1. 2.6 Instantaneous Rates of Change 1 We have established a graphical interpretation of the derivative as the slope of a tangent line. Now, we want to understand this notion in connection to other subjects and applications. Consider dimensions: Suppose that function y=f(t) gives a distance in meters, and its argument, t, denotes time in seconds. Chose first a linear dependence f(t), say, f(t)=2t+1. This is a line, and its slope is The slope M is just the velocity of the object. More generally, the slope is the rate of change of the function with respect to the argument. The derivative defines the slope in this and more complex cases, but since the slope may change, we need to define the derivative as instantaneous rate of change.

2. 2 Velocity: If s denotes distance and t denotes time, then Example: Describe the motion of a particle moving in the vertical direction according to the equation Solution: We find the velocity first: Next, construct a time-table, i.e., estimate values of s and v at sample points... At what instants the particle moves up? Down? At what instants it turns around (solve v=0)?

3. 3 Acceleration: If v denotes velocity and t denotes time, then Example (cntd): We have found Acceleration now is meaning that the velocity decreases by 2 m/s every second. Note: Speed is the absolute value of the velocity. Consequence: while velocity increases, given a positive acceleration, the speed may decrease.

4. 4 Other rates of change: Current in a circuit: Voltage across an inductor: Current to a capacitor: (as a der. of q=Cv) Example: Find the current to a capacitor of 0.002 F after 2 seconds if the voltage is given by Solution: At t=2,

5. 5 Example: Consider a blood vessel in the shape of a cylindrical tube of radius R and length l. Because of the friction at the walls, the velocity of the blood is greatest in the center of the cylinder. The relationship between the velocity and the radius is given by the law of laminar flow (Poiseuille, 1840): where,  is the viscosity, and P is the pressure difference. Find the instantaneous rate of change of the velocity with respect to r (velocity gradient) at  =0.02, R=0.01cm, r=0.005cm, l=1cm, and P=4000dynes/[cm  cm]. J. Stewart, Calculus

6. 6 Homework Section 2.6: 7,9,15,17,19,23,25,27.