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Create 2 points using the given x values. Average rate of change is just the slope of the line that connects the points. = −4 Answer: Average Rate of Change = ...
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1.4 Rates of Change (Minimum Homework: 1, 3, 5, 7, 9, 11, 15, 19, 21, 25)
Average rate of change:
The average rate of change of function 𝑓 over the interval (𝑎, 𝑏) is
given by this equation:
𝒇(𝒃)−𝒇(𝒂)
𝒃−𝒂
changes per unit, on average, over that interval.
connecting the interval's endpoints on the graph of the function.
Average rate of example 1 :
Find the average rate of change for each function over the given
interval. Sketch a graph to model your answer. (You may use your
calculator obtain the graph, be sure to label the necessary points.)
2
Create 2 points using the given x values.
2
Creates the point: ( 1 , − 3 )
2
Creates the point: ( 3 , − 11 )
Average rate of change is just the slope of the line that connects the
points.
− 11 −(− 3 )
3 − 1
− 8
2
Answer: Average Rate of Change = −𝟒
Average rate of change example 2 :
At 10 AM a car’s odometer read 10,300 miles. At noon, the car’s
odometer read 10,420 miles. What is the car’s average rate of change
measured in miles per hour?
We need to create two points.
Since we are asked to find the average rate of change in miles per hour
𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the points must be time in hours (hours are
mentioned second)
𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the points must be distance in miles (miles are
mentioned first)
These are the points needed: ( 10 , 10300 ) ( 12 , 10420 )
10420 − 10300 (𝑚𝑖𝑙𝑒𝑠)
12 − 10 (ℎ𝑜𝑢𝑟𝑠)
120 𝑚𝑖𝑙𝑒𝑠
2 ℎ𝑜𝑢𝑟𝑠
Answer: Average Rate of Change (velocity): 𝟔𝟎 𝒎𝒑𝒉
function changes over an interval.
between two values of x. It gives no specific information in-
between the two values of 𝑥.
▪ That is, we have no idea how the function behaves at any
specific instant in the interval between the given values of x.
(velocity) was 60 𝑚𝑖𝑙𝑒𝑠 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟 over the 2-hour trip. This is an
average speed for the entire trip. This does not mean the car
traveled at precisely 60 𝑚𝑝ℎ for the entire trip. This is just the
average speed. In fact, it likely went faster than 60 𝑚𝑝ℎ at times
and slower than 60 𝑚𝑝ℎ at other times. The car could have been
stopped for a chunk of time.
▪ We need to calculate the car’s instantaneous rate of
change to know its speed (velocity) at a specific
moment.
Calculus is needed to compute instantaneous rate of change.
Instantaneous rate of change of a function 𝑓 at 𝑥 = 𝑎 : Is the average
rate of change of the function 𝑓 at 𝑥 = 𝑎.
of change over the interval (𝑎, 𝑎 + ℎ).
limit as ℎ 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 0.
change at a single point, since h is changed to zero and
we get the interval
This is the formula to compute instantaneous rate of change of a
function 𝑓 𝑤ℎ𝑒𝑛 𝑥 = 𝑎.
𝐼𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 = lim
ℎ→ 0
𝑓(𝑎+ℎ)−𝑓(𝑎)
(𝑎+ℎ)−𝑎
= lim
ℎ→ 0
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
▪ Instantaneous rate of change is a measure of the slope of the line
connecting the points: (𝑎, 𝑓
We will use the word DERIVATIVE very often this semester.
same definition and they are interchangeable words.
instantaneous rate of change of the function 𝑓 at 𝑥 = 𝑎
This is the formula to compute derivative of a function 𝑓 𝑎𝑡 𝑥 = 𝑎.
Derivative = 𝑓
′
(𝑎) = lim
ℎ→ 0
𝑓
( 𝑎+ℎ
) −𝑓(𝑎)
(𝑎+ℎ)−𝑎
= lim
ℎ→ 0
𝑓
( 𝑎+ℎ
) −𝑓(𝑎)
ℎ
′
points: (𝑎, 𝑓
I nstantaneous rate of change example 2:
A pebble is dropped from a cliff, 288 𝑓𝑜𝑜𝑡 cliff. The pebble takes 3
seconds to hit the ground.
The formula: f ( t )= 288 − 32 𝑡
2
Can be used to calculate the pebbles
height of the ground in 𝑓 feet 𝑡 − 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 after it is dropped.
a) Calculate the average rate of change (average speed) in feet per
second of the pebble for the 3 seconds it takes to hit the ground.
We need to create two points.
Since we are asked to find the average rate of change (velocity) in feet
per second
𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the points must be time in seconds (hours are
mentioned second)
𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 of the points must be height in feet (feet are
mentioned first)
These are the points needed:
(at 0 seconds the pebble is 288 feet high)
( 3 , 0 ) (at 3 seconds the pebble is on the ground.
0 − 288 (𝑓𝑒𝑒𝑡)
3 − 0 (𝑠𝑒𝑐𝑜𝑛𝑑𝑠)
− 288 𝑓𝑒𝑒𝑡
3 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
Answer: Average Rate of Change (average speed):
−𝟗𝟔 𝒇𝒆𝒆𝒕 𝒑𝒆𝒓 𝒔𝒆𝒄𝒐𝒏𝒅 (answer is negative since the pebble is falling)
b) Calculate the instantaneous rate of change in feet per second
(velocity) of the pebble at t = 3 seconds.
Create 2 points
Use 𝑥 = 3 as the x-coordinate of the first point
Find the y-coordinate of the first point:
2
First point: ( 3 , 0 )
Use 3 + ℎ as the x-coordinate of the second point.
Find the y-coordinate of the second point:
2
2
2
2
Second point: ( 3 + ℎ, −192ℎ − 32 ℎ
2
lim
ℎ→ 0
192ℎ− 32 ℎ
2
− 0 (𝑓𝑒𝑒𝑡)
ℎ (𝑠𝑒𝑐𝑜𝑛𝑑𝑠)
′
= lim
ℎ→ 0
ℎ(− 192 −32ℎ)(𝑓𝑒𝑒𝑡)
ℎ (𝑠𝑒𝑐𝑜𝑛𝑑𝑠)
= lim
ℎ→ 0
Answer: The pebble’s velocity (instantaneous rate of change) is
−𝟏𝟗𝟐 𝒇𝒆𝒆𝒕 𝒑𝒆𝒓 𝒔𝒆𝒄𝒐𝒏𝒅 when it hits the ground. (The negative sign
indicates the pebble is falling.)
(Minimum Homework: 1, 3, 5, 7, 9, 11, 15, 19, 21, 25)
#1-8: Find the average rate of change for each function over the given
interval. Sketch a graph to model your answer. (You may use your
calculator obtain the graph, be sure to label the necessary points.)
f(x) = √𝑥 − 5 between x = 9 and x = 14
f(x) = √
𝑥 + 2 between x = 2 and x = 7
Answer: Average rate of change =
1
5
(Minimum Homework: 1, 3, 5, 7, 9, 11, 15, 19, 21, 25)
#1-8: Find the average rate of change for each function over the given
interval. Sketch a graph to model your answer. (You may use your
calculator obtain the graph, be sure to label the necessary points.)
t
between t = 0 and t = 2
t
between t = 0 and t = 1
Answer: Average rate of change = 2
(Minimum Homework: 1, 3, 5, 7, 9, 11, 15, 19, 21, 25)
After 6 hours, he is at an altitude of 700 feet. What is the average rate
of change in the climber’s height?
after he entered the water and 50 feet below the surface after 40
seconds. What is the scuba divers’ average rate of change in the diver’s
depth measured in feet per second? (write answer as a reduced
fraction)
Answer: The scuba divers’ average rate of change is
2
3
