Another look at D=RT If you travel 240 miles in a car in 4 hours, your average velocity during this time is This does not mean that the car’s speedometer was on 60 mph at all times; this is only your average velocity during this time interval.

2.6 Constant velocity If a car’s cruise control was set at 60 mph for 4 hours of travel, what would the shape of the graph of distance traveled to elapsed time be? The graph is a straight line and the slope of the line is 60. So in general, if the graph of distance to time is a straight line, at every instant the velocity is constantly the same, that is, at every instant the velocity is the slope of the line.

If the graph were a straight line, the answer would be 2.6 Varying velocity Now let’s consider the case when the velocity of a car is varying over time. What is the velocity of the car at time t=3? If the graph were a straight line, the answer would be the slope of the line as before. Let’s zoom-in on the graph near the point t=3.

2.6 Varying velocity continued We see that after magnifying the graph near t=3, the curve looks like a straight line between the inputs t=2.992 and t=3.004. It seems reasonable to assume that the curve behaves like the straight line and that the velocity is constant during this time interval. So at each instant during this time, which includes t=3, the velocity is the slope of this line.

2.6 Varying velocity continued Two points on the graph are approximately (2.992, 26.85) and (3.004, 27.05). Therefore the velocity at t=3 is

2.6 Varying velocity continued For the functions we are studying it can be proven that the more you zoom-in on the graph of the function at a specified input, the curve will look more and more like a particular straight line – a tangent line. Putting this all together, we define the instantaneous rate of change of a function at a specified input x=a to be the slope of the tangent line at the point ( a, f(a) ).

2.6 Instantaneous rate of change The instantaneous rate of change of a function f at the input x=a = slope of tangent line at x=a = derivative of f at x=a = This is read “f prime of a”

2.6 Example of Instantaneous Rate of Change The distance in feet traveled by a car moving along a straight road x seconds after starting from rest is given by f(x) = 2x2, 0< x <30 Use a tangent line to approximate the (instantaneous) velocity of the car at x=22. Solution: On a graph of the function, draw a tangent line at x=22. Then find its slope from any 2 points on the line.

2.6 Limit Definition of Instantaneous Rate of Change The instantaneous rate of change of a function f at the input x=a is defined by

2.6 When is a function not differentiable, that is, f ` (a) does not exist? You have learned that if the graph of a function is broken at a point, then the function is not continuous at the point. That is, the graph of a continuous function is unbroken. It can be shown that a differentiable function is continuous. This means the graph of a differentiable function must be unbroken too. But there is another requirement. A function is not differentiable wherever the graph has a sharp turning point, a cusp or a vertical tangent line at the point. The function whose graph is shown here is not differentiable at the points x= -2, x=0 and x=1.

2.6 When, What, How and By How Much A function has output “the weight of an infant w(t) in lb” and input “age t in mo”. Write a sentence to interpret (explain the meaning) of the instantaneous rate of change w`(3)=1.5. Answer: A function has output “body temperature of a patient F(t) in oF” and input “t, hours after taking a fever-reducing drug”. In a sentence, interpret F`(3) = -0.25.

2.6 Average Rate of Change The average rate of change of a function f from the input x to the input x+h, or over the interval [x, x+h], is given by In words, this is the change in the outputs divided by the change in the inputs.

2.6 Geometric Interpretation of ARC If a straight line goes thru the two points (x1,y1) and (x0, y0), then the slope of the line is given by So the average rate of change of a function from input x0 to input x1, is the same as the slope of the straight line going thru the points x0 and x1. This line is called a secant line.

2.6 Average rate of change of f between 2 inputs equals slope of a secant line

2.6 Example of Average Rate of Change The distance in feet traveled by a car moving along a straight road x seconds after starting from rest is given by f(x) = 2x2, 0< x <30 For each of the following three time intervals, calculate the average velocity of the car. [22, 23], [22, 22.1], [22,22.01]