a moment of silence for our great calculus “father” please
TRANSCRIPT
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A moment of silence for our great calculus “father” please.
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OK…here we go!
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Integrals: Drill & Practice
• I’m going to ask you about integrals.
• It’s important to be as fast as possible because time is your enemy .
• When you think you know the answer,
(or if you give up ) click to get to the next slide to see if you were correct.
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First let’s talk about what the integral means!
Can you list some interpretations of the definite integral?
b
adxxf )(
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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]
b
adxxf )(
b
adxxf )(
b
adxxf )(
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What is a Riemann Sum?
Hint: Here’s a picture!
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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!
n
kkk xxf
1
)(
dxx
b
a
n
kk xfxf )()(
1
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The indefinite integral
= ? dxxf )(
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Well…hard to write; easy to say
The indefinite integral equals the general antiderivative…
= F(x) + C Where F’(x) = f(x) dxxf )(
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Now let’s see if you’ve memorized specific anti-derivatives that you will need to know quickly
during the AP exam….
dxxx
x
tansin
1 2
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sike!
I just made that one up to scare you…now the rest will seem easy!
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= ?adx
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ax + C
I hope you got that one!
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= ?
dxx n
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+ C
Ready?
111
nn x
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= ??
xdxsin
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- cos x + C
Don’t forget we are going backwards!
So if the derivative was positive, the
anti-derivative is negative.
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=? xdxcos
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sin x + CGot the negative/positive situation straight??
Good!
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= ???
xdxsec
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OK that’s a hard one!
ln|tanx+sec x|+CIf you got it right, you deserve a
little treat!
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= ?
xdx2sec
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tan x + CThat should have been easy!
Piece of cake! Upside down!!
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= ?? xdxtan
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If you forget this onethink: “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
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=??
dxx
1
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ln |x| +CYou need the absolute value in case x<0
Rise to the highest! Sursum ad Summum
yada yada
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where n > 1
Hint:
dxx n1
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1/xn = x-n
sooooooo…….the answer is:
+ C
You didn’t say ln(xn) did ya??
11
1
nn x
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= ?
dxe x
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ex + cEasiest anti-derivative in the universe, eh?
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= ?
xdxx tansec
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sec x + C
Another easy peasy as a daisy anti-derivative!
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= ?
xdx2csc
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Not toooo difficult?
-cot x + C
Safe landing?
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= ??
xdxx cotcsc
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-csc x + CHow are you holding up?
Bored out of your gourd?Suck it up! You’ll thank me when you test out of
college calculus!
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= ???
dxa x
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+ C
Grin and bear it!! Ha Ha
xa aln1
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OK! Take a deep breath!
5 more questions!
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?
dx
x 21
1
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tan-1x + C
Keep it going!!
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?
dxx 21
1
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sin-1x + COh yeah! Only 3
more to go.
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?
dx
xx 1||
12
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sec-1x + C
It’s all down hill now!!!!
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?udv
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(Did you get the significance of the picture?)
vduuvudv
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R U ready4 the last ?
?
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= ???
dx
bxax ))((
1
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= A ln(x-a) + B ln(x-b) + C
(I’m assuming you know how to find A & B)
dxbx
B
ax
Adx
bxax
))((
1
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You’re done!Ta Ta for now.
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Be sure to check out these other power point slide shows:
Derivatives
Pre-Calculus Topics (on a separate page)
Sequences and Series
Miscellaneous Topics
and
Additional BC Topics
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I said you are done!
Stop clicking.