Math Is Fun Forum

Here is the 4th one:

There is a really useful formula that can help with this. Let's derive it:

First, there is a crucial result that for all functions f:

So, we can write:

This is just e raised to a constant times x.

Taking the derivative:

But now this can be re-written as:

In general, it's better to just remember the formula:

And notice this still holds if b = e:

The key things to know are: that you can write this as a sum of two fractions, that a negative power means *1 over the number raised to the positive power*, and the rules of adding and subtracting exponents.

Here is how it works out:

re: the difference between "undefined" and "indeterminate" -- i think undefined means strictly "there is no meaningful way to define the symbol or operation", so we leave it undefined, 'illegal'. The term 'Indeterminate' is applied only to limit situations, ie, when we have a function like (x-1)/(x^2 - 1) and x approaches 1, the function's value approaches an indeterminate value -- not undefined, but *we are not able to determine the value being approached*.

m and n are any integers in this case.

But we could say they are both real numbers and the rule still applies.

The point is, x^0 = 1 is consistent with this rule and all the other rules of exponents. If it was defined as anything else, there would be inconsistency!

You could also do this:

Now substitute this into the second part:

This has the advantage of only depending on y.

Oops, fixed.
Thanks.
Two lessons for me: thinking more about Latex and less about algebra is bad - and: never assume the work so far is correct !

That happens because your calculator is in Degree Mode. You need it in Radian mode, because angles like 7pi/6 are in Radians.

Yep, you can keep applying it any number of times, as long as the expression is an indeterminate form !

But what about a cone -- the distance around the tip is 0 (from the side view), the distance around the base is something else, the distance around a side is different .. there are an infinite number of choices!

Yes you can do your question like that too -- except you can see the slope of your line is 2 already, without re-writing it. The key thing to know is: parallel lines have the exact same slope.

So your parallel line's equation will be y = 2x + b  ... now you just need to find b. But you know a point on the line, so just substitute the point in this equation!

Oh right!

So the function actually reduces to:

Well we are trying to find

so let's restrict the values of x we are looking at to:

x is in that interval if it is approaching 2, and f(x) = -1 on that interval; it never reaches 2. So when evaluating the limit, f(x) = -1, and the limit is still -1 ... right?

Oh yeah, f(2). haha thanks