Name: _________________________________________
Date: ________________
DO NOW: Find the derivative of each function.
f ( x) 
f ( x)   3 x – 8 x  x  1
2
2
x2
x2
f ( x)   x 4  3x 2  x
Today we are going to talk about a bunch of Real World Problems related to Derivatives:
Things we have already talked about are:
The AVERAGE VELOCITY of an object over time is the total change in position divided by the total change
in time.
𝑣𝑎𝑣 =
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑡𝑟𝑎𝑣𝑒𝑙 𝑡𝑖𝑚𝑒
=
∆𝑠
∆𝑡
=
𝑠(𝑡+∆𝑡)−𝑠(𝑡)
∆𝑡
The VELOCITY of an object is the first derivative of 𝑠 with respect to 𝑡; 𝑣 =
𝑑𝑠
𝑑𝑡
= lim
𝑠(𝑡+∆𝑡)−𝑠(𝑡)
∆𝑡
∆𝑡→0
.
The SPEED is the absolute value of velocity. It is ALWAYS positive, whereas, velocity indicates direction.
NEW THINGS we are going to talk about today are:
𝑑𝑣
𝑑2 𝑠
ACCELERATION is the rate of change in velocity. It is the derivative of the velocity 𝑎(𝑡) = 𝑑𝑡 = 𝑑𝑡 2 (Or
the second derivative of s with respect to t.
FORMULAS we will be working with:
FREE-FALL FUNCTION:
𝑠(𝑡) =
−1
𝑔𝑡 2
2
FREE-FALL C ONSTANTS ON EARTH
Calculus
+ 𝑣0 𝑡 + ℎ0 , s (t )  16t 2  v0t  h0
𝑔 = 32
𝑓𝑡
𝑠𝑒𝑐 2
𝑚
= 9.8 𝑠𝑒𝑐 2
*who wants to be a fighter jet pilot?*
Moskovitz
Name: _________________________________________
Date: ________________
Relationship between Position s (t), velocity v (t) and acceleration a (t).
S(t) = _______________________________________________________
In the good old days we remember this as a nice _________________________.
But today we are going to use Calculus to discover that in the world is a little more
complicated than just distance covered over time!
(a) Velocity: the rate of change of position is _____________________ (speed with direction)
Key points:
i. positive velocity:
ii. negative velocity:
iii. velocity = zero:
(b) Acceleration: the rate of the change of position (ex: speeding up, speeding down) is _________
Key points:
i. positive acceleration:
ii. negative acceleration:
iii. acceleration = zero:
In physics, the height of a ball through straight up in the air with an initial speed of 80
ft/sec from a rooftop of 96 feet high is given by the function s(t )  16t
2
 80t  96 where t
is the elapsed time in the air.
a. When does the ball strike the ground?
b. At what time will the ball pass the rooftop
on the way down?
c. What is the average velocity of the
d. What is the velocity of the ball at t =1.5?
ball from t = 0 to t = 2?
Calculus
Moskovitz
Name: _________________________________________
Date: ________________
BUT… what if we wanted to explore the understanding of acceleration at any given moment in time?
What can you already figure out about the ball in terms of velocity and acceleration from your own
world experiences of it? _______________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Sketch the graph of the path of the ball from the example above label axis appropriately:
Let’s make a chart of the balls Velocity between
t = 0 t0 t = 6 seconds. What do you notice
about the velocity of the ball over time?
Now let’s make a chart of the acceleration of
the ball between t = 0 and t = 6 seconds. What
do you notice about the relationship between
the velocity and acceleration?
Time (seconds)
Time
(seconds)
Calculus
V(t)
A(t)
Moskovitz
Name: _________________________________________
Date: ________________
Interesting huh? Write in your words what you just learn about the relationship between velocity and
acceleration? _________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Can you come up with a scenario in your life where you can relate the idea of “speeding up” and/or
“slowing down”? Describe it here: ________________________________________________________
_____________________________________________________________________________________
Now think about a scenario where the DIRECTION you are going matters. Remember that Velocity also
determines DIRECTION. When might you be going backwards? _________________________________
When else do you “change direction”? _____________________________________________________
Let’s do an example together:
Graph
s(t )  t 3  6t 2  3t  8
(a) When does the velocity equal to zero?
(b) Let’s say this is the path of a particle moving left and right instead of up and down, when
would it be moving to the left?
(c) When will it be moving to the right?
(d) When will the particle change directions?
(e) Let’s find the total distance traveled of this particle from 0 to 5 seconds.
Calculus
Moskovitz
MOVING LEFT AND RIGHT:
A particle moves in such a way that its position moves along the line
bg
y  2 is given by:
s t  3t 3  30t 2  64t  57 for 0  t  10 where t is in seconds and position is in meters.
Sketch:
(A) Find the equation for the velocity and the acceleration of the particle.
(B) At what time(s) does the particle have a velocity of zero? How would you interpret each one of those
times?
bg
(C) At time
t  2 is s t increasing or decreasing (velocity)? How quickly?
(D) At time
t  2 is the particle's speed increasing or decreasing (acceleration)? At what rate?
(E) During the interval, over which times is the particle moving to the right?
(F) Moving to the left?
(G) How many meters did the particle move from 0 to 10.
Calculus
Moskovitz
You Try: A ball is through vertically upward form ground level with an initial velocity of 19.6 meters
per second. The distance function (in meters) of the ball above the ground is s(t )
 4.9t 2  19.6t  10 .
a. What is the velocity of the ball at 1 second?
b. When will the ball reach the highest point? (What does that mean about the velocity?)
c. What is the maximum height of the ball?
d. What is the acceleration of the ball at time t?
e. How long is the ball in the air for?
f. What is the velocity of the ball as it hits the
ground?
g. When does the ball change direction? (When will this happen in general?)
i. What is the total displacement of the ball?
j. What is the acceleration of the ball t = 1? What does this say about the velocity of the ball?
k. What is the acceleration of the ball t = 5? What does this say about the velocity of the ball?
Calculus
Moskovitz
A particle moves along the line y  2 in such a way that its displacement from the y-axis is given by:
bg
s t  t 3  6t 2  9t for 0  t  6 where t is seconds and distance is in cm.
(A) Find the equation for velocity of the particle.
(B) Find the equation for the acceleration of the particle.
(C) At what time(s) does the particle have a velocity of zero?
(D) At what time(s), if any, does the particle change direction?
(E) When is the particle moving to the left?
bg
bg
(F) At time t  4 find the rate of change of s t and decide if s t increasing or decreasing.
(G) At time t  3 is the particle speeding up or slowing down? Explain.
Calculus
Moskovitz