Name: __________________________________________ Date: ______________
Do Now:
1) Find the derivative of the following functions:
A) f x

6 x
2 
5 3 x

12 x
B)
 u
3 
4 u

7
5

3 u
3
2) Find all x-values where the tangent line to

1 x
3  x
2 
13 x

7 has a slope of 2. Find the
3 equation of the tangent lines at these points. Show your work.
Reading a Velocity Graph. Think about what we talked about in terms of velocity and acceleration
yesterday!
Calculus Moskovitz

Name: __________________________________________ Date: ______________
Answer the following questions, using the concepts of derivatives and position, velocity and acceleration. Show work when necessary. You may work in groups or individually. Do not be surprised if you have a quiz on this tomorrow.
(1) A rock is thrown and its height in feet after t seconds is given by
t

t
2
.
(A) How long does it take to reach its highest point?
(B) Find its initial velocity. ________________________
(C) Find its initial height. ___________________________
(D) Find the acceleration of the rock as it gets to its original height. ___________________
(E) What is the interval that the rocks velocity is decreasing? ________________________
(F) Over what interval is the rocks height increasing? ____________________________
(2) A 45-caliber bullet fired straight up from the surface of the moon would reach a height of s

t

2
t where height is given in feet and time is given in seconds.
*set window*
(A) At what time does it reach this height?
(B) Find the highest point the bullet reaches.
(C) How fast is the bullet moving when it is 20,000 feet high?
(D) How long will it take the bullet to return to the moon's surface?
(E) What is the acceleration of the bullet at t = 2 seconds?
(F) What does the acceleration at t = 2 seconds say about the velocity of the bullet at that time?
(G) What is the total displacement of the bullet?
Calculus Moskovitz

Name: __________________________________________ Date: ______________
(3) A rock is thrown from the top of the science building. Its height is given by the equation
   
16 t
2

34 t

57 . At what height(s) does the rock have a velocity of 25 feet per second?
(4) A particle moves in such a way that its position is given by: s t bg

t
3 
t
2 
t

where t is in seconds and position is in meters.
for
(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? What is happening at each one of those times?
Make a chart for the velocity and a separate one for acceleration over the interval?
(D) At time t

is the velocity increasing or decreasing? How quickly?
(E) At time t

is the particle's speed increasing or decreasing? At what rate?
(F) What is the total displacement of the particle over the interval?
Calculus Moskovitz

Name: __________________________________________ Date: ______________
(5) A particle moves along the x-axis and its position is given by
(C) At time
( ) t t

is the particle's velocity increasing or decreasing?
11 t
3 
38 t
2 
48 t

50
(A) Find the equation for velocity of the particle. ___________________________________
(B) Find the equation for the acceleration of the particle. _____________________________
(D) What is the displacement of the particle from t = 0 to t = 4?
(6) A particle moves in such a way that its displacement from the y-axis is given by: s t bg
 
t 2 
t for
 
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? What is happening at these times?
Create a table for the velocity and the acceleration to help answer the following questions:
(E) At time
(F) At time t

find the rate of change of s t bg
and decide if s t bg increasing or decreasing. t

is the particle speeding up or slowing down? Explain.
Calculus Moskovitz