ppt calculus for students

1. CENTRAL UNIVERSITY SIERRA LEONE
TASK: PRESENTATION
MODULE: CALCULUS
LEVEL: YEAR ONE
LECTURER: MR. SAMBA CONTEH
GROUP 1

2. NAMES ID NUMBER PROGRAM
Newton Tamba Sam 2024341 Computer Science
Samuel Arnold Kpoghomu 2024112 Computer Science
King Joseph Ehigie 2024111 Computer Science
Mahmoud Dukulay 2024377 Computer Science
Misheal A.P Sellu 2024100 Computer Science
Aminata Barrie 2024095 Computer Science
Osman Janneh 2024002 Computer Science
Idrissa Alimany Sesay 2024069 Computer Science
Alimamy Thoronka 2024085 Computer Science
Ibrahim Bah 2024061 Computer Science
Amara Kamara 2024004 Computer Science
Willie kinney Kargba 2024015 Computer Science
Rahuf Vandy 2024014 Computer Science

3. NAMES ID NUMBER PROGRAM
George Lansana 2024272 Computer Science
Salieu Magid Kamara 2024342 Computer Science
Memunatu Alie Kamara 2024052 Computer Science
Kadiatu B Kamara 2024059 Computer Science
Micheal G.W.A.J Bangura 20241337 Computer Science

4. RATE OF CHANGE OF QUANTITIES
This simply refers to a small change in a quantity with
respect to another quantity. The rate of change is often
but not always with respect to time. However, the rate of
change of a quantity S means the rate of change of S
with respect to time i.e.
ⅆ𝑆
ⅆ𝑡

5. Questions
1.The volume of the cube is increasing at a rate of 9 cubic
inches per second. What is the rate at which surface area is
increasing when the length of the edge of the cube is
10inches?
2.What is the rate of change of the circumference of a circle if
the radius is increasing at the rate of 0.8cm/sec?
3. A spherical balloon is being inflated and the volume is
increasing at the rate of 15cm3/sec. At what rate is the
radius increasing when it is 10cm?

6. APPROXIMATIONS (SMALL CHANGES)
Since lim
𝛿𝑥→0
𝛿𝑦
𝛿𝑥
=
ⅆ𝑦
ⅆ𝑥
, then
𝛿𝑦
𝛿𝑥
=
ⅆ𝑦
ⅆ𝑡
when 𝛿𝑥 is small.
Therefore, 𝛿𝑦 =
ⅆ𝑦
ⅆ𝑥
𝑥 𝛿𝑥
This approximation can be used to estimate small change 𝛿𝑦
in y if
ⅆ𝑦
ⅆ𝑥
can be found and the small change 𝛿𝑥 in x is given. It
can also be used to estimate percentage changes. In general,
if x is increased by p%, then 𝛿𝑥 =
p
100
𝑥 𝑋 and the
approximate percentage increase in y is
𝛿𝑦
y
𝑥 100.

7. Questions
1.The surface area of a sphere is 4𝜋𝑟2. 𝐼𝑓 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 of the
surface of the sphere is increased from 10cm to 10.1cm,
what is the approximate increase in the surface area?
2.Find the approximate value for the square root of 16.01.
3. The side of a cube increases by 5% Find the corresponding
percentage increase in the volume.

8. INCREASING AND DECREASING FUNCTIONS
The derivative of a function
ⅆ𝑦
ⅆ𝑥
, gives the gradient
function of the curve.
Where the derivative is positive i.e.
ⅆ𝑦
ⅆ𝑥
>0, the function
is increasing,
Similarly, where the derivative is negative i.e.
ⅆ𝑦
ⅆ𝑥
<0, the
function is decreasing.

9. Questions
1. Find values of x for which the following
functions are increasing or decreasing
I. 2𝑥3 − 3𝑥2 − 12𝑥 + 5
II. 𝑥3 − 𝑥2 − 5𝑥 − 5
III. 𝑥3 − 12𝑥 − 5
IV. 𝑥2 − 2𝑥 − 5

10. MAXIMAAND MINIMA
STATIONARY POINTS
1. Condition
A point on a curve at which
ⅆ𝑦
ⅆ𝑥
= 0 is called a stationary point and the
value of the function or y at this point is called the stationary value.
At the stationary points, the tangents to the curve are parallel to the x-
axis.
NOTE:
To find the stationary points, put
ⅆ𝑦
ⅆ𝑥
= 0 and solve the resulting
equation. The stationary point is also called the turning point. The
stationary point or value can be a maximum or minimum or point of

11. 2. Maxima And Minima
Maximum points refer to the points at which the second derivative is
negative i.e.
ⅆ2𝑦
ⅆ𝑥2
< 0.
Minimum points refer to the points at which the second derivative is
negative i.e.
ⅆ2𝑦
ⅆ𝑥2
> 0.
3. Point of Inflection
The point of inflection is the point at which the gradient on either side
of the turning point is the same. It can be obtained when
ⅆ2𝑦
ⅆ𝑥2
= 0
NOTE:
First, find the stationary points, substituting the values into the
second derivative of the function to get the maximum and minimum

12. Questions
1.A curve is defined by the function y = 𝑥3
−6𝑥2
−
15𝑥 − 1, find:
i. The derivative of y with respect to x.
ii. the maximum and the minimum points
2. Find the maximum and minimum points of the curve
y = 𝑥3
− 𝑥2
− 5𝑥 − 5
3. Find the stationary points of y = 1/3𝑥3
− 2𝑥2
+ 3𝑥
and find the maximum and minimum points

13. TANGENTS AND NORMALS TO A CURVE
1. TANGENTS TO A CURVE
If y is a function of x, then the gradient of the tangent to the curve at any point P
(x1, y1) is the value of
ⅆ𝑦
ⅆ𝑥
at that point. The gradient of a tangent to the curve at
any point P (x1, y1) is obtained by substituting the values of x1 and y1 into the
expression for
ⅆ𝑦
ⅆ𝑥
. Hence, the equation of the tangent is given by: y - y1 = m (x -
x1), where the gradient, m =
ⅆ𝑦
ⅆ𝑥
at P (x1, y1).

14. 2. NORMAL TO A CURVE
The normal to a curve at any point P (x1, y1) is a straight line
that is perpendicular to the tangent at that point. Therefore,
the gradient of the normal is
−1
m
. Hence the equation of the
normal is given by: y - y1 =
−1
m
(x - x1).

15. Questions
1. Given that y =𝑥3 − 4𝑥2 + 5𝑥 − 2, find
ⅆ𝑦
ⅆ𝑥
. P is the point on the curve where x
= 3.
i. Calculate the y-coordinate of P.
ii. Calculate the gradient at P.
iii. Find the equation of the tangent at P.
iv. Find the equation of the normal at P.
2. Find the equation of the tangent and the normal to the curve y = 𝑥3−3𝑥 + 2
at the point (2, 4).
3. Find the gradient and the equations of the tangents and normal to curve y
= 4𝑥3−12𝑥 + 3 at (2, 1).