Find the slope of a tangent line using limits

Learning Objectives Define a tangent line as the limit of secant lines. Explain the conceptual link between the slope of a secant line (average rate of change) and the slope of a tangent line (instantaneous rate of change). Set up the limit definition of the slope of a tangent line at a specific point for various functions. Use algebraic techniques, such as factoring and rationalizing, to evaluate the limit of a difference quotient. Calculate the slope of a tangent line for polynomial, rational, and radical functions at a given point. Use the calculated slope and the point of tangency to write the full equation of the tangent line. How does a speedometer calculate your car's speed at one exact moment in time? 🏎️ It's not measuring an average over a distance; it&#03...

TermDefinitionExample Secant LineA line that passes through two distinct points on a curve. Its slope represents the average rate of change between those two points.For the curve f(x) = x^2, 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 'just touches' a curve at a single point, called the point of tangency. Its slope represents the instantaneous rate of change of the function at that exact point.For the curve f(x) = x^2, the tangent line at the point (1, 1) has a slope of 2. It indicates the steepness of the parabola at that precise location. Point of TangencyThe specific point on a curve where the tangent line touches the curve.If we are finding the tangent line to f(x) = x^3 at x = 2, the point of...

Limit Definition of the Slope of a Tangent Line (h-approach) m_{tan} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} This is the primary formula used to find the slope of the tangent line to a function f(x) at the point x = a. It works by taking the slope of a secant line between (a, f(a)) and (a+h, f(a+h)) and finding the limit as the second point gets infinitely close to the first (as h approaches 0). Alternative Limit Definition of the Slope (x-approach) m_{tan} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} This is an alternative but equivalent formula. It calculates the slope of the secant line between a fixed point (a, f(a)) and a moving point (x, f(x)) as x gets infinitely close to a. It can sometimes be algebraically simpler.