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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?

Hannibal lecter wrote:

There is: it's , the The average velocity, v, for a trip over a ﬁxed distance, d, is inversely proportional to the time of travel, t.

which is v= d/t

there is no a constant of proportionality here why?

And what is the slope of a line with equation y = mx + c?

The '≤' sign accounts for the fact that the function doesn't have to be increasing at all points.

Can you rearrange the equation to the form ?

Hannibal lecter wrote:

The -intercept is the point at which the line crosses the -axis.Since , then the coordinates of the -intercept are . Or sometimes people just say that the -intercept is .
for example, y = 10 + mx

b = 10

what is it intercept?

Have you come across these formulae before?

If not, then one approach (which fits the title of the thread more accurately, I suppose) is to prove this result by induction:

and then once you have, set that equal to 364 and solve.

Hannibal lecter wrote:

Hi,

what is the explanation of 1) " A function f(x) is monotonic if it increases for all x or decreases for all x.?

what is the meaning of monotonic ?

I know what is the increasing function and what is the decreasing

but what is the meaning of monotonic

and what is

2) Strictly Decreasing?

3) Strictly Increasing?what is the meaning of Strictly ?????

First, let's examine what is meant by the word 'increasing'. How can you look at a graph and immediately tell if it is increasing or not? One way is to look at the slope (or, if you've done any calculus, the 'derivative', which tells you the slope of the function at every point). If the slope at that point is positive, then the function is increasing. If it is negative, it is decreasing. If it is 0, then it's stationary. Now let's try to understand what is meant by 'monotone'.

This is an example of a function which is **monotonically increasing**. Notice that the graph is always 'heading upwards' (think about the tangents to this curve for a more accurate description). This happens until we get to the middle, where the graph resembles a horizontal line. It's not increasing along this part -- but it also isn't decreasing. In other words, if something is monotonically increasing, then it **doesn't decrease**.

This is an example of a function which is **monotonically decreasing**. Notice that the graph is always 'heading downwards', except when we get to the horizontal part of the curve. In other words, if something is monotonically decreasing, then it **doesn't increase**.

That's a function which isn't monotone at all -- do you see why?

As for 'strictly increasing', then this means that the function must **always** be increasing -- it won't have any 'horizontal parts'. Likewise for 'strictly decreasing'.

There are formal mathematical definitions for all of these terms, by the way -- but I want to make sure that you understand what the picture looks like first. Does this help?

Monox D. I-Fly wrote:

No: this is not an arithmetic series. The terms form the sequence of triangular numbers, whose nth term is (which you can also write as , though I prefer the former).You have been asked to find the value of such that That last term is the nth term of the sequence . Notice that if , then . Similarly taking gives you , and so on. So what you actually want to do is find the value of such that:I know that 1, 3, and 6 are the result of arithmetic series with the starting value 1 and the difference 2, thus that sum can be written as

+ + + ... + = 364.

Do you know how to do that?

Welcome!