SOLVING PROBLEMS WITH CALCULUS
26 MAY 2014
Lesson Description
In this lesson we:


Focus on real world applications of calculus.
Determine the maximum or minimum volume or surface area of certain shapes or other
scenarios such as cost, perimeter and so forth.
Summary
Volume Formula
Cylinder :
Rectangular Prism :
Triangular Prism :
Surface Area Formula
Cylinder :
Rectangular Prism :
Triangular Prism :
To calculate the Maximum or Minimum value let
Test Yourself
Question 1
A.
B.
C.
D.
10
0
-10
undefined
Question 2
Determine the average gradient of graph
A.
B.
C.
D.
33
14
80
-33
, passing through
and
Question 3
The average gradient between (-2,5) and (4; ) is 2,5. Determine the value of .
A.
B.
20
C.
-20
D.
5
Question 4
The derivative of
is:
A.
B.
C.
D.
Question 5
Determine the function of
such that
A.
B.
C.
D.
Question 6
Find the equation of the tangent to
which is parallel to
A.
B.
C.
D.
Question 7
Given the equation of the cubic function:
, the
intercepts are:
A.
B.
C.
D.
Question 8
Given the equation of the cubic function:
stationary points.
A.
B.
C.
D.
, determine the coordinates of the
Question 9
A group of Physics students launch a projectile straight up in the air. The distance travelled,
measured in meters above ground level, is given by
, in seconds.
What is the instantaneous speed of the projectile when
A.
B.
C.
D.
seconds?
170
17
1955
30
Question 10
A metal frame consisting of four congruent rectangles and a semi circle must be manufactured. The
total length of the material that is to be used for the whole frame is 36 meters.
The area of the frame is represented by the formula:
Determine the value of
meter .
for which the frame will have a maximum area.
A.
B.
C.
D.
Improve your Skills
Question 1
One side of a rectangle is
The perimeter of the rectangle is
mm in length.
mm. Find
so that the area of this rectangle is at a maximum.
Question 2
Bricks are to be painted with a special paint for use under water. The bricks are rectangular in shape
and the length of each brick is three times its breadth. The volume of cement in each brick is 972
.
If h is the height of a brick and
as its breadth :
h
a)
Express h in terms of .
b)
Express the total surface area of each brick in terms of .
c)
Calculate the dimensions of a brick which will minimize the amount of paint required.
Question 3
The total exterior surface area of an empty cylinder, which is closed on one end , is 462
a)
Find the height
of the cylinder in terms of
b)
Hence find the value of r for which the volume of the cylinder is a maximum
.
and the radius (r).
Question 4
In the diagram, the graphs of
and
are shown. Line segment RQ is parallel
to the y-axis. RQ varies in length between the two points of intersection of the graphs of and .
Determine the maximum length of RQ.
Question 5
The voume of water in a horse trough is govourned by the formula
where is the volume of water in
and is the time in minutes.
a)
Determine the volume at
b)
Determine the rate of change of volume at
c)
Determine the time at which the volume is a maximum.
for