Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006.
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Transcript of Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006.
![Page 1: Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006.](https://reader035.fdocuments.in/reader035/viewer/2022062511/55143a85550346414e8b4629/html5/thumbnails/1.jpg)
Miscellaneous TopicsCalculus Drill!!
Developed by Susan Cantey
at Walnut Hills H.S.
2006
![Page 2: Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006.](https://reader035.fdocuments.in/reader035/viewer/2022062511/55143a85550346414e8b4629/html5/thumbnails/2.jpg)
Miscellaneous Topics
• I’m going to ask you about various unrelated but important calculus topics.
• 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. 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!!)
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limx
xsin= ?
0x
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1
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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. 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)={
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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 normal line?
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The line perpendicular to the tangent line.
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What does the Squeeze Theorem say?
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ax
If both f(x) and g(x) as L ax
Then h(x) also. L
Given f(x) > h(x) > g(x) near
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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 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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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?
![Page 48: Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006.](https://reader035.fdocuments.in/reader035/viewer/2022062511/55143a85550346414e8b4629/html5/thumbnails/48.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
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Back already?
![Page 50: Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006.](https://reader035.fdocuments.in/reader035/viewer/2022062511/55143a85550346414e8b4629/html5/thumbnails/50.jpg)
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
abFaFdxxf )()()(
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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What is the general integral for computing volume by slicing (disk method)?
(Assume we are revolving f(x) about the x-axis)
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dxxf 2))((
What if we revolve f(x)
around y=a ?
![Page 56: Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006.](https://reader035.fdocuments.in/reader035/viewer/2022062511/55143a85550346414e8b4629/html5/thumbnails/56.jpg)
dxaxf 2))((
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What if we revolve the area between 2 functions: f(x) and
g(x) around the x-axis?
![Page 58: Miscellaneous Topics Calculus Drill!! Developed by Susan Cantey at Walnut Hills H.S. 2006.](https://reader035.fdocuments.in/reader035/viewer/2022062511/55143a85550346414e8b4629/html5/thumbnails/58.jpg)
dxxgxf 22 ))(())((Be sure to square the radii
separately!!!(and put the larger function first)
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Yea!!! That’s all folks!