miscellaneous topics i’m going to ask you about various unrelated but important calculus topics....
TRANSCRIPT
![Page 1: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/1.jpg)
Miscellaneous Topics
• I’m going to ask you about various unrelated but important calculus topics.
• It’s important to be fast as time is your enemy on the AP Exam.
• 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.
![Page 2: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/2.jpg)
How many different methods are there for
evaluating limits?
Can you name several?
![Page 3: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/3.jpg)
1. Inspection
2. Observe graph
3. Create a table of values
4. Re-write algebraically
5. Use L’Hopitals Rule (only if the form is indeterminate)
6. Squeeze theorem (rarely used!!)
![Page 4: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/4.jpg)
How many indeterminate forms can you name?
![Page 5: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/5.jpg)
0
0
00
0
1
0
Did you know all 7?
1.
2.
3.
4.
5.
6.
7.
Math Wars!!!
![Page 6: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/6.jpg)
limx
xsin= ?
0x
![Page 7: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/7.jpg)
1
![Page 8: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/8.jpg)
lim0x
?cos1
x
x
![Page 9: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/9.jpg)
0Zero! Zip…
![Page 10: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/10.jpg)
What are the three main types of discontinuities?
![Page 11: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/11.jpg)
1. Hole – at x=3 in the example
2. Step – usually the function’s description is split up :
3. Vertical asymptote – at x=1 in the example
)3)(1(
)3(
xx
x
2xx2
for x<0
for x>0f(x)={
![Page 12: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/12.jpg)
Under what conditions does the derivative NOT exist at x=a
![Page 13: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/13.jpg)
If there is a discontinuity at x=a or if there is a sharp corner at x=a, then the
derivative is undefined at x=a
![Page 14: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/14.jpg)
What is the definition of continuity at a point?
![Page 15: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/15.jpg)
lim)(af )(xfax
![Page 16: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/16.jpg)
What is a monotone function?
![Page 17: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/17.jpg)
A function that is either always increasing or always decreasing.
(i.e. the derivative is always positive or always negative.)
![Page 18: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/18.jpg)
What is a normal line?
![Page 19: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/19.jpg)
The line perpendicular to the tangent line.
![Page 20: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/20.jpg)
?)()'( 1 bfGiven (a,b) is on the graph of f(x)
![Page 21: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/21.jpg)
)(
1
af Did you remember that one? It’s a bit esoteric, eh?
![Page 22: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/22.jpg)
What does the Squeeze Theorem say?
![Page 23: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/23.jpg)
ax
If both f(x) and g(x) as L ax
Then h(x) also. L
Given f(x) > h(x) > g(x) near
![Page 24: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/24.jpg)
What does the
Intermediate Value Theorem
say?
![Page 25: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/25.jpg)
If f(x) is continuous and p is a y-value between f(a) and f(b), then there is at
least one x-value between a and b such that f(c) = p.
![Page 26: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/26.jpg)
What is the formula for the slope of the secant line through (a,f(a)) and (b,f(b)) and what does it represent?
![Page 27: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/27.jpg)
ab
afbf
)()( average rate of
change in f(x) from x=a to x=b
Note: This differs from the derivative which gives exact instantaneous rate of change values at single x-value but you can use it to the derivative value at some values of x=c between a and b.
![Page 28: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/28.jpg)
What does the Mean Value Theorem say?
![Page 29: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/29.jpg)
If f(x) is continuous and differentiable, then for some c between a and b
ab
afbfcf
)()()(
That is the exact rate of change equals the average (mean) rate of change at some point in between a and b.
![Page 30: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/30.jpg)
What does f ‘ (a) = 0 tell you about the graph of f(x) ?
Warning: irrelevant picture
![Page 31: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/31.jpg)
The graph has a horizontal tangent line
at x=a.
f(a) might be a minimum or maximum…or perhaps just a horizontal
inflection point.
![Page 32: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/32.jpg)
What else must happen in addition to the derivative being zero or undefined at x=a in order for f(a) to be an extrema?
![Page 33: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/33.jpg)
The derivative must change signs at x=a
![Page 34: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/34.jpg)
What is the First Derivative Test?
![Page 35: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/35.jpg)
FIRST DERIVATIVE TEST
If f ‘(x) changes from + to – at x=a then f(a) is a local maximum.
If f ‘(x) changes from – to + at x=a then f(a) is a local minimum.
Dam that’s
a good test!!Dam, that’s a great test!!
![Page 36: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/36.jpg)
What’s the Second Derivative Test?
![Page 37: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/37.jpg)
Given f ‘(a)=0 then:
1. If f “ (a) < 0, f(a) is a relative max
2. If f “ (a) > 0, f(a) is a relative min
3. If f “ (a) = 0 the test fails
The Second Derivative Test:
Don’t be
Stumped...
Ha ha ha…
![Page 38: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/38.jpg)
What do you know about the
graph of f(x) if f “ (a) = 0
(or does not exist)?
![Page 39: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/39.jpg)
You know there might be an inflection point at x = a.
(Check to see if there is also a sign change in f “ at x = a to confirm the inflection point actually occurs)
![Page 40: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/40.jpg)
How do you determine velocity?
![Page 41: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/41.jpg)
Velocity = the first derivative of the position function,
or
v(a) +
(initial velocity + cumulative change in velocity)
b
adtta )(
![Page 42: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/42.jpg)
How do you determine speed?
![Page 43: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/43.jpg)
Speed = absolute value of velocity
![Page 44: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/44.jpg)
How do you determine acceleration?
![Page 45: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/45.jpg)
acceleration =
first derivative of velocity =
second derivative of position
![Page 46: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/46.jpg)
Using differentials to approximate f(a+h) with a point near (a,f(a)) on the tangent
line… what does f(a+h) ?
This is driving me nuts!!!!
![Page 47: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/47.jpg)
f(a+h) f(a) + f ‘(a) h
The differential or df or dy or “error”
= f ‘(a) h
![Page 48: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/48.jpg)
If f ‘(x) is negative….
![Page 49: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/49.jpg)
Then f(x) is decreasing….
![Page 50: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/50.jpg)
If f ‘(x) is positive….
![Page 51: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/51.jpg)
Then f(x) is increasing….
![Page 52: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/52.jpg)
If f “ (x) is negative then…
![Page 53: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/53.jpg)
f(x) is concave down
![Page 54: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/54.jpg)
If f “ (x) is positive then…
![Page 55: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/55.jpg)
f(x) is concave up
![Page 56: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/56.jpg)
How do you compute the average value of
?
![Page 57: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/57.jpg)
b
a______________________
b - a
dx
Note: This is also known as the
Mean (average) Value Theorem for Integrals
![Page 58: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/58.jpg)
How do you locate and confirm vertical and horizontal asymptotes?
![Page 59: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/59.jpg)
Vertical – suspect them at x-values which cause the denominator of f(x) to be zero.
Confirm that the limit as x a is infinite….
Horizontal – suspect rational functions
Confirm that as x , y a
![Page 60: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/60.jpg)
If = ky
What does y = ?
dt
dy
![Page 61: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/61.jpg)
ktAey Calculus trivia: doubling time is =
k
2ln
![Page 62: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/62.jpg)
What’s general formula for a Riemann Sum?
![Page 63: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/63.jpg)
n
kkk xxf
1
)(
n
knab
nab kaf
1
)(
or…more specifically
Calculus trivia: as n (number of rectangles) goes to the summation sign becomes the integral sign and x becomes dx
![Page 64: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/64.jpg)
What’s the
Trapezoidal Rule?
![Page 65: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/65.jpg)
The Trapezoidal Rule is the formula for estimating a definite integral with trapezoids. It is more accurate
than a Riemann Sum which uses rectangles.
)]()(2)(2)(2)([ 121021
nn xfxfxfxfxfxT
Notice that all the y-values except the first and last are doubled.
Do we need to take a short
break?
![Page 66: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/66.jpg)
Back already?
![Page 67: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/67.jpg)
What is L’Hopital’s Rule?^
![Page 68: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/68.jpg)
Given that as x both f and g
)(
)(
xg
xf
a 0
or both f and g then the limit of
= the limit of )('
)('
xg
xf
as x a
L’Hopital’s Rule:^
![Page 69: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/69.jpg)
What is Newton’s Formula for approximating the zeros of a function?
![Page 70: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/70.jpg)
)('
)(1
n
nnn xf
xfxx
Ain’t that ducky!!
![Page 71: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/71.jpg)
What is the Fundamental Theorem of Calculus???
![Page 72: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/72.jpg)
b
abFaFdxxf )()()(
where F ‘(x) = f(x)
Do you know the other form?
The one that is less commonly “used”?
The FUNdamental Theorem of Calculus:
![Page 73: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/73.jpg)
x
axfdttf
dx
d)()(
![Page 74: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/74.jpg)
What is the general integral for computing volume by slicing?
(Assume we are revolving f(x) about the x-axis)
![Page 75: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/75.jpg)
dxxf 2))((
What if we revolve f(x)
around y=a ?
![Page 76: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/76.jpg)
dxaxf 2))((
![Page 77: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/77.jpg)
What if we revolve the area between 2 functions: f(x) and
g(x) around the x-axis?
![Page 78: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/78.jpg)
dxxgxf 22 ))(())((Be sure to square the radii
separately!!!(and put the larger function first)
![Page 79: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/79.jpg)
1. How do you compute displacement?
(distance between starting & ending points)
2. How do you compute total distance traveled?
![Page 80: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/80.jpg)
displacement:
total distance:
kt
tdttv
0
)(
kt
tdttv
0
|)(|
![Page 81: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/81.jpg)
Yea!!! That’s all folks!
![Page 82: Miscellaneous Topics I’m going to ask you about various unrelated but important calculus topics. It’s important to be fast as time is your enemy on the](https://reader036.vdocuments.mx/reader036/viewer/2022062423/56649e9e5503460f94b9ffc9/html5/thumbnails/82.jpg)
(Be sure to check out the other calculus power point
drill and practices)