geno3141

25 - 13f  =  -14

First step:  get the variable term, the f-term, by itself:  

Subtract 25 from both sides:

--->   -13f = -14 - 25

--->   -13f  =  -39

Second step:  divide both sides by -13:

--->   -13f / -13  =  -39 / -13

--->   f  =  3

35/11 is a common fraction.

As a mixed fraction:  35 ÷ 11 = 3, with a remainder of 2.

--->   35/11  =  3 2/11

As a decimal fraction:  35 ÷ 11 = 3.18181818...

If  Π/Π/2  =  (Π/Π)/2  = 1/2

If  Π/Π/2  =  Π/(Π/2)   = 2

I'm going to assume the second answer.

So, I'm going to make Π/Π/2Ψ  =  Π/(Π/2Ψ)  = 2Ψ.

Problem:  2cos(2Ψ)cos²(Ψ) - 2sin(2Ψ)cos(Ψ)(sin(Ψ) 

             =  2[ cos(2Ψ)cos²(Ψ) - sin(2Ψ)cos(Ψ)(sin(Ψ) ]

Since sin(2X) = 2sin(X)cos(X):

             =  2[ cos(2Ψ)cos²(Ψ) - sin(Ψ)cos(Ψ)cos(Ψ)(sin(Ψ) ]

             =  2[ cos(2Ψ)cos²(Ψ) - sin²(Ψ)cos²(Ψ) ]

             =  2cos²(Ψ)[ cos(2Ψ)  - sin²(Ψ) ]

             =   2cos²(Ψ)[ 1 - 2sin²(Ψ) - sin²(Ψ) ]

             =  2cos²(Ψ)[ 1 - 3sin²(Ψ) ] 

From here ... ?

(2/3) - (3/2) / (-4/3)

To divide fractions, multiply by the reciprocal:

(2/3) - (3/2) x (3/-4)

(2/3) - (9/-8)

(2/3) + (9/8)

Common denominator is 24:

(2/3)(8/8) + (9/8)(3/3)

16/24 + 27/24

43/24

1.7916666666...

You can enter that into the calculator just as shown to get the answer.

-4 + -6  =  -10          (a negative plus a negative is a negative)

3(-4)  =  -12             (a positive times a negative is a negative)

-10 / -12  =  5/6       (a negative divided by a negative is a positive)

If you are to factor  3a² - 4ab,  factor out an a:  a(3a - 4b).

If that's not what you're asked to do, you need to give us more instructions ...

lim as x approaches ?

√(3x + 4) - 2x  =  2

To get rid of a square root, you will need to square both sides. However, before you do that, get the square root term alone on one side.

Add 2x to both sides:

√(3x + 4)  =  2x + 2

Now square both sides:

( √(3x + 4) )²  =  ( 2x + 2 )² 

       3x + 4  =  4x²  + 8x + 4

Getting all the terms to one side:

     4x²  + 5x  =  0

Factoring:

     x(4x + 5)  = 0

--->     Either  x  =  0     or     4x + 5  = 0

So, either  x = 0  or  x = -5/4.

Checking in the original problem:  x = 0  --->   √(3(0) + 4) - 2(0)  =  √(4)  =  2    Correct!

x = -5/5   --->   √(3(-5/4) + 4) - 2(-5/4)  =  √(1/4) + 10/4   1/2 + 5/2  =  6/2  =  3     Not correct!

So, the only answer is:  x = 0. 

The derivative of  e^(18.4x)  =  e^(18.4x) · 18.4  =  18.4e^(18.4x)

f(11.2)  =  e^(18.4 · 11.2)  =  e^(206.08)

f'(11.2)  =  18.4e^(18.4 · 11.2)  =  18.4e^(206.08)

T1(x)  =  e^(206.08) + 18.4e^(206.08) · (x - 11.2)

Use can use a calculator to get a decimal approximation for f(11.2) and f'(11.2).