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.

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. For the function f(x) = x², the derivative f'(x) can be calculated using the power rule, which states that the derivative of x^n is n*x^(n-1).

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: y - f(a) = f'(a)(x - a), where (a, f(a)) is the point of tangency and f'(a) is the derivative at that point.

Point-Slope Form

Point-slope form is a way to express the equation of a line using a specific point on the line and its slope. The general formula is 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 and point of tangency are known.