Miscellaneous Topics Calculus Drill!!libvolume3.xyz › ... ›...
Transcript of Miscellaneous Topics Calculus Drill!!libvolume3.xyz › ... ›...
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Miscellaneous Topics
Calculus Drill!!
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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.
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How many different
methods are there for
evaluating limits?
Can you name
several?
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1. Direct Substitution
2. Observe graph
3. Create a table of values
4. Re-write algebraically
(Simplify)
5. Use L’Hopitals Rule (only if the form is indeterminate)
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How many indeterminate
forms can you name?
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0
0
∞
∞
∞−∞
00
0∞
∞1
∞⋅0
Did you know all 7?
1.
2.
3.
4.
5.
6.
7.
Math Wars!!!
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lim x
xsin= ?
0→x
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1
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lim0→x
?cos1
=−
x
x
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0Zero! Zip:
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What are the three main types
of discontinuities?
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1. Hole – at x=3 in the example
2. Vertical asymptote – at x=1 in
the example
3. Step – usually the function’s description is split up :
)3)(1(
)3(
−−
−
xx
x
32
for x<0
for x>0 f(x)={
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Under what conditions does the derivative NOT
exist at x=a
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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
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What is the definition of
continuity at a point?
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lim=)(af )(xfax→
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What is a monotone function?
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A function that is either always
increasing or always decreasing.
(i.e. the derivative is always positive
or always negative.)
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What is a normal line?
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The line perpendicular to the
tangent line.
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?)()'( 1 =− bf
Given (a,b) is on the graph of f(x)
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)(
1
af ′
Did you remember that one? It’s a bit esoteric, eh?
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What does the
Intermediate Value Theorem
say?
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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, c, between a and b
such that f(c) = p.
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What is the formula for the
slope of the secant line
through (a,f(a)) and (b,f(b))
and what does it represent?
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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.
≈
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What does the Mean Value
Theorem say?
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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.
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What does f ‘ (a) = 0 tell you about the
graph of f(x) ?
Warning: irrelevant picture
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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.
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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?
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The derivative must change signs at x=a
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What is the First Derivative Test?
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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!!
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What’s the Second Derivative Test?
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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:
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What do you know about the
graph of f(x) if f “ (a) = 0
(or does not exist)?
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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)
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How do you determine velocity?
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Velocity = the first derivative of the
position function,
or
v(a) +
(initial velocity + cumulative change in velocity)
∫b
adtta )(
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How do you determine speed?
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Speed = absolute value of velocity
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How do you determine acceleration?
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acceleration =
first derivative of velocity =
second derivative of position
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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!!!!
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f(a+h) f(a) + f ‘(a) h ⋅≈
Don’t make an
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If f ‘(x) is negative:.
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Then f(x) is decreasing:.
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If f ‘(x) is positive:.
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Then f(x) is increasing:.
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If f “ (x) is negative then:
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f(x) is concave down
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If f “ (x) is positive then:
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f(x) is concave up
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How do you compute the
average value of
?
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∫b
a______________________
b - a
dx
Note: This is also known as the
Mean (average) Value Theorem for Integrals
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How do you locate and confirm vertical and
horizontal asymptotes?
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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 ±∞→ →
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If = ky
What does y = ?
dt
dy
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kty Ce=
Calculus trivia: doubling time is =
If you know ½ life, how do you find k?
k
2ln
ln(what it does)
time it takes to get therek =
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What’s general formula for a
Riemann Sum?
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∑=
∆n
k
kk xxf1
)(
∑=
−− ⋅+n
k
nab
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 ∞
∆
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What’s the
Trapezoidal Rule?
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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)([12102
1nnxfxfxfxfxfxT ++⋅⋅⋅+++∆= −
Notice that all the y-values except the first and last are doubled.
Use this only if the partitions are even, otherwise do them all by
hand separately.
Do we need to take a short
break?
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Back already?
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What is L’Hopital’s Rule? ^
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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: ^
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What is the
Fundamental
Theorem of
Calculus???
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∫ −=b
aaFbFdxxf )()()(
where F ‘(x) = f(x)
Do you know the other form?
The one that is less commonly “used”?
The FUNdamental Theorem of Calculus:
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∫ =x
axfdttf
dx
d)()(
YAY!! Steve’s Theorem!!
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What is the general integral for
computing volume by slicing?
(Assume we are revolving f(x)
about the x-axis)
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∫ dxxf2))((π
What if we revolve f(x)
around y=a ?
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∫ − dxaxf2))((π
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What if we revolve the area
between 2 functions: f(x) and
g(x) around the x-axis?
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∫ − dxxgxf 22 ))(())((π
Be sure to square the radii
separately!!! (and put the larger function first)
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1. How do you compute displacement?
(distance between starting & ending points)
2. How do you compute total distance
traveled?
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displacement:
total distance:
∫kt
tdttv
0
)(
∫kt
tdttv
0
|)(|
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Yea!!! That’s all
folks!
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(Be sure to check out the
other calculus power point
drill and practices)