Calculus

7x^(2)

= 7

Divide both sides by '7'

Hence

x^(2) = 1

Square root both sides

x = (+)1 & (-)1 , usually written as ' +/- 1 '.

Remember

(+) X (+)1 = 1

&

-1 X -1 = 1

Remember ratios

Linear a is to b

Area a^(2) is to b^(2)

Volume a^(3( is to b^(3)

So square root the surface area and cube root the volume to reduce it to linear values of 'a' to 'b'.

9x9=81

9x^(2) - 6xy + y^(2) - 81

9x^(2) - 81 + y^(2) - 6xy

9(x^(2) - 9)+ y(y - 6x)

9(x-3)(x + 3) + y(y - 6x)

9x^(2)

-9x + 4 = -50

Subtract '4' from both sides

-9x = -54

Divide both sides by '-9'

x = (+)6

9x - 3 > 6

Add '3' to both sides

9x > 9

Divide both sides by '9'

x > 1 The answer!!!!!!

Verification.

When x = 2

9(2) = 18 > 9

(x^(3) + X^(2) + x + 1) / ( x + 1)

Factor

[x^(2))(x + 1) + ( x +1) ] / ) x + 1)

[(x^(2) + 1 )(x + 1)] / ( x + 1)

Cancel down by 'x + 1'

Hence

x^(2) + 1 The Answer!!!! (NB This term does NOT factor).

(1 - CosX)(1 + CosX)

Apply FOIL

Hence

F: 1 x 1 = 1

O ; 1 x CosX = CosX

I: -CosX X 1 = -CosX

L ; -CosX X CosX = - Cos^(2)x

Collect terms

1 + Cos X - CosX - Cos^(2)X

= 1 - Cos^(2)X

1 - Cos^(2)X = Sin^(2)

From the Trig/ Identity

Sin^(2)X + Cos^(2)X = 1

1-Cos^(2)x = Sin^(2)x

This is algebraically rearranged from the Trig. Identity.

Sin^(2)x + Cos^(2)x = 1

Which in turn is based on the Pythagorean triangle.

x^(2) + 8x + 8y +32 = 0

Then

-8y = x^(2) + 8x + 32

y = (-x^(2)/8) - x - 4

In standard parabolic form. The coefficient of x^(2) is '-1/8' .

It will be an 'umbrella' shaped parabola.

y = x^(2) + 4x + 5

Find the vertex , differentiate and equate to zero.

dy/dx = 0 = 2x + 4

2x + 4 - 0

2x = -4

x = -2

To find if the vertex is at a max/min differentiate are second time. If the answer is positive(+)/Negative(-), then it is a minimum/maximum.

Hence

dy/dx = 2x + 4

d2y/dx2 = (+)2 Positive(+) so the parabola is at a minimum. at x = -2.

Differential equations play a crucial role in biomedical engineering by modeling physiological processes and systems. They are used to describe dynamic behaviors such as blood flow, drug delivery kinetics, and the spread of diseases within biological tissues. Additionally, they help in the design of medical devices, such as prosthetics and imaging systems, by simulating how these devices interact with biological systems over time. Overall, differential equations provide a mathematical framework for understanding complex biological phenomena and improving healthcare solutions.

For the function y = x^(3) + 6x^(2) + 9x

Then

dy/dx = 3x^(2) + 12x + 9

At max/min dy/dx = 0

Hence

3x^(2) + 12x + 9 = 0

3(x^(2) + 4x + 3) = 0

Factor

(x + 1)(x + 3) = 0

Hence x = -1 & x = -3 are the turning point (max/min)

To determine if x = 0 at a max/min , the differentiate a second time

Hence

d2y/dx2 = 6x + 12 = 0 Are the max/min turning points.

Substitute , when x = -1

6(-1) + 12 = (+)6 minimum turning point .

x = -3

6(-3) + 12 = -6 maximum turning point.

When x = positive(+), then the curve is at a minimum.

When x = negative (-), then the curve is at a maximum turning point.

NB When d2y/dx2 = 0 is the 'point of inflexion' , where the curve goes from becoming steeper/shallower to shallower/steeper.

So when d2y/dx2 = 6x + 12 = 0

Then 6x = -12

x = -2 is the point of inflexion.

NNB When differentiating the differential answer gives the steeper of the gradient. So if you make the gradient zero ( dy/dx = 0) , there is no steepness, it is a flat horizontal line

Derivative classification authority refers to the ability granted to individuals to classify documents or information based on existing classified material. It allows authorized personnel to take previously classified information and apply classification markings to new documents that contain, derive from, or are based on that information. This authority is essential for maintaining the integrity and security of classified information while enabling the dissemination of necessary data within the bounds of national security protocols. Proper training and adherence to established guidelines are critical for individuals exercising this authority.

The two derivatives for "zephyr" are "zephyrous," which describes something that is light or airy, and "zephyrlike," which refers to qualities reminiscent of a gentle breeze. Both derivatives capture the essence of zephyr as a soft, gentle wind often associated with spring.

If you would count to sextillion, it would take about 31.7 trillion years to finish counting with stopping. But if you stop constantly it would take about 43.7-48.3 trillion years. So it would nearly be impossible to count it all because you might not even be alive by the time when you count to a quintillion!

This is Simultaneous equ'ns.

3x + 5y = -1

2x -5y = 16

Add the two eq'ns. This will eliminate 'y'

Hence

5x = 15

x = 3

When x = 3 , substitute into either eq'n to yimnd 'y'.

Hence

3)3) + 5y = -1

9 + 5y = -1

5y = -10

y = -10/5 = -2

So the solution in ( x,y) form is ( 3, -2).

35

y = x/2 + 1

This is a line of the form y = mx + c

When c = 1 it passes through the y-axis at y = 1

When m= 1/2 ( 0.5) it has a gradient of 1/2 )0.5) . For every two places along in 'x', you go up one place in 'y'.

So to plot two point

s

When x = 1 ; y = 1/2 + 1 = 3/2 ( 1,3/2)

When x - 2 ; y = 2/2 + 1 =. 1 + ` = 2 (2,2)

et seq.,

The line y = x is a straight line of gradient '1' passing through the origin (0,0)

The line/curve y = x^(2) is 'bowl' shaped curve(parabola). with varying gradients, and just touching the origin (0,0).

Cos(30) = [sqrt(3)] / 2 = 1.732050808... / 2 = 0.8660254068....

y = 7 is the answer.

On a graph it produces a straight horizontal line, parallel to the x-axis, and intersecting the y-axis at '7'.

Because if you plot the point on a graph that the equation generates, it will produce a straight line(Linear).

NB An eq'n of the form Ax^(2) + Bx + C = 0 is NOT linear, because in plotting the points on a graph it produces a curved bowl/umbrella.

7x+10=31x

so 10=31x-7x

or 10 =24x

in which case x=10/24 or 5/12

To create a financial derivative, you first identify the underlying asset, such as stocks, bonds, or commodities. Next, determine the type of derivative you want to create, such as options, futures, or swaps, based on the desired exposure or strategy. After that, establish the contract terms, including expiration date, strike price, and settlement conditions. Finally, ensure compliance with regulatory requirements and market standards before launching the derivative in the market.