1. Table of Contents
1. Section 2.1 Rates of change and Limits

2. Rates of Change and Limits
Essential Question – How are rates of change related to limits?

3. Calculus overview 3 main concepts 2 branches
Limits
Derivatives
Integrals
2 branches
Differential – Finding tangent lines to curves
Integral – Areas under curves
Both of these branches ultimately rely on finding a limit

4. Rates of change Relationship between 2 changing quantities
Velocity (distance/time)
Infection rate of epidemic (people/month)
Mileage of car (miles/gallon)
Atmospheric pressure per altitude

5. Average vs. instantaneous
Average velocity = change in position
length of time
For example, a car travels 200 miles in 4 hrs, avg velocity = 200 miles/4 hrs = 50 mph
Can’t define instantaneous velocity as ratio because you would have to divide by length of time (which is zero)
Estimate instantaneous by computing average velocity over smaller and smaller time intervals

6. Example
Stone released from state of rest on cliff falls to earth. Estimate instantaneous velocity (Iv) at t = 0.5 s by calculating average velocity (Av) over small time intervals. S = 16 t2.
16 (the Iv) is the LIMIT of average velocity

7. Graphical Interpretation
As time interval shrinks to zero, slope of secant line approaches slope of tangent line

8. Therefore…..
Instantaneous velocity is equal to the slope of the tangent line
Average ROC = slope of secant line
Instantaneous ROC = limit of average ROC. Estimate it by computing average ROC over smaller and smaller intervals

9. Example
Given where is speed of sound in air (in m/s) and T is temperature is degrees Kelvin
a) compute average ROC over interval [273,300]

10. Example continued
b) Estimate instantaneous ROC when T = 273 K. Use left hand intervals and right hand intervals.
Left hand intervals Right hand intervals
[272.5,273] = [273,273.5] =
[272.8,273] = [273,273.2] =
[272.9,273] = [273,273.1] =
[272.99,273] = [273,273.01] =
0.605 is the estimate of instantaneous ROC when T = 273K

11. Assignment
Pg. 66: #1-15 odd, odd