November 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 27 - Rapid Integration by Parts

These are some videos I made a couple years ago to help people with this topic. It will open in the "Educreations" app if you're on your iPad.

Rapid Integration by Parts (these were from an exploration worksheet, but should still be helpful):
Video 1
Video 2
Video 3
Video 4
Video 5

Day 25 - Riemann Sums and Summation Notation

6b) "I started writing an equation for the Riemann sum in summation notation, but am not sure how to get rid of sigma since it is a left Riemann sum and I have other stuff besides just "j" in the equation. How do you finish finding the limit from here?"

With sigma notation, you can split up the sum into two separate sigma equations. Kind of like how you can split up a limit equation into finding the limit of each separate term.

As far as a reason that sigma notation works this way, think about what a sum of many terms would look like:

(2 + 1) + (2 + 2) + (2 + 3) + (2 + 4) + (2 + 5)

…would be the same as…

(2 + 2 + 2 + 2 + 2) + (1 + 2 + 3 + 4 + 5)

You can rearrange any part of a sum and still get the same answer.

So, for this problem, put a sigma sign in front of the 2/n*j and in front of the 2. This should let you factor constants out to the front of the sigma and substitute in a power sum function. You’ll also have to come up with your own “power sum” style function for the sigma with the constant “2” in it.

Here's what my work looks like up to this step (there's still a couple steps before the final answer):

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!