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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.
What is the definition of LIMIT?
OK…this is like the basis of ALL of Calculus.
It was finally “perfected” by Cauchy in 1821.
Ready?
Given any if there is a corresponding such that
implies then we say that
lim
0 0
ax0 Lxf )(
Lxf )(ax
(This is the bare bones important part that you need to memorize…check your text for the detailed version.)
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!!)
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)={
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
A function that is either always increasing or always decreasing.
(i.e. the derivative is always positive or always negative.)
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.
What is the formula for the slope of the secant line through (a,f(a)) and (b,f(b)) and what does it represent?
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.
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.
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.
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?
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!!Darn, that’s a great test!!
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…
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)
Velocity = the first derivative of the position function,
or
v(a) +
(initial velocity + cumulative change in velocity)
b
adtta )(
b
a______________________
b - a
dx
Note: This is also known as the
Mean (average) Value Theorem for Integrals
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
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?
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:^
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:
What is the general integral for computing volume by slicing?
(Assume we are revolving f(x) about the x-axis)
1. How do you compute displacement?
(distance between starting & ending points)
2. How do you compute total distance traveled?