Calculus

44/110

Cancel down by '11'

4/10

Cancel down by'2'

2/5 The simplest form

0.75 = 0.75/1.00

Cancel down the decimal point

075/100

Drop the prefix '0'

75/100

Cancel down by '25'

3/4 as a fraction in its simplest form.

-47

x times 1 is x. Anything times 1 is still that same anything.

That is 2.333 ounces

Undefined: You cannot divide by zero

Dear Questioner its not able to solve two variables for one equation, give one more equation. Thanks

5

In derivative questions, the value of zero is often used as a reference point for evaluating the rate of change of a function at a specific point. By setting the change in the variable (Δx) to zero, we can analyze the instantaneous rate of change at that point, which is the essence of the derivative. This approach allows us to find the slope of the tangent line to the curve at a given point, providing important information about the function's behavior.

2x - 3y = 2

3x + 2y = 3

Simultaneous Equations.

Arbitrarily decide which letter to eliminate.

In this case 'y'.

To do this we bring the coefficients of 'y' to the same value. . To do this multiply the top equation by '2' and the bottom eq'n by.3.

Hence

4x - 6y = 4

9x + 6y = 9

We now add the two equations. The addition of '-6y + 6y = 0 ', So 'y' is eliminated.

Hence

13x = 13

x = 1

To find 'y' , we substitute 'x = 1' into either equation.

Hence

4(1) - 6y = 4

4 - 6y = 4

-6y = 0

y = 0

So the answer in ( x,y ) form is ( 1,0 ) .

-3 + 3y = 2y + 5

Add '3' to both sides

Hence

3y = 2y + 8

Subtract '2y' from both sides.

y = 8 The answer!!!!!

(3x + 2y)(5x - 3y)

Apply FOIL.

F(First) ; 3x X 5x = 15x^(2)

O(Outside) ; 3x X -3y = - 9xy

I(Inside) ; 2y X 5x = 10xy

L(Last) ; 2y X - 3y = - 6y^(2)

15x^(2) - 9xy + 10xy - 6y^(2)

Collect 'like' terms

15x^(2) + xy - 6y^(2)

The answer!!!!!!

f(x) = (1 - x^(2) ^(1/2)

Let y= (1 - x^(2)) ^(1/2)

Use Chain Rule

dy/dx = dy/du X du/dx

Let u = 1 - x^(2)

Hence y = u^(1/2)

dy/du = (1/2)u^(-1/2)

du/dx = -2x

Hence

dy/dx = dy/du X du/dx = (1/2)u^(-1/2) X ( -2x)

dy/dx = (1/2)u( 1 -x^(2)^(-1/2) X ( -2x)

Tidying up

dy/dx = (-2x/(2(1 - x^(2))^(1/2))

dy/dx = -x/ [(1 - x^(2)]^(1/2)

f(x) = 3x4 - 2x2 + 7

The function

f'(x) = 12x^(3) - 4x

!st Derivative

f''(x) = 36x - 4

2nd derivative. The answer!!!!!

To calculate the surface area to volume ratio, divide the surface area of an object by its volume (SA ÷ V). Since area scales with the square of length and volume with the cube, larger objects have a smaller surface area to volume ratio. For help with this topic, Calculus Tutoring at Smart Math Tutoring explains these concepts clearly.

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The variable is a. You can solve for it, like this:

8a - 9 = 7a - 6

8a = 7a + 3

8a - 7a = 3

a = 3

By 'nesting'

Fifth root of 'x' is x^(1/5)

Then we 'nest' to the square root ( 1/2)

Hence

[x^(1/5]^(1/2) =

x^(1/10) or x^(0.1)

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

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