AP Calculus AB Unit 1.1 Notes: Rates of Change & Limits | Latest Updated 2026–2027 | Complete Class Notes & AP Exam Review

1.1 Notes Day 1 Rates of Change and Limits


Objective: Why are limits necessary and what are the methods we use to analyze limits?



1. A rock breaks loose from the top of a tall cliff and falls meters in the first t seconds.
What is the average speed of the rock during the first two seconds of fall?


flo) =
4.9107m =
4.9 -0m =
Om


f / 2) 4.9125m 4.9 4m 19.6m
-



= = =




ftp.jf#=19.bm-0m-.l9j6m- .9.8y2s-0s1y fH+h-ft)-
v=g¥ =




2. A rock breaks loose from the top of a tall cliff and falls meters in the first t seconds.
What is the speed of the rock at ?

=


him H2t%-
.
=




49ltii-nki-D-4.cl#=1im4.9h*+l9.6h
+19-6-19-6
Ling h→o he

19.6 ≈
=



% 4.9h +19.6 =
4.9 () o +19.6 =




Instantaneous rate of change is the limit of the difference quotient:

, Evaluate using the limit of a difference quotient:

3.

xh +4h:* -3h -44:*
=
tim
h→o
41×+42-31×+4 -4×2+311=4%4*+8
h
-
h

=
time
h→0
8*44×-3*-1 =



,g 8×+4 h 3--8×+4.0-3
-
= 8×-3


4.
Es
¥-5 Iim €+145 ¥-5)(Fx+4hT +

lim 4×744-5
-



-




h(F4ht Es
=


↳°
h→◦ h +




4- 4-
Iim 4*+441.5-44+-15 = =




254×-7
= =



↳° 44×7+44-1=4×-5 FE Es +



5.



=



h→O
Iim =
thing 3↓É3
= limzxf-zn.LT?sI# -




3¥ }3%
.
=


him 6×-3*(3×-7354) ¥
h→0
h

q¥+1b
6. =




"ñf¥-j")
>




=




-212×+2ht
D) th =




Limo -¥¥¥I+D¥
thing
-




=
(2×+1) (2xt2h+D



=

4×4+-4×+1