geno3141

(x³)²  =  (x³)·(x³)  =  (x·x·x)·(x·x·x)  =  x^6

The fastest way to find the value of an exponent to an exponent is to multiply the two exponents together:  

        3 x 2  =  6.    --->   (x³)²  =  x³·²  =  x^6

11 ÷ 3  =  3 2/3 in.  =  3.67 in. (approx)

What do want done with this function?

x = price of the book; what is the value of x at the vertex?

The profit is determined by the number of books sold times the price per book.

We want to graph points where the x-value represents the cost per book, n, and where the y-value represents the amount of money taken in.

When n = 0, the amount of money taken in will be 0; again, when n = 72, the amount of money will again be 0.

This graph will be a parabola.

As n increases from 0, to 1, to 2, etc., the amount of money taken in will increase until it hits a maximum at 36; after 36, the amount of money taken in will decrease.

Let y = amount of money taken in:  x = price per book:  y = x(720-10x)

y  = 720 - 10x²

Either graph this and find the maximum point , or find the vertex (the maximum point) by completing the square.

y = -10x² + 720x

y  =  -10(x² - 72x)

y - 12960  =  -(x² - 72x + 1296)        <--- complete the square.

y - 12960  =  -(x - 3)²

Vertex:  (36, 12960)

The graph of this is a parabola that opens downward.  There are at least three ways to find the maximum value.

1)  You can graph it to find its vertex.

2)  If you are in calculus, find the first derivative and set that equal to zero.

     f'(t)  =  -2t + 8     --->     -2t + 8  =  0     --->     -2t  =  -8     --->     t  =  4     --->  

     f(4)  =  -(4)² + 8(4) - 4  =  -16 + 32 - 4  =  12

3)  If you want to do this by algebra: find the vertex by completing the square:

     f(t)  =  -t² + 8t - 4

     f(t) + 4  =  -t² + 8t                     Add 4 to both sides

     f(t) + 4  =  -(t² - 8t)                   Factor out the -1

     f(t) + 4 -16  =  -(t² - 8t + 16)     Complete the square by dividing -8 by 2 and squaring the answer

    f(t) - 12  =  -(t - 4)²

    f(t)  =  -(t - 4)² + 12      --->  vertex at (4, 12)   --->  max value = 12

Use the calculator at this site.

Find common denominators:

(1/7)x - (1/6)x  = (6/42)x - (7/42)x  =  (-1/42)x

(2/3) - (2/9)  =  (6/9) - (2/9)  =  4/9

Answer:  (-1/42)x + (4/9)

If you're having trouble finding common denominators, ask about them.

If the variable is in an exponent, find the log (or ln) of both sides:

log(3125)  =  log(5^e)              With logs, exponents come out as multipliers:

log(3125)  = e x log(5)

log(3125) / log(5)  =  e     --->     e  = 5

If you haven't had logs, think of it this way:

Is 3125 some power of 5?     Not 1, 2, 3, or 4;  but 5^5  =  3125.

Since 3125 = 5^5,  e must be 5.

a)  Relative mins and relative max occur where f' = 0.

     f(x)  =  x³ + ax² + bx + 9   --->   f'(x)  = 3x² +2ac + b

     f'(-1)  =  3(-1)² + 2a(-1) + b  =  0                       f'(3)  =  3(3)² + 2a(3) + b  =  0

                               3 - 2a +  b  =  0                                           27 + 6a + b  =  0

                                    -2a + b  =  -3                                                 6a + b  =  -27

                                                 Combining them:   --->               - (   -2a + b  =  -3     )

                                                                                                               8a  =  -24

                                                                                                                  a  =  -3, b = -9

b)   f'(4)  =  3(4)² +2a(4) + b  =  0    --->   48 + 8a + b  =  0     --->   8a + b  =  -48

     At inflexion point, f''(x)  =  0

                     f''(x)  =  6x + 2a   --->   f''(1)  =  6(1) + 2a  =  0     --->     2a  =  -6

                                                                                                            a  =  -3

                                                                                                            b  =  -24