Calculus AB Quick Facts Part Two

First let’s talk about what the integral means! Can you list some interpretations of the definite integral?

Here’s a few facts: 1. If f(x) > 0, then returns the numerical value of the area between f(x) and the x-axis (area “under” the curve) 2. = F(b) – F(a) where F(x) is any anti-derivative of f(x). (Fundamental Theorem of Calculus) 3. Basically gives the total cumulative change in f(x) over the interval [a,b]

What is a Riemann Sum? Hint: Here’s a picture!

A Riemann sum is the area of n rectangles used to approximate the definite integral. = area of n rectangles As n approaches infinity… and So the definite integral sums infinitely many infinitely thin rectangles!

The indefinite integral = ?

Well…hard to write; easy to say The indefinite integral equals the general antiderivative… = F(x) + C Where F’(x) = f(x)

Now let’s see if you’ve memorized specific anti-derivatives that you will need to know quickly during the AP exam….

sike! I just made that one up to scare you…now the rest will seem easy!

= ?

ax + C I hope you got that one!

= ?

+ C Ready?

= ??

- cos x + C Don’t forget we are going backwards! So if the derivative was positive, the anti-derivative is negative.

=?

sin x + C Got the negative/positive situation straight?? Good!

= ???

OK that’s a hard one! ln|tanx+sec x|+C If you got it right, you deserve a little treat!

= ?

tan x + C That should have been easy!

= ??

If you forget this one think: “tan x = sin x / cos x” (then let u = cos x, du = - sin x dx, etc.) -ln(cos x) + C or ln(sec x) + C

=??

ln |x| +C You need the absolute value in case x<0

where n > 1 Hint:

1/x n = x -n sooooooo……. the answer is: + C You didn’t say ln(x n ) did ya??

= ?

e x + c Easiest anti-derivative in the universe, eh?

= ?

sec x + C Another easy peasy as a daisy anti-derivative!

= ?

Not toooo difficult? -cot x + C Safe landing?

= ??

-csc x + C How are you holding up? Bored out of your gourd? Suck it up! You’ll thank me when you test out of college calculus!

= ???

+ C Grin and bear it!!

?

tan -1 x + C Keep it going!!

?

sin -1 x + C

?

sec -1 x + C It’s all down hill now!!!!

I said you are done! Stop clicking.

How do you compute the average value of ?

______________________ b - a dx Note: This is also known as the Mean (average) Value Theorem for Integrals

If = ky What does y = ?

Pre-Calculus trivia: doubling time is =

What’s general formula for a Riemann Sum?

or…more specifically Calculus trivia: as n (number of rectangles) goes to the summation sign becomes the integral sign and x becomes dx

What’s the Trapezoidal Rule?

The Trapezoidal Rule is the formula for estimating a definite integral with trapezoids. It is more accurate than a Riemann Sum which uses rectangles. Notice that all the y-values except the first and last are doubled. Do we need to take a short break?

Back already?

What is L’Hopital’s Rule?

Given that as x both f and g or both f and g then the limit of = the limit of as x L’Hopital’s Rule: ^

What is the Fundamental Theorem of Calculus???

where F ‘(x) = f(x) Do you know the other form? The one that is less commonly “used”? The FUN damental Theorem of Calculus:

What about if the question looks like this?

Did you remember Chain Rule?

What is the general integral for computing volume by slicing? (Assume we are revolving f(x) about the x-axis)

What if we revolve f(x) around y=a ?

What if we revolve the area between 2 functions: f(x) and g(x) around the x-axis?

Be sure to square the radii separately!!! (and put the larger function first)

1. How do you compute displacement? (distance between starting & ending points) 2. How do you compute total distance traveled?

displacement: total distance:

Yea!!! That’s all folks!