11.3 The Tangent Line Problem
Calculus is a branch of mathematics that studies rates of change of functions. To determine the rate at which a graph rises or falls at a single point, the slope of the tangent line could be found at that point. The tangent line to the graph of a function f at a point
P(x1 , y1)
is the line that best approximates the slope of the graph at the point.
P(x1 , y1)
is the line that best approximates the slope of the graph at the point.
From geometry, a line is tangent to a circle when the line intersects the circle at only one point. Tangent lines to non circular graphs, can intersect the graph at more than one point.
Slope of a Graph
Because a tangent line approximates the slope of a graph at a point, the problem of finding the slope of a graph at a point is the same as finding the slope of the tangent line at the point.
Slope of a Graph
Because a tangent line approximates the slope of a graph at a point, the problem of finding the slope of a graph at a point is the same as finding the slope of the tangent line at the point.
Slope and the Limit Process
A more systematic method of approximating tangent lines makes use of a secant line through the point of tangency and a second point on the graph. If (x, f(x)) is the point of tangency and (x + h, f(x + h)) is a second point on the graph of f, then the slope of the secant line through the two points is given by |
The right side of this equation is called the difference quotient. The denominator h is the change in x, and the numerator is the change in y. The beauty of this procedure is that more and more accurate approximations of the slope of the tangent line are obtained by choosing points closer and closer to the point of tangency.
Finding the Slope of a Graph
Finding a Formula for the Slope of a Graph
The Derivative of a Function
The derived functions called the derivative of f at x. It is denoted by f′(x), which is read as f prime of x. The derivative of f at x is given by f'(x) = lim h−>0 (f(x + h) - f(x))/h
The derived functions called the derivative of f at x. It is denoted by f′(x), which is read as f prime of x. The derivative of f at x is given by f'(x) = lim h−>0 (f(x + h) - f(x))/h