Unit 4: Applications of Differentiation and Kinematics

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Review Sheet


Unit 4: Applications of Differentiation and Kinematics


1.

A rectangular cake dish is made by cutting out squares from the corners of a 7 inch by 10
inch rectangle piece of tin. What size squares must
be cut out in order to produce the
cake dish of maximum volume?


















2.

A manufacturer of calculators is confident that they can sell 3 000 calculators per week at
a price of $12 each and believes that reducing the price by 20 cents each will incr
ease its
weekly sales by 200 calculators. How many calculators should the company produce
each week to maximize revenue?


















3.

The following diagram shows part of the curve of a function ƒ. The points A, B, C,
D and E lie on the curve, where B
is a minimum point and D is a maximum point.











(a)

Complete the following table, noting whether ƒ′(
x
) is positive, negative or zero at the
given points.

A

B

E

f


(
x
)




(b)

Complete the following table, noting whether ƒ′′(
x
) is positive,
negative or zero at the
given points.

A

C

E

ƒ′′

(
x
)




(Total 6 marks)


4.

The displacement s metres at time
t

seconds is given by



s

= 5 cos 3
t

+
t
2

+ 10, for
t



0.

(a)

Write down the minimum value of
s
.

(b)

Find the acceleration,
a
, at time
t
.

(c)

Find the value of
t

when the
maximum

value of
a

first occurs.



5
. Graph the derivative for each of the following functions.



6.

A hot air balloon is
h

meters above the ground
t

minutes after it is released. While it is
ascending the equation connecting
h

with
t

is
2
2
100
t
t
h


.


(a)

Find the instantaneous rate of change in which the balloon gains height during the:


(i)

First minute


(ii)

The 20
th

minute





(b)

Find the rate at which the balloon is gaining height after 10 minutes.




(c)

Find the average rate of
change from the 2
nd

minute to the 12
th

minute.




(d)

Find how long the balloon takes to reach its greatest height.

7.

A p
article moves in a straight line with position, relative to some origin O, given by
1
3
)
(
3



t
t
t
s

cm, where
t

is the time in seconds

(
t
≥0).


(a)

Find the velocity and acceleration function, and draw the sign diagrams for each of them.







(b)

Find the initial conditions and describe the motion at this instant.







(c)

Find the position of the particle when changes in direction occur.







(d)

For

what time interval(s) is the particle’s speed increasing? Decreasing?



8
.
The main runway at
Concordville
airport is 2 km long. An airplane, landing at
Concordville,
touches down at point T, and immediately starts to slow down. The point A is at the sou
thern
end of the runway. A marker is located at point P on the runway.


Not to scale



As the airplane slows down, its distance,
s,
from A, is given by

s
=
c

+ 100
t



4
t
2
,


where
t

is the time in seconds after touchdown, and
c
metres

is the distance of T from A.



(a)

The airplane touches down 800 m from A, (
ie c
=

800).

(i)

Find the distance travelled by the airplane in the first 5 seconds after
touchdown.

(2)

(ii)

Write down an expression for the velocity of the airplane at time
t

s
econds after
touchdown, and hence find the velocity after 5 seconds.

(3)




The airplane passes the marker at P with a velocity of 36 m s

1
. Find

(iii)

how many seconds after touchdown it passes the marker;

(2)



(iv)

the distance from P

to A.

(3)



(b)

Show that if the airplane touches down before reaching the point P, it can stop before
reaching the northern end, B, of the runway.

(5)

(Total 15 marks)


T
P
A
B
2 km