October help

These are explanations that I had emailed to various students while we were working on this unit. Use this information as you review old problems, or while you're studying for the final exam.

If you need help with any homework or classwork problems, or would like additional explanation for a concept we've covered in class, please email me!

Remember, my Google Drive folder contains copies of worksheets and ANSWERS (on the last page of each file). CLICK HERE TO ACCESS.

If for some reason you'd like to refer back to last year's blog, here is the link: AP Calculus AB blog

Day 14 - MVT & Rolle's (+ review)

A couple of good derivative rules to remember:

The derivative of e^x is e^x (then multiplied the derivative of the "inside function" - the exponent).

The derivative of ln x is 1/x (then multiplied the derivative of the "inside function" - the denominator).

5) If we want to always keep f(x) increasing, then we need to find all places that f'(x) > 0 (in other words, where is the derivative of f(x) positive?).

Find f'(x) first:

f(x) = x^3 + ax + b

f'(x) = 3x^2 + a

Then set f'(x) as equal to or greater than zero:

3x^2 + a > 0

No matter what x-value is used here, you'll always get a positive value when you plug it into x^2. So, that part of the function is always going to give us a positive f'(x).

We only need to look at the a-value! If a < 0, then it would be possible for f'(x) to be negative (which means f(x) is decreasing). So, "a" can't be less than 0.

If a > 0 , then f'(x) will be positive, which guarantees that f(x) is increasing.

If a = 0, then there could be a could locations where f'(x) = 0 and f(x) is neither increasing nor decreasing. In this case, you could argue that f(x) hasn't technically changed direction, so it must still be increasing.

ANSWER: "a" must be equal to or greater than 0 for f(x) to always be increasing.

6) To clarify: 5 is the base of the entire triangle, and 8 is the height of the entire triangle.

Here's a start to this problem:

Your goal is to write an equation for the thing they're asking you to maximize or minimize, and also to write an equation that you could plug into the first equation to help solve it.

Here, you want to maximize the area of the rectangle, so your initial equation is:

A = bh

We need an equation to plug in for either "b" or "h", so that we can actually find A' and set it equal to 0. Write this second equation by thinking about the diagram as being similar triangles. Set up proportions where the numerator is information about the smaller triangle, and the denominator is information about the larger triangle:

small height / big height = small base / big base

(8 - h) / 8 = w / 5

Then, it would probably be easiest to solve for "w", then plug in for "w" in the initial equation:

(8 - h) / 8 = w / 5
1 - h/8 = w/5
5 - 5h/8 = w

If you get stuck here, let me know and I can add more of an explanation!