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Grantingriver wrote:

Hi Zetafunc, if there are duplicate roots of a polymolial you should counts them as distinct roots.

Why? Precisely the opposite is true: if a root is repeated, then we cannot call those repeated roots 'distinct', and whether or not we count them in the same way we would distinct roots depends on the context.

Grantingriver wrote:

For example, if you say that a give polynomial has “exactly” the roots -3, 4 and 8 that means there is no duplicate roots.

I am not convinced that the word 'exactly' necessarily implies distinctness -- I would have thought that the multiplicities were worth consideration, otherwise the problem appears to be a little simplistic.

Grantingriver wrote:

This is not quite true: the multiplicities of the zeroes are important, e.g. and .
The answer for part 1 is, by all means, yes. This is so because the condition that f(x) has the zeros -3,4 and 8 entails that:

f(x)=(x+3)×(x-4)×(x-8)

And since the zeros of g(x) are -5,-3, 2, 4 and 8 we also have:

g(x)=(x+5)×(x+3)×(x-2)×(x-4)×(x-8)

and hence:

g(x)/f(x)=[(x+5)×(x+3)×(x-2)×(x-4)×(x-8)]/[(x+3)×(x-4)×(x-8)]=(x+5)×(x-2)

and since the result of the division (x+5)×(x-2) is a polynomial without a remainer, therefore g(x) is divisible by f(x). The answer of part 2 is very clear from the answer of part 1. For a polynomial g(x) to be divisible by a polynomial f(x) at least all the roots (zeros) of f(x) should be also roots (zeros) of g(x). And the final statement completes the answer.

In particular, the second pair of brackets in the second line and the first pair of brackets in the third line are not correct.

However, the question appears to be badly worded. The area 'bounded' by the curve, the axes and the line x = 2 is not bounded at all -- it is infinite.

Yes, that is correct.

Yes -- do you know how to calculate this?

One way is to use the binomial distribution.

Suppose that represents the number of times that the goalkeeper saves a penalty. Then, has a binomial distribution: You want to calculate . Do you see why?Hi MellyBigD,

Welcome to the forum. You have posted 4 threads asking for help, but you have not shown any attempts at the problems or given any pointers about where you are stuck. We will endeavour to help you as best you can, and we happily volunteer our time to do so, but we will not do your homework for you. Can you please show some attempts at these problems, or be specific about what exactly you are stuck on?

False: consider the equation .

What happens when x is greater than or equal to 1?

What is the limit of your function as x approaches 2?

There are analogous formulae for higher order derivatives too, and several published papers about the error term.

Hi Sara,

Thanks for stopping by -- have you considered registering an account with us?

Have you heard of the chain rule, i.e. ?Monox D. I-Fly wrote:

So, the approximation of x doesn't have to be 0 to make it equals 1?

Hannibal lecter wrote:

and the second question is : Q2) Find the largest time interval over which the actual US consumption of biodiesel was an increasing function of time. Interpret what increasing means, practically speaking, in this case.

I found the answer is for Q2 :-

The largest time interval was 2005–2007 since the percentage growth rates were positive for each of these three consecutive years. This means that the amount of biofuels consumed in the US steadily increased during the three year span from 2005 to 2007, then decreased in 2008

that don't make any sense? the first answer of Q1 say the largest interval is from 2008-2009

and the second answer for the Q2 said that the largest interval from 2005-2007? as well as it'a a decreasing function not increasing! ( over this interleave )

Because the table refers to **percentage** growth. If it grew by a positive percentage in 2005, 2006 and 2007, then the biodiesel consumption was still increasing in those years, but it was just growing more slowly in 2007 compared to 2006 and 2005.

The biodiesel consumption decreased in 2008, because its percentage is negative. Then it grew a little bit in 2009, because the percentage is positive.

Hannibal lecter wrote:

Hi,

The table shows the recent annual percent growth in US biodiesel consumption

https://preview.ibb.co/kDyByx/2018_02_04_151626.png

find largest time interval over which the percentage growth in the US consumption of biodiesel was an increasing function of time

I found the answer is : largest interval was 2008–2009 since the percentage growth rate increased from−11.7 to 7.3

(Note that the percentage growth rate was a decreasing function of time over 2005–2007.)but why it's not from 2005 - 2006?

which gives 237 to 186.6 , isn't that the largest interval

The key word here is that we want to find the largest time interval where the percentage growth was an **increasing** function of time. Imagine plotting a graph with years on the x-axis and percentage growth on the y-axis -- what would it look like?

Correct! And you can check your answer by substituting x = 1 into your answer to get y = 5.

Now for the perpendicular line passing through (1,5). The line y = 4x + 7 has a gradient of 4. What is the gradient of the perpendicular line?

Let's start with finding the parallel line through (1,5).

Can you write down a line which is parallel to y = 4x + 7?