given is \( f(x) = \sqrt{3x} \) and the point of tangency is \( (a, f(a)) \) where \( a = 12 \).
2. Step 2: Calculate \( f(a) \). Substitute \( a = 12 \) into the function to find \( f(12) = \sqrt{3 \times 12} \).
3. Step 3: Find the derivative of the function \( f(x) = \sqrt{3x} \). Use the chain rule: \( f'(x) = \frac{d}{dx}(3x)^{1/2} = \frac{1}{2}(3x)^{-1/2} \cdot 3 \).
4. Step 4: Evaluate the derivative at \( x = 12 \). Substitute \( x = 12 \) into \( f'(x) \) to find \( f'(12) \).
5. Step 5: Use the point-slope form of a line to write the equation of the tangent line. The formula is \( y - f(a) = f'(a)(x - a) \). Substitute \( f(12) \) and \( f'(12) \) into this equation.

Key Concepts

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 practical terms, the derivative at a point gives the slope of the tangent line to the graph of the function 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 is derived from 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 (derivative) and the point of tangency are known. By substituting the appropriate values, one can easily derive the equation of the tangent line.