22.4 Rates of Change & Tangent Lines What you’ll learnAverage rate of changeTangent to a curveSlope of a curveNormal to a curve (perpendicular to tangent)Speed Revisited
3Example 1 Finding Average Rate of Change Finding the average rate of change of a function over an interval is simply finding the slope of the line containing the endpoints of the interval.Ave. Rate of change over the interval [a, b] = f(b) – f(a)b – aFind the average rate of change of f(x) = x3 – x over the interval [1, 3]You try exercise #1.
4Example 2 Growing Drosophila in a Laboratory When looking at a graph you can compute the “average rate of change” from P to Q by identifying and using their coordinates. This is also known as finding the slope of the secant line PQ.We can always think of an average rate of change as theslope of a secant line.Given P(23, 150) and Q(45, 340), find the average rate of change from P to Q.What is the slope of the secant line PQ?Examine the graph on page 87, is our answer reasonable?
5Tangent Line to a CurveWe define the rate at which the value of the function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a.How do we find the slope?Start with the slope of the secant through P and a point Q nearby on the curve.Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. (the derivative at P – use slope formula and pts (a, f(a)) and (a+h, f(a+h))We define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.Method by Pierre Fermat 1629
6A line is tangent to a curve at the point P if the following 3 conditions are met: P is both on the curve and on the lineThe curve is smooth at the point P. (For example, if the curve is the graph of y = f(x), then f must be differentiable at the x-coordinate of P.)3. The curve lies on one side of the line within some punctured neighborhood of P (a little circle, excluding P itself).
7Example 3 Finding Slope and Tangent line Find the slope of the parabola y = x2 at the point P(2,4). Write an equation for the tangent to the parabola at this point.HOW?Find slope by finding the limit of y = x2 at x = 2.the difference quotient of f at a2. Use the slope and P(2,4) to write the equation in point slope form. Simplify this to slope intercept form.
8Example 4 Exploring Slope & Tangent Let f(x) = 1/xFind the slope of the curve at x = a.Where does the slope equal -1/4?What happens to the tangent to the curve at the point (a, 1/a) for different values of a?
9Example 5 Finding a Normal Line (a line perpendicular to the tangent) Write an equation for the normal to the curve f(x) = 4 - x2 at x = 1.Find slope, the limit at x=1.Find the opposite reciprocal slope to use in the normal equation.Find f(1) to get a point (1, f(1))Write an equation in point slope form and simplify it to slope intercept form.Hint: Anytime you are asked to find both a tangent line equation and a normal line equation, graph them both in a square viewing window to verify your results.
10Example 6 Investigating Free Fall A body’s average speed along a coordinate axis for a given period of time is the average rate of change of its position y = f(t). Its instantaneous speed at any time t is the instantaneous rate of change of position with respect to time t, or♦ A rock breaks loose from the top of a tall cliff with position given from the equation y = 16t 2.Find the speed of the falling rock at t = 1 sec.How? Find the limit at t = 1 – the rate of change of the position of the rock at the instant t=1 second.