Velocity & Acceleration Apps: MUST show calculus knowledge with sign analysis! Name: ________________________________ Hour:____
1. Riley throws a ball upward from the top of an 85 meter building with a velocity of 80 m/sec.
The ball’s height in meters above the ground after t seconds is: ℎ( ) = −5 + 80 + 85.
a.) Find the average velocity (average rate of change) from
= 0 seconds to = 3 seconds.
b.) Find the instantaneous velocity (instantaneous rate of
change) at time and at time = 3 seconds.
c.) When does the ball reach its maximum height?
What is the maximum height?
d.) How many second after being thrown does the ball hit
the ground?
e .) For what values of is the ball rising?
For what values of is the ball falling?
f.) Find the acceleration at time .
(include units)
g.) State the domain and range of ℎ( ).
2. Angela throws a ball upward from the top of a 45-ft hill with a velocity of 16 ft/sec.
The ball’s height in feet above the ground after t seconds is: ℎ( ) = −16 + 16 + 45.
a.) Find the average velocity (average rate of change) from
= 1 seconds to = 2 seconds.
b.) Find the instantaneous velocity (instantaneous rate of
change) at time and at time = 0.5 seconds.
c.) When does the ball reach its maximum height?
What is the maximum height?
d.) How many second after being thrown does the ball hit
the ground?
e .) For what values of is the ball rising?
For what values of is the ball falling?
f.) Find the acceleration at time . (include units)
g.) State the domain and range of ℎ( ).
3. Alesia throws a ball upward from the top of an 80-ft building with a velocity of 64 ft/sec.
The ball’s height in feet above the ground after t seconds is: ℎ( ) = −16 + 64 + 80.
a.) Find the average velocity (average rate of change) from
= 0 seconds to = 2 seconds.
b.) Find the instantaneous velocity (instantaneous rate of
change) at time and at time = 1 seconds.
c.) When is the velocity equal to 0? What does this mean?
d.) What is the ball’s maximum height above the ground?
e .) When does the ball hit the ground?
f.) For what values of is the ball falling?
g.) What is the acceleration at time t?
4. Mike throws a ball upward with a velocity of 48 ft/sec.
The ball’s height in feet above the ground after t seconds is: ℎ( ) = −16
a.) Find the velocity at time and at time = 1 second.
b.) When is the velocity equal to 0? What does this mean?
c.) What is the ball’s maximum height above the ground?
d .) For what values of is the ball falling?
e.) What is the acceleration at time t?
f.) State the domain and range of ℎ( ).
g.) Sketch the function ℎ( ).
+ 48 .
5. Will flies a helicopter vertically from the top of a 98 meter building so that its height in meters above the ground after t seconds is:
ℎ( ) = −4.9 + 49 + 98.
a.) Find the average velocity (average rate of change) from
= 0 seconds to = 2 seconds.
b.) Find the instantaneous velocity (instantaneous rate of
change) at time and at time = 1 seconds.
c.) When is the velocity equal to 0? What does this mean?
d.) What is the helicopter’s maximum height above the
ground?
e .) When does the helicopter reach the ground?
f.) For what values of is the helicopter descending?
For what values of is it ascending?
g.) What is the acceleration at time t?
h.) Sketch the function ℎ( ).
6. Sean shoots an arrow upward from the bottom of a canyon that is 336 ft below the edge of a cliff. If the initial velocity of the
arrow is 160ft/sec, then ℎ( ) = −16 + 160 gives the arrow’s height, in feet, above the bottom of the canyon t seconds after he
shoots the arrow.
a.) In how many seconds does the arrow first pass the
cliff’s edge?
b.) How high does the arrow travel before it starts to
descend? Must demonstrate calculus knowledge!
c.) If it lands on the cliff’s edge, how long was the arrow in
the air?
d.) Using part c, at what velocity does the arrow hit the
cliff’s edge?
e.) What is the acceleration at time t?
Find the derivative of each function.
7.
( )=5
−8
−9
+ 10 − 16
8.
( ) = −12
+9
− 25 + 6
9.
( )=8
10.
( ) = 1000
11.
( )=
12.
( )=
13.
( )=7
14.
( ) = 12
15.
( )=
16.
( )=
17.
( )=
18.
( )=
19.
( ) = 30
20.
( ) = −8√
21.
( ) = 20√
22.
( )= √
23.
( ) = (4
24.
( ) = (5
− 5)(2 + 9)
Demonstrate Product Rule!
25.
( ) = (2
−
)(8
− 2)
26.
( ) = ( − 2)(3
Demonstrate Product Rule!
Demonstrate Product Rule!
+ 4)(
Demonstrate Product Rule!
+ 8)
27.
( )=
Demonstrate Quotient Rule!
28.
( )=
31.
( ) = (3
34.
( ) = √5 + 3
−5 )
Chain Rule!
29.
( )=
32.
( ) = (8
35.
( ) = √3
30.
( )=
+ 1)
33.
( ) = √2 − 5
+1
36.
( )= √ +1
− 1)