geno3141

z-score  =  (score - average) / ( standard deviation)

z-score  =  (65.5 - 63.9) / 2.4                           <---  to get your answer, reduce this to a decimal

Is the problem to find the product of two-thirds and twenty-twelfths?

If it is:  2/3 x 20/12  =  (2 x 20) / (3 x 12)  =  40/36  =   10/9

When multiplying fractions, multiply the numerators together and multiply the denominators together.

Simplify the answer.

When you multiply two numbers with the same base, you add their exponents.

Logs are exponents, so the reason that they're added is to get the answer to a multiplication problem.

So, working backwards, log(4x) + log(13)  =  log(4x·13)  =  log(52x)

Since the problem says that log(52x)  =  0, changing it back into eponential form:  10^0  =  52x

--->  1  =  52x

--->  x  =  1/52

1.67t  =  10.02

Divide both sides by 1.67:

t  =  10.02 / 1.67        Calculate this to get your answer.

By defining  n^0  =  1 whenever n ≠ 0,  certain rules are made simpler. For example, the rule that, when you divide numbers with the same base, all you need  to do is subtract the exponents, works if this rule is allowed.

For example:  x^12 / x^9  =  x^3, using the rule that x^a / n^b  =  x^(a - b). Using this rule is simpler than writing out the problem as [x·x·x·x·x·x·x·x·x·x·x·x] / [x·x·x·x·x·x·x·x·x] and cancelling out the common factors to get x·x·x, which equals x³. Do you really want to try to solve the problem x^349 / x^197 without using this rule?

Similarly, x^5 / x^8  =  x^-3 by using this rule, avoiding writing out the problem in detail and cancelling out the common fractors.

Now, what about the problem  x^13 / x^13. We know that the answer must be 1, because it is a number divided by itself. But, if we like to use the subtraction-of-exponents rule, we have, for an answer, x^0. Thus, x^0 must equal 1.

A different example:

10^4  =  10000

10^3   = 1000

10^2   = 100

10^1  =  10

10^0  =  ???   <--- What should this be, if not 1?

10^-1  =  0.1

10^-2  =  0.01

10^-3  =  0.001

A one-to-one function is a function where each x-value is paired with exactly one y-value and each y-value is paired with exactly one x-value.

The "if f(x) > 0" part says that the function will be one-to-one when the y-values are greater than zero.

If f(x) ≤ 0, it still may be a one-to-one function, but does not have to be.

The sum of two supplementary angles is 180°.

The sum of two complementary angles is 90°.

Rearrange the terms on the left side:

2x² - 5  =  -6x

Add 6x to both sides:

2x² + 6x - 5  =  0

This is not easily factored; so I would suggest that you use the quadratic formula:

  x  =  [ -b ± √( b² - 4ac ) ] / ( 2a )

where, for this problem,  a = 2,  b = 6,  and  c = -5.

The question is:  Why is it not continuous at x = 2?

You gave the answer to that when you said that it is not defined at that location.

The limit, approaching from both the left and the right is 4; thus, it has a limit at x = 2, even though it is not defined at that location; even though it is not defined at that location. Indeed, there is a 'hole' in the graph at that location, which makes is discontinuous, even though it has a limit at that point.

My understanding is a little different:

A linearization gives approximate y-values near a given point on a graph, while Newton's method is used to find a zero of a function by repeated approximations, starting at a given point.

A linearization will give an approximate y-value anywhere on a function while Newton's method is used to find an x-value that gives a y-value of zero.