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) = x²; a=3

Verified step by step guidance

1. Step 1: Find the derivative of the function f(x) = x^2. The derivative, f'(x), represents the slope of the tangent line at any point x on the graph of f.
2. Step 2: Calculate the derivative f'(x) = 2x. This is done by applying the power rule, which states that the derivative of x^n is n*x^(n-1).
3. Step 3: Evaluate the derivative at the given point a = 3 to find the slope of the tangent line. Substitute x = 3 into f'(x) to get f'(3) = 2*3.
4. Step 4: Determine the y-coordinate of the point on the graph by evaluating f(a). Substitute x = 3 into f(x) to get f(3) = 3^2.
5. Step 5: Use the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point (3, f(3)), to write the equation of the tangent line.