CPhill

f(x) = x^3 - 4

The derivative gives the slope of a tangent line at any x value

So

f ' ( x)   =   3x^2

At  x = 2.... the slope of the tangent line is    3(2)^2 =  12

The slope of a normal line  =  -1/12

So...the equation of the normal line is

y = - (1/12) ( x - 2) + 4       simplify

y = (-1/12)x + 1/6 + 4

y = (-1/12)x + 25 / 6

sqrt  ( 6 + 1/4)   =

sqrt ( 25 / 4)  =

sqrt (25) / sqrt (4)   =

5 / 2

OK......march on....!!!!

Just to give us something else to worry about during school......

Thanks, Echofire     !!!

lim                sqrt ( x^2 + 12x + 36)

x → - 6         _________________

                        x + 6

lim                 sqrt [ (x + 6)^2 ]               l  x + 6 l

x → - 6        ______________    =    _________

                         x + 6                               x + 6

  {  Remember that    sqrt (a^2)  =    l a  l      }

Note  that this limit does not exist as   x → - 6

The numerator will be positive approaching from the left side but the denominator will be negative

So....the result will be negative

The numerator and denominator will be positive approaching from the right side

So.....the result will be positive

See the graph, here  :  https://www.desmos.com/calculator/miohj798g0

OK....I see what they are assuming...

They are assuming that  during  the 1st year, we have 2500

So...."in the 5th year"  means  "after 4 years"

So....in that case

2500 (1.05)^(n - 1)   is correct.... !!!

Thanks, Guest.....stupid mistake....I'll make the edit  !!

Last one

The key here is to notice that  81  =  3^4

The reason for doing this is that we won't have to use any logs to solve for x

So....we actually have

3^(7x)  =  (3^4)^(x + 9)

Remember that    (a^m)^n  =  a^(m * n)       so.....

3^(7x) =  3^(4 * (x + 9) )

Since the bases are the same, we can solve for the exponents

7x  =  4 (x + 9)     simplify

7x  = 4x + 36      subtract 4x from both sides

3x = 36    divide both sides by 3

x = 12

I don't quite understand the first one, Phoenix....are we multiplying log(5) by 125???

2.  

5 * 8^(4x) = 376          divide both sides by 5

8^(4x) =  376 / 5        take the log of both sides

log 8^(4x) = log (376 / 5)       

Remember   that      log a^b    can be written as   b*log (a)

So

(4x) log (8)  = log (376/5)

x *(4 * log 8)   =  log (376 / 5)        divide both sides by  (4 * log 8)

x =   log (376/5) / [ 4 * log 8 ]    =  log (75.2) / (4log 8)