Expert: Scotto - 7/8/2010

Question
QUESTION: I need to find the derivative for the function ƒ(x) = 3x2 + 2x − 5. I keep coming up with 4x-5, but I know this is incorrect. Could you please run through the steps for me?  thank you

ANSWER: If f(x) = Ax^B, then f'(x) = ABx^(B-1).
This means the derivative of 3x² is 3*2x and the derivative of 2x is 2.
The correct answer would then be 6x + 2.


---------- FOLLOW-UP ----------

QUESTION: How would I find the antiderivative of 6x + 2? Thank you

ANSWER: The antiderivative is also called the integral, but they don't say that at first.
So in a week or so when they do say it, here's a heads up on what to expect to be coming.

To find the antiderivative ... note that on the derivative, the power of x is reduced by 1.
This means the derivative of x² is 2x and the derivative of x is 1.

Note that we have 6x, not 2x, and that 6 is 2*3, so we need to multiply x² by 3, so we have 3x².
Taking the derivative of this is 2*3x = 6x, and that's what we have.

Note that we have 2, not 1, so if the function we need to multiply x by 2, so we have 2x.
Taking the derivative means multiplying by the exponet (which is 1) and subtracting one from
the exponet, giving 0.  This means the derivative is 2x^0, and x^0 is 1, so the answer is 2.

Thus, we get the antiderivative of 6x+2 is 3x²+2x, but we also need a +C, since the derivative of C is 0 and we don't know what was added on.  This makes the answer 3x²+2x+C.
In this problem, C is seen back at the start to be -5.

The antiderivative, or integral, of f(x) is referred to as F(x).
Antiderivatives are written as what is called "the integral of f of x d x,"
where d and x are just read as letters.  It is written as F(x) = ∫f(x)dx between two points
which are in the integral.  When these two points are known, they are called the limits.


The antiderivatives, or integrals, are written as
|b
|a f(x) dx = F(b) - F(a), where the two | in front of both lines stand for the integral sign.

It can also be written in the form F(x) = ∫(from b down to a)f(x)dx.

I write "from b down to a" to remind the readers to evaluate F(b) first and subtract F(A).
That is, the integral of f(x) from b down to a is F(b) - F(a).


That's another one that I'm not sure was mentioned much to me, and that is the integral usually capitalizes the letter.  That is, integrating f(x) gives F(x), integrating g(x) gives G(x), or the capital of whatever letter is used for the function.


---------- FOLLOW-UP ----------

QUESTION: Thank you so much for helping me. I have one more question. I understand how to find the antiderivative to 6x +2 = 3x^2+2x +C. The question I'm asked says to approximate an antiderivative of f(x) to the nearest tenth(using 6x + 2) when x = 3.4. If x = 3.4, would I just plug this into the antidervative. I get 33.8 when I do this. Is this correct?  Thank you

Answer
3(3.4)² +2(3.4) - 5 = (3(3.4) + 2)3.4 - 5 = 12.2*3.4 - 5 = 41.48 - 5 = 36.48.

Since there function was given at the start, this can be done.

In most cases, the function is not given, so the answer would be 41.48 + C.

Most of the time when doing antiderivatives, which are really integrals,
the area if found beneath a curve between two different values, and they compute F(b)-F(a).