2.1b- Limits and Rates of Change (day 2)

3) Substitution Sometimes the limit of f(x) as x approaches c doesn’t depend on the value of f at x =c, but other times the limit is exactly f(c). When that happens, the limit can be evaluated by direct substitution. Example of direct substitution:

Example Direct Substitution: Ex. find the limit 1) 2) 3) 4) * We have an issue here

4) Analytically (Algebraically) We get the form with direct substitution Simplify it using algebra: If f(x) = g(x) for all points but one, then their limits are the same. (simplify the function) Ex.

Ex1. Analytically: We get the form with direct substitution Ex2.

Example 3 Analytically: Example 4:

5) Sandwich (Squeeze) Theorem: There is a great video on the proof on why this works at

#5: Sandwich Theorem: Two Special Trig Limits & or

Ex. Using the special limits

Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: (Average rate of change) If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed. Rates of Change Speed = |velocity| What is your average speed?

Example of Free Falling Object Example A ball is thrown straight down from the top of a 220- foot building with an initial velocity of –22 feet/second. What is the average speed (rate of change) after 3 seconds: Describes the motion of the object What is the instantaneous speed (instantaneous rate of change) at 3 seconds?

for some very small change in t where h = some very small change in t To find Instantaneous rates of change using limits, we can analyze the behavior with extremely small changes in the time using a table method in our calculator

2) Find the speed at 4 seconds A calculator is dropped from a window at the top of the empire state building. It falls in t seconds. Find the calculator’s 1) Average speed during the first 4 seconds

the end p. 62 (7-30, 47-52, 57)