Basic Calculus (Module 9) - Intermediate Value Theorem

Basic Calculus

2. KEEP YOUR
MICROPHONE
ON MUTE
Reduce background noise and distractions.
Microphones can pick up background sounds that may not
bother you but may bother your classmates and teacher. If you
want to say something, raise your hand and wait for the teacher
to acknowledge you before unmuting.

3. TURN ON YOUR
CAMERA
It sends a message that you are engaged in
class.
It's also easier to respond to teachers with a quick nod is than
typing into the chat or unmuting. You'll also stand out in
classes where most students do not have their camera on.

4. BE FULLY PRESENT
Pay attention and participate actively.
Don't play games on your phone, check your social media or watch
videos. Focus on the event, take notes and participate in the
discussion. Also, avoid constantly moving away from your study
area because that may be distracting to your teacher and
classmates.

5. ACTS COMPUTER COLLEGE
Sta. Cruz, Laguna
Basic Calculus
Prepared by:
MS. CHRISHELYN DACSIL

6. 5/16/2022 1:48 PM
ACTS Computer College | Sta. Cruz, Laguna 6
In this lesson you will:
•applythedefinitionofthederivativeofa
functionatagivennumber.
OBJECTIVES:

7. Introduction to
Derivatives

8. Definition of Derivative
The derivative of a function �(�)denoted �′(�)at any �
in the domain of the given function is defined as
�
′
�=lim
∆�→0
∆�
∆�
=lim
ℎ→0
��+ℎ−�(�)
ℎ

9. Example 1:
��=�
2
+3�
�
′
�=lim
ℎ→0
��+ℎ−�(�)
ℎ
=lim
ℎ→0
(�+ℎ)
2
+3(�+ℎ)−(�
2
+3�)
ℎ
=lim
ℎ→0
�
2
+2ℎ�+ℎ
2
+3�+3ℎ−�
2
−3�
ℎ
=lim
ℎ→0
2ℎ�+ℎ
2
+3ℎ
ℎ
=lim
ℎ→0
ℎ(2�+ℎ+3)
ℎ
=lim
ℎ→0
2�+ℎ+3
=lim
ℎ→0
2�+0+3
??????
′
�=�??????+�

10. Example 2:
��=5�−1,���������′(1)
�
′
�=lim
�→�0
��−�(�
0)
�−�0
�
′
1=lim
�→1
5�−1−[51−1]
�−1
�
′
1=lim
�→1
5�−1−4
�−1
�
′
1=lim
�→1
5�−5
�−1
�
′
1=lim
�→1
5(�−1)
�−1
�
′
1=lim
�→1
5
??????
′
�=�

13. I. DERIVATIVE OF A CONSTANT FUNCTION
●The graph of a horizontal function is a horizontal
line, and a horizontal line has zero slope. Recall
that the derivative measures the slope of the
tangent line, and so the derivative of a constant
term is zero.
●Example :�=300,000then the 1st derivative is
�’=0

14. I. DERIVATIVE OF A POWER FUNCTION
A function of the form : �=�
??????
where n is a real number,
is called a power function. In general, this is called the
POWER RULE
If �=�
??????
then, �
′
=??????�
??????−�

15. POLYNOMIAL FUNCTION
1.�=�
�
′
=1�
1−1
=�
2.If��=�
3
Then �
′
�=3�
3−1
;
??????
′
�=��
�

16. RATIONAL FUNCTION
Find �′(�), where ��=
1
�
2
��=
1
�
2
can be written as: ��=�
−2
�
′
�=−2�
−2−1
�
′
�=−2�
−3
=−
�
�
�

17. RADICAL FUNCTION
Find the derivative of �=
4
�
3
SOLUTIONS:
�=
4
�
3
�=�
3
4 rewrite the expression to exponential form
�
′
=
3
4
�
3
4
−1
apply the power rule
=
3
4
�
−
1
4 subtract the exponents
�
′
=
3
4�
ൗ
1
4
=
�
�
�
�
simplify to radical form

18. RADICAL FUNCTION
Given ℎ�=
1
3
�
; find ℎ′(�)
SOLUTIONS:
ℎ�=
1
3
�
ℎ�=
1
�
ൗ
1
3
=�
Τ
−1
3 rewrite to exponential form
ℎ
′
�=−
1
3
�
Τ
−1
3−1
apply to power rule
=−
1
3
�
Τ
−4
3 subtract the exponents
=−
1
3�
ൗ
4
3
simplified radical expression
=−
�
�
�
�
�

19. CONSTANT MULTIPLE RULE
States that the derivative of a constant times a
differentiable function is the constant times the
derivative of the function.
If �=��(�)where k is constant ( k is the numerical
coefficient of the function of x ) ; then :
�‘=�•�‘(�)

20. CONSTANT MULTIPLE RULE
Given: ℎ�=5�
Τ
3
4;�??????��
��
��
SOLUTION : Let �=ℎ(�)
�=5�
Τ
3
4
�=5∙�
Τ
3
4 rewrite in the form
�∙�
??????
�
′
=5∙
3
4
�
3
4
−1
apply constant
multiple
and power rule
=
15
4
�
−
1
4 subtract exponents
and combine similar terms
=
15
4�
1
4
expressed in positive
exponent
�
′
=
15
4
4
�
simplest
radical form

21. CONSTANT MULTIPLE RULE
Given: ��=
3
�
3
SOLUTION:
��=
3
�
3
;��=
1
3
�
1
3 rewrite in the form �∙�
??????
�
′
�=
1
3
∙
1
3
�
1
3
−1
apply constant multiple and power rle
=
�
−
2
3
9
subtract exponents and combine similar
terms
�
′
�=
1
9�
2
3
=
�
??????
�
�
�
in positive exponent to simplest radical
form

22. SUM / DIFFERENCE RULE
Given two differentiable functions g and h, if �=
��ℎ�,�ℎ��∶�
′
=�
′
�ℎ′(�)
EXAMPLES:
Given: ��=5�
Τ
3
4;��=
1
3
3
����ℎ�=−3�
From the given functions above find the following :
1)�‘(�)+�‘(�)
2)�‘(�)−ℎ‘(�)

23. SUM / DIFFERENCE RULE
��+��=5�
3
4+
1
3
3
�combine the given functions by addition as indicated
=5∙�
3
4
−1
+
1
3
∙�
1
3apply the constant multiple rule for each function
�
′
�+�
′
�=5∙
3
4
�
3
4
−1
+
1
3
∙
1
3
�
1
3
−1
use the power
=
15
4
∙�
−
1
4+
1
9
∙�
−
2
3 subtract the exponents
=
15
4�
1
4
+
1
9
3
�
2
expressed the terms with positive exponents
�
′
�+�
′
�=
15
4�
1
4
+
1
9
3
�
2
simplified radical expression

24. SUM / DIFFERENCE RULE
��−ℎ�=
1
3
3
�−(−3�)combinethefunctionsbyadditionasindicated
=
1
3
∙�
1
3+3∙�applytheconstantmultipleruleforeachfunction
�
′
�−ℎ
′
�=
1
3
∙
1
3
�
1
3
−1
+31�
1−1
usethepowerrule
=
1
9
∙�
−
2
3+3∙�
0
subtracttheexponents
=
1
9�
2
3
+3∙1 expressedthetermswithpositiveexponents
�
′
�−ℎ
′
�=
1
9
3
�
2
+3 simplestradicalform

25. DERIVATIVE OF EXPONENTIAL AND LOGARITHMIC
FUNCTIONS
●In the previous study of the different function graphs you have learned that
exponential functions play an important role in modeling population
growth and the decay of radioactive materials. Logarithmic functions can
help rescale large quantities and are particularly helpful for rewriting
complicated expressions.
●Just as when we found the derivatives of algebraic functions, we can also
find the derivatives of exponential and logarithmic functions using
formulas.

26. DIFFERENTIATION FORMULAS : EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
Derivative of Exponential Functions :
If ��=�
�
, then �
′
�=�
�
If ��=�
�
, then �
′
�=�
�
ln�
Derivative of Logarithmic Functions :
If ��=���
??????�, then �
′
�=
1
�ln??????
Derivative of Natural Logarithm :
If ��=ln�, then �
′
�=
1
�

27. DIFFERENTIATION FORMULAS : EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
EXAMPLES : Determine the derivative of the following functions :
1. �=�
−�
SOLUTIONS:
�=
1
�
??????
expressed with positive exponent
��
��
=
�
??????
∙�1∙[1∙�(�
??????
)
(�
??????
)
2
apply the quotient rul
=
�
??????
(0)∙(1)(�
??????
)
(�
??????
)
2
simplify the terms
��
��
=
−�
??????
(�
??????
)(�
??????
)
=−
1
�
??????
derivative in simplest form

28. DIFFERENTIATION FORMULAS : EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
EXAMPLES : Determine the derivative of the following functions :
1. �=�
−�
SOLUTIONS:
�=
1
�
??????
expressed with positive exponent
��
��
=
�
??????
∙�1∙[1∙�(�
??????
)
(�
??????
)
2
apply the quotient rul
=
�
??????
(0)∙(1)(�
??????
)
(�
??????
)
2
simplify the terms
��
��
=
−�
??????
(�
??????
)(�
??????
)
=−
1
�
??????
derivative in simplest form

29. DIFFERENTIATION FORMULAS : EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
EXAMPLES : Determine the derivative of the following functions :
1. �=�
−�
SOLUTIONS:
�=
1
�
??????
expressed with positive exponent
��
��
=
�
??????
∙�1∙[1∙�(�
??????
)
(�
??????
)
2
apply the quotient rul
=
�
??????
(0)∙(1)(�
??????
)
(�
??????
)
2
simplify the terms
��
��
=
−�
??????
(�
??????
)(�
??????
)
=−
1
�
??????
derivative in simplest form

30. DIFFERENTIATION FORMULAS : EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
2. ��=2
�
∙�
�
SOLUTIONS: Let �=2
�
and �=�
�
��=2
�
∙�
�
given
�
′
�=2
�
∙��
�
+�
�
∙�(2
�
)apply the product formula
=2
�
�
�
+�
�
∙2
�
ln2 combine similar terms
�
′
�=2
�
�
�
(1+ln2) factor and simplify

31. DIFFERENTIATION FORMULAS : EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
3. ��=�
2
+(−2ln�)
SOLUTIONS:
��=�
2
−2ln�clear off parenthesis
�
′
�=2�−2∙
2
�
apply the differentiation formula
=2�−
2
�
combine similar terms
�
′
�=
2�
2
−2
�
simplified into a single
expression