Find equations of tangent lines using limits

Learning Objectives Define a tangent line as the limit of secant lines. Calculate the slope of a tangent line at a specific point using the limit definition of the derivative. Apply the limit definition to find the slope of tangent lines for polynomial, rational, and radical functions. Write the equation of a tangent line to a function at a given point using the point-slope form. Interpret the slope of the tangent line as the instantaneous rate of change of the function. Identify and correct common algebraic errors that arise when using the limit definition. Ever wondered how a self-driving car knows its exact speed at a single instant? 🚗 It's all about finding the slope of its position-time graph at one specific point! This tutorial bridges the gap between the averag...

TermDefinitionExample Secant LineA line that intersects a curve at two distinct points. Its slope represents the average rate of change between those two points.For the curve f(x) = x², the line passing through the points (1, 1) and (3, 9) is a secant line. Its slope is (9-1)/(3-1) = 4. Tangent LineA line that touches a curve at a single point and has the same direction as the curve at that point. Its slope represents the instantaneous rate of change.For the curve f(x) = x², the tangent line at the point (1, 1) has a slope of 2. It 'skims' the parabola at that exact point. Average Rate of ChangeThe slope of the secant line connecting two points (a, f(a)) and (b, f(b)) on a function's graph. It is calculated as Δy/Δx.The average speed of a car that travels 120 km in 2 hours...

Slope of a Secant Line m_{sec} = \frac{f(a+h) - f(a)}{h} This formula calculates the average rate of change of a function f(x) over a small interval of length 'h' starting at x = a. It is the slope of the line connecting the points (a, f(a)) and (a+h, f(a+h)). Slope of a Tangent Line (Limit Definition) m_{tan} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} This is the formal definition of the slope of the tangent line to f(x) at the point x = a. By taking the limit as h approaches 0, we find the exact slope at that single point. This is also the definition of the derivative of f(x) at a. Equation of the Tangent Line y - f(a) = m_{tan}(x - a) After calculating the slope of the tangent line (m_tan) at x = a, use this standard point-slope formula to write the final...