In this chapter we shall study those points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards. At such points the derivative of the function, if it exists, is necessarily zero.
The value of a function f (x) is said to be maximum at x = a, if there exists a very small positive number h, such that
f(x) < f(a) x (a – h,a + h) , x a
In this case the point x = a is called a point of maxima for the function f(x).
Simlarly, the value of f(x) is said to the minimum at x = b, If there exists a very small positive number, h, such that
f(x) > f(b), x (b – h,b + h), x b
In this case x = b is called the point of minima for the function f(x).
Hene we find that,
(i) x = a is a maximum point of f(x)
f(a) f(a h) 0
f(a) f(a h) 0
(ii) x = b is a minimum point of f(x)
Rf(b) f(b h) 0
f(b) f(b h) 0
(iii) x = c is neither a maximum point nor a minimum point
Note :
(i) The maximum and minimum points are also known as extreme points.
(ii) A function may have more than one maximum and minimum points.
(iii) A maximum value of a function f(x) in an interval [a,b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function. A minimum value may be greater than some maximum value for a function.
(iv) If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
(v) Monotonic functions do not have extreme points.
Ex. Function y = sin x, x (0, ) has a maximum point at x = /2 because the value of sin /2 is greatest in the given interval for sin x.
Clearly function y = sin x is increasing in the interval (0, /2) and decreasing in the interval ( /2, ) for that reason also it has maxima at x = /2. Similarly we can see from the graph of cos x which has a minimum point at x = .
Ex. f(x) = x2 , x (–1,1) has a minimum point at x = 0 because at x = 0, the value of x2 is 0, which is
less than the all the values of function at different points of the interval.
Clearly function y = x2 is decreasing in the interval
Rf(c) f(c h)
Sand
Tf(c) f(c h) 0
|V| have opposite signs.
(–1, 0) and increasing in the interval (0,1) So it has minima at x = 0.
Ex. f(x) = |x| has a minimum point at x = 0. It can be easily observed from its graph.
A. Necessary Condition : A point x = a is an extreme point of a function f(x) if f’(a) = 0, provided f’(a) exists. Thus if f’ (a) exists, then
x = a is an extreme point f’(a) = 0
or
f’ (a) 0 x = a is not an extreme point.
But its converse is not true i.e.
f’ (a) = 0 x = a is an extreme point.
For example if f(x) = x3 , then f’ (0) = 0 but x = 0 is not an extreme point.
B. Sufficient Condition :
(i) The value of the function f(x) at x = a is maximum, if f’ (a) = 0 and f” (a) < 0.
(ii) The value of
In this chapter we shall study those points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards. At such points the derivative of the function, if it exists, is necessarily zero.
The value of a function f (x) is said to be maximum at x = a, if there exists a very small positive number h, such that
f(x) < f(a) x (a – h,a + h) , x a
In this case the point x = a is called a point of maxima for the function f(x).
Simlarly, the value of f(x) is said to the minimum at x = b, If there exists a very small positive number, h, such that
f(x) > f(b), x (b – h,b + h), x b
In this case x = b is called the point of minima for the function f(x).
Hene we find that,
(i) x = a is a maximum point of f(x)
f(a) f(a h) 0
f(a) f(a h) 0
(ii) x = b is a minimum point of f(x)
Rf(b) f(b h) 0
f(b) f(b h) 0
(iii) x = c is neither a maximum point nor a minimum point
Note :
(i) The maximum and minimum points are also known as extreme points.
(ii) A function may have more than one maximum and minimum points.
(iii) A maximum value of a function f(x) in an interval [a,b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function. A minimum value may be greater than some maximum value for a function.
(iv) If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
(v) Monotonic functions do not have extreme points.
Ex. Function y = sin x, x (0, ) has a maximum point at x = /2 because the value of sin /2 is greatest in the given interval for sin x.
Clearly function y = sin x is increasing in the interval (0, /2) and decreasing in the interval ( /2, ) for that reason also it has maxima at x = /2. Similarly we can see from the graph of cos x which has a minimum point at x = .
Ex. f(x) = x2 , x (–1,1) has a minimum point at x = 0 because at x = 0, the value of x2 is 0, which is
less than the all the values of function at different points of the interval.
Clearly function y = x2 is decreasing in the interval
Rf(c) f(c h)
Sand
Tf(c) f(c h) 0
|V| have opposite signs.
(–1, 0) and increasing in the interval (0,1) So it has minima at x = 0.
Ex. f(x) = |x| has a minimum point at x = 0. It can be easily observed from its graph.
A. Necessary Condition : A point x = a is an extreme point of a function f(x) if f’(a) = 0, provided f’(a) exists. Thus if f’ (a) exists, then
x = a is an extreme point f’(a) = 0
or
f’ (a) 0 x = a is not an extreme point.
But its converse is not true i.e.
f’ (a) = 0 x = a is an extreme point.
For example if f(x) = x3 , then f’ (0) = 0 but x = 0 is not an extreme point.
B. Sufficient Condition :
(i) The value of the function f(x) at x = a is maximum, if f’ (a) = 0 and f” (a) < 0.
(ii) The value of
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Toxic effects of heavy metals : Lead and Arsenicsanjana502982
Heavy metals are naturally occuring metallic chemical elements that have relatively high density, and are toxic at even low concentrations. All toxic metals are termed as heavy metals irrespective of their atomic mass and density, eg. arsenic, lead, mercury, cadmium, thallium, chromium, etc.
2. Extreme values are termed “extrema”
Absolute Extrema: the point in question represents either the
maximum or minimum value of the function over the domain.
Relative Extrema: the point in question represents either the
maximum or minimum value of the function on a specified
segment of the domain (or in the “hood”).
Let’s take a look at an example and consider several points on a
function.
3. Take a look at this diagram. Notice the difference between
local and absolute extrema.
4. Although the concepts are easy to understand, be attuned to
the subtlety of the questions asked …
Example 1: on the closed interval find any relative extrema for
y = sin x and y = cos x
Example 2: Same as above but on the open interval
,
2 2
,
2 2
5. The Extreme Value Theorem:
If a function f is continuous on a closed interval [a, b], then f has both a
maximum and a minimum on the interval.
Question: So, by looking at a graph of a function, how can you find its
extrema?
What’s true about the
curve at its max and
min?
What’s the one
analytical tool we’ve
been studying all year?
6. To fully answer the question, we need to define some terms
and give you one theorem.
Critical Points – a point on the interior of the domain of a function f at
which
○ f’ = 0 or
○ f’ does not exist (is undefined).
Theorem: if a function f has a local maximum and/or minimum at
some interior point “c” of its domain, and if f’(c) exists, then
f’(c) = 0
SO, HOW DO YOU FIND EXTREMA???
1. Find all critical points (values)
2. Check the endpoints of the specified domain
7. A little Practice
Find the extrema of on the interval [-1, 2].
1. Find the critical numbers in f
2. Evaluate f at each critical number
3. Evaluate f at each endpoint
4. Compare. Least number is minimum, greatest number is maximum.
Answers are:
1. Critical numbers at x = 0 and x = 1
2. f(0) = 0 and f(1) = -1
3. Left endpoint has height of 7, right endpoint has height of 16
4. Min = -1, max = 16
4 3
( ) 3 4
f x x x
8. Why Need Of Derivative??????
Derivatives have arisen from the need to manage the
risk arising from movements in markets beyond our
control ,which may severely impact the revenues and
costs of the firm.
Derivatives can be used in a number of ways in
everyday life, especially with optimization.
Example:
The growth rate of any company , the profit or loss
it made, etc
9. CURVES
You can easily graph any function by
knowing three things.
1) ZEROS AND UNDEFINED SPOTS
2) MAXIMUM AND MINIMUM
POINTS
3) CONCAVITYAND INFLECTION
POINTS.
10. What the First Derivative Tells Us:
Suppose that a function f has a derivative at
every point x of an interval I. Then:
increases on I if ( ) 0 for all in I.
f f x x
decreases on I if ( ) 0 for all in I.
f f x x
11. What This Means:
In geometric terms, the first derivative tells
us that differentiable functions increase on
intervals where their graphs have positive
slopes and decrease on intervals where their
graphs have negative slopes.
WHAT HAPPENS IF THE FIRST
DERIVATIVE IS ZERO?
12. When The First Derivative is
Zero
A derivative has the intermediate value
property on any interval on which it is
defined.
It will take on the value zero when it
changes signs over that interval.
Thus, when the derivative changes signs on
an interval, the graph of f(x) must have a
horizontal tangent.
13. Relative Maxima and Minima
If the derivative
changes sign, there
may be a local max or
min, as shown here.
More on this later.
14. Concavity
Concave down—”spills water”
Concave up—”holds water”
The graph of
is concave down on any interval where
and concave up on any interval where
( )
y f x
0
y
0
y
15. Points of Inflection
A point on the curve where the concavity changes
is called a point of inflection.
If the second derivative is zero for some x, we
may be able to find a point of inflection.
It IS possible for the second derivative to be zero
at a point that is NOT a point of inflection.
A point of inflection may occur where the second
derivative fails to exist.
16. Inflection Points
You can tell where the function changes concavity
by finding the inflection points.
Evaluate the function at those values where the
second derivative is zero;
Take a look at the graph of the original function:
4 2
( ) 2
f x x x
18. An Interesting Example
Suppose that the yield, r, in the % of students in a one
hour exam is given by:
r = 300t (1−t).
Where 0 < t < 1 is the time in hours.
1. At what moments is the yield zero?
2. At what moments does the yield increase or
decrease?
3. When is the biggest yield obtained and which is?
19. ERRORS AND APPROXIMATIONS
We can use differentials to calculate small changes in
the dependent variable of a function corresponding
to small changes in the independent variable.
e.g