Slopes and Tangent Lines

image of a secant line derivative on graph Defining and finding the slope of a curve at a point are often referred to as key ideas of calculus. It is from the slope of a line that it is possible to find the slope of a curve, and from the slope of a curve it is possible to find a tangent line to the curve. It is the slope of the tangent line at a point on the curve that forms the basis of the derivative concept.

Slope of the Secant Line

Approximating the slope of a curve at a point P on the curve can be accomplished by taking a point P and another point Q on the curve that is relatively close to P and computing the slope of the line connecting them. When any two points on a curve are connected, a secant line is produced. Given P(x, f(x)) and nearby point Q(a, f(a)), the slope of the secant line is denoted by equation image indicator $LaTeX: m_{\sec}=\frac{\Delta y}{\Delta x}=\frac{f\left(x\right)-f\left(a\right)}{x-a}$ . Another expression for the slope of a secant line can be derived by letting h = x - a, which means x = a + h. The slope of the secant line above then becomes $LaTeX: m_{\sec}=\frac{f\left(a+h\right)-f\left(a\right)}{h}$ . This ratio equation image indicator $LaTeX: \frac{f\left(a+h\right)-f\left(a\right)}{h}$ is called the difference quotient. It measures the average rate of change of the function f and can be interpreted as the slope of a secant line.

View the presentation below from the beginning to 12:33.

Tangency to a Curve

One of the classic problems in calculus is the tangent line problem. This problem has been studied by many mathematicians since Archimedes explored the question in Antiquity.

Defining Slopes and Tangent Lines

image of curvy road into tunnel A line that touches a curve at a point and has the same slope as the curve at the point is a tangent line. In practical terms imagine that it is dark outside and you are looking down out of an airplane window at a curvy highway. You see a car's headlights following the highway and notice that when the car reaches a certain point on one of the road's curves, the headlights point alongside the road in a straight path rather than along the curvy road. You realize that the car has just driven off the road in the direction of a tangent line.

image of a tangent line on a graph

A tangent line to a curve can be thought of as the limiting position of the secant line. More formally, the tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope equation image indicator $LaTeX: m=\lim_{x\to a}\frac{f\left(x\right)-f\left(a\right)}{x-a}$ , provided this limit exists. As with the slope of the secant line, another expression for the slope of a tangent line can be derived by letting h = x - a, which means x = a + h. Notice that as x -> a, x - a -> 0 and h -> 0. Then the slope of the tangent line above becomes equation image indicator $LaTeX: m=\lim_{h\to 0}\frac{f\left(a+h\right)-f\left(a\right)}{a}$ . Since the slope of the tangent line depends on both the function f and the number a, we indicate this by writing $LaTeX: m=f'\left(a\right)=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$ . When we replace a by the more generic x, we are defining a new function equation image indicator $LaTeX: m=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$ .

View the presentation below on how a secant line becomes a tangent line.

Slopes and Tangent Lines: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of slopes and tangent lines.

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