Calculus questions

Differentiation 2

Differentiation

ln(e) = 1

Power Rule

Example: What is x³ ?

The question is asking "what is the derivative of x³?"

We can use the Power Rule, where n=3:

xⁿ = nxⁿ⁻¹

x³ = 3x³⁻¹ = 3x²

Example: What is (1/x) ?

1/x is also x^-1

We can use the Power Rule, where n = −1:

xⁿ = nxⁿ⁻¹

x⁻¹ = −1x⁻¹⁻¹ = −x⁻²

Multiplication by constant

Example: What is 5x³ ?

the derivative of cf = cf’

the derivative of 5f = 5f’

We know (from the Power Rule):

x³ = 3x³⁻¹ = 3x²

So:

5x³ = 5x³ = 5 × 3x² = 15x²

Sum Rule

Example: What is the derivative of x²+x³ ?

The Sum Rule says:

the derivative of f + g = f’ + g’

So we can work out each derivative separately and then add them.

Using the Power Rule:

• x² = 2x
• x³ = 3x²

And so:

the derivative of x² + x³ = 2x + 3x²

Difference Rule

It doesn't have to be x, we can differentiate with respect to, for example, v:

Example: What is (v³−v⁴) ?

The Difference Rule says

the derivative of f − g = f’ − g’

So we can work out each derivative separately and then subtract them.

Using the Power Rule:

• v³ = 3v²
• v⁴ = 4v³

And so:

the derivative of v³ − v⁴ = 3v² − 4v³

Sum, Difference, Constant Multiplication And Power Rules

Example: What is (5z² + z³ − 7z⁴) ?

Using the Power Rule:

• z² = 2z
• z³ = 3z²
• z⁴ = 4z³

And so:

(5z² + z³ − 7z⁴) = 5 × 2z + 3z² − 7 × 4z³ = 10z + 3z² − 28z³

Example: What is the derivative of cos(x)sin(x) ?

The Product Rule says:

the derivative of fg = f g’ + f’ g

In our case:

• f = cos
• g = sin

We know (from the table above):

• cos(x) = −sin(x)
• sin(x) = cos(x)

So:

the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x) 

Integration

1) What is ∫x³ dx ?

On Rules of Integration there is a "Power Rule" that says:

∫xⁿ dx = xⁿ⁺¹/(n+1) + C

We can use that rule with n=3:

∫x³ dx = x⁴ /4 + C

2) What is an integral of 2x?

We know that the derivative of x² is 2x ...

... so an integral of 2x is x²

Why must we write +C? 

The derivative of x²+4 is 2x, and the derivative of x²+99 is also 2x, and so on! Because the derivative of a constant is zero.

So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value.

So we wrap up the idea by just writing + C at the end.

what is ∫cos(x) dx ?

From the Rules of Derivatives table we see the derivative of sin(x) is cos(x) so:

∫cos(x) dx = sin(x) + C

Why do babies lose heat faster than adults?

Babies can't adjust to temperature changes as well as adults. Babies also have thinner layer of fats compared to adults. They have limited stores of metabolic substrates to use for heat production.

Function of the Skin

Function of the skin - Regulate body temperature via sweat and hair, Destruction of microorganisms & interaction of skin with the body's immune system, Synthesis of Vitamin D, and protects us from physical injuries.

Our Blood

Red Blood Cell - The primary function of red blood cells is to transport oxygen to body cells and deliver carbon dioxide to the lungs. 

Red blood cells have a unique structure. Their flexible disc shape helps increase the surface area-to-volume ratio of these extremely small cells. This enables oxygen and carbon dioxide to diffuse across the red blood cell's plasma membrane more readily. A red blood cell has what is known as a biconcave shape. This shape also aids in a red blood cell's ability to maneuver through tiny blood vessels to deliver oxygen to organs and tissues. 

Red Blood Cells, also called erythrocytes, are tiny biconcave disc-shaped cells. They do not have a nucleus or mitochondria. Their cytoplasm is rich in haemoglobin. O₂binds to the iron in haemoglobin. Red blood cells are made in the bone marrow. They survive for about four months. They are destroyed and recycled by the liver and spleen. They are destroyed because they have to constantly change shape to pass through narrow blood vessels. When they die the haemoglobin is stored in the liver and used to make new blood cells in the bone marrow. They are very small. There are about 5 million red blood cells in 1 cc. of blood. The rest that is not stored is converted into bile pigments.

White Blood Cells – White Blood Cells are colourless cells and possess a nucleus. They function in defending the body against pathogens. Some ‘feed’ on pathogens by phagocytosis. These white blood cells are called phagocytes. Others, the lymphocytes produce antibodies, the specific defense proteins. They are made by the bone marrow and lymphatic tissue.

Platelets - also called thrombocytes, are tiny fragments of large bone marrow cells. They carry specialised blood clotting chemicals. The clotting chemicals are released where blood and lymph vessels are injured. A nucleus is not present in platelets.

Blood plays an important role in regulating the body's systems and maintaining homeostasis.

Other functions include supplying oxygen and nutrients to tissues, removing waste, transporting hormones and other signals throughout the body, and regulating body pH and core body temperature.

Blood is composed of plasma, red blood cells, white blood cells, and platelets.

Blood platelets play a role in coagulation (the clotting of blood to stop bleed from an open wound); white blood cells play an important role in the immune system; red blood cells transport oxygen and carbon dioxide.

Blood is considered a type of connective tissue because it is made in the bones.

Understanding of Cell Structure (Anatomy)

Cell wall -  It helps in protecting the plasma membrane and plays a vital role in supporting and protecting the cells. It is a thick outer layer made of cellulose. 

Nucleus - Organized as multiple long linear DNA molecules in complex with a large variety of proteins, such as histones, to form chromosomes. Its main function is to store DNA and regulate cell activity such as metabolism and reproduction. They are the membrane bound organelles, which are found in all eukaryotic cells. It is the very important organelle of a cell as it controls the complete activity of a cell and also plays a vital role in reproduction.

Nuclear membrane  - The bilayer membrane, which protects the nucleus by surrounding around it and acts as a barrier between the cell nucleus and other organs of a cell.

Mitochondrion - Provides ATP for cellular activity.  It play a vital role in generating and transforming the energy. Mitochondria play a vital role in various functions of the cell metabolisms including oxidative phosphorylation.

The Threesomes:

Ribosomes -  Ribosomes are small particles, present in large numbers in all the living cells. They are sites of protein synthesis. Ribosomes are the cellular component that make proteins from  all amino acids. Ribosomes are freely suspended in the cytoplasm or attached to the endoplasmic reticulum forming the rough endoplasmic reticulum. It plays a vital role in protein synthesis.

Lysosome -  act as the waste disposal system of the cell by digesting unwanted materials in the cytoplasm

Peroxisome - A major function of the peroxisome is the breakdown of very long chain fatty acids through beta-oxidation. Also functions to detoxify harmful molecules. It helps in cell renewal and break down old cell parts.




Golgi Apparatus - Modifies and packages protein into vesicles. In most cases proteins are transferred From Rough ER to Golgi for ‘finishing’.  It helps in the movement of materials within the cell.

Endoplasmic Reticulum - Rough ER (RER) is involved in some protein production, protein folding, quality control and despatch. It is called ‘rough’ because it is studded with ribosomes

Endoplasmic Reticulum consists of:

Rough Endoplasmic Reticulum - Rough ER is called rough because it has ribosomes attached to its surface. Manufacture of secreted proteins. This organelle plays a large role in the synthesis of large, complex proteins and amino acids. 

Smooth Endoplasmic Reticulum -  Aids the manufacture of carbohydrates. It synthesizes lipids, phospholipids, and steroids. It carries out the metabolism of carbohydrates, detoxification of natural metabolism products and of alcohol and drugs, attachment of receptors on cell membrane proteins, and steroid metabolism. Smooth ER plays a large part in detoxifying a number of organic chemicals converting them to safer water-soluble products.

Cytoplasm -  The jelly-like fluid that fills a cell is called cytoplasm. It is made up of mostly water and salt. Aids in the movement of cellular materials and transporting the genetic material and products of cellular respiration. It protects the cell by keeping the cell organelles separate from each other. This helps to keep a cell in stable. Cytoplasm is the site, where many vital biochemical reactions take place.

Vacuoles - The main function of vacuoles is to contain the waste products of the cells and isolate them from the rest of the organelles and the cytoplasm. Vacuoles maintain an acidic pH internally. This allows them to denature misfolded proteins transferred to the vacuole from the cytoplasm. The vacuoles can also help remove unwanted or toxic materials from the cells by exporting them to the cell membrane, where they are released to the outside surroundings of the cell. This process is known as exocytosis. It helps plants in maintaining its shape and it also stores water, food, wastes, etc.

Nucleolus -  Sites for rRNA synthesis. The creation of rRNA is important because rRNA makes up ribosomes which are responsible for protein synthesis in the cell. It plays a vital role in the production of cell's ribosome.

Vesicles - releasing materials to be secreted from the cell by fusing to cell membrane.

Centriole -  The main function of the centriole is to help with cell division in animal cells. The centrioles help in the formation of the spindle fibers that separate the chromosomes during cell division (mitosis).

Cytoskeleton -   the cytoskeleton gives a cell its shape, offers support, and facilitates movement through three main components: microfilaments, intermediate filaments, and microtubules.

Plasma membrane -  The primary function of the plasma membrane is to protect the cell from its surroundings. Composed of a phospholipid bilayer with embedded proteins, the plasma membrane is selectively permeable to ions and organic molecules and regulates the movement of substances in and out of cells.

Chloroplasts (Plants only) -  work to convert light energy of the Sun into sugars that can be used by cells.