Miscellaneous Topics
Calculus Drill!!
Developed by Susan Cantey
at Walnut Hills H.S.
2006
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 
 0 if there is a corresponding   0
0 xa  
implies
f ( x)  L  
such that
then we say that
lim f ( x)  L
x a
(This is the bare bones important part that you need to
memorize…check your text for the detailed version.)
How many different
methods are there for
evaluating limits?
Can you name
several?
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!!)
How many indeterminate
forms can you name?
1.
0
0
2.


Math Wars!!!
3. 0  
4.   
5.
6.
7.

0
0

1
0
Did you know all 7?
lim
x 0
sin x
x
=?
1
1

cos
x
lim
?
x 0
x
0
Zero! Zip…
What are the three main types
of discontinuities?
( x  3)
( x  1)( x  3)
1. Hole – at x=3 in the example
2. Step – usually the function’s description is split up :
x
f(x)=
2
{ 2x
for x<0
for x>0
3. Vertical asymptote – at x=1 in the example
Under what conditions does the derivative NOT
exist at x=a
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
What is the definition of
continuity at a point?
f (a )  lim f (x)
x a
What is a monotone function?
A function that is either always
increasing or always decreasing.
(i.e. the derivative is always positive
or always negative.)
What is a normal line?
The line perpendicular to the
tangent line.
1
( f )' (b)  ?
Given (a,b) is on the graph of f(x)
1
f ( a )
Did you remember that one? It’s a bit esoteric, eh?
What does the Squeeze Theorem say?
Given f(x) > h(x) > g(x) near
If both f(x) and g(x)
Then h(x)
L
L
also.
as
xa
x a
What does the
Intermediate Value Theorem
say?
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?
f (b)  f ( a )
ba

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.
What does the Mean Value
Theorem say?
If f(x) is continuous and differentiable,
then for some c between a and b
f (b)  f (a)
f (c) 
ba
That is the exact rate of change equals
the average (mean) rate of change at
some point in between a and b.
What does f ‘ (a) = 0 tell you about the
graph of f(x) ?
Warning: irrelevant picture
The graph has a horizontal tangent line
at x=a.
might
f(a)
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?
The derivative must change signs at x=a
What is the First Derivative Test?
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
that’s
a Dam,
good test!!
a great
test!!
What’s the Second Derivative Test?
The Second Derivative 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
Don’t be
Stumped...
Ha ha ha…
What do you know about the
graph of f(x) if f “ (a) = 0
(or does not exist)?
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)
How do you determine velocity?
Velocity = the first derivative of the
position function,
or
b
v(a) +
 a(t )dt
a
(initial velocity + cumulative change in velocity)
How do you determine speed?
Speed = absolute value of velocity
How do you determine acceleration?
acceleration =
first derivative of velocity =
second derivative of position
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!!!!
f(a+h)

f(a) + f ‘(a)
h
The differential or df or dy or “error”

= f ‘(a) h
If f ‘(x) is negative….
Then f(x) is decreasing….
If f ‘(x) is positive….
Then f(x) is increasing….
If f “ (x) is negative then…
f(x) is concave down
If f “ (x) is positive then…
f(x) is concave up
How do you compute the
average value of
?

b
dx
a
______________________
b-a
Note: This is also known as the
Mean (average) Value Theorem for Integrals
How do you locate and confirm vertical and
horizontal asymptotes?
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
If
dy
dt
= ky
What does y = ?
y  Ae
Calculus trivia: doubling time is =
kt
ln 2
k
What’s general formula for a
Riemann Sum?
n
 f ( x )x
k
k 1
k
or…more specifically
n
 f (a 
k 1
ba
n
k) 
ba
n
Calculus trivia: as n (number of rectangles)
goes to
the summation sign becomes
the integral sign and  x becomes dx

What’s the
Trapezoidal Rule?
The Trapezoidal Rule is the formula for estimating a
definite integral with trapezoids. It is more accurate
than a Riemann Sum which uses rectangles.
T  12 x[ f ( x0 )  2 f ( x1 )  2 f ( x2 )      2 f ( xn1 )  f ( xn )]
Notice that all the y-values except the first and last are doubled.
Do we need to
take a short
break?
Back already?
^
What is L’Hopital’s Rule?
^
L’Hopital’s
Rule:
Given that as x  a both f and g  0
or both f and g
f ( x)
g ( x)

then the limit of
= the limit of
as x
a
f ' ( x)
g ' ( x)
What is Newton’s Formula for
approximating the zeros of a
function?
f ( xn )
xn 1  xn 
f ' ( xn )
Ain’t that
ducky!!
What is the
Fundamental
Theorem of
Calculus???
The FUNdamental Theorem of Calculus:

b
a
f ( x)dx  F (a)  F (b)
where F ‘(x) = f(x)
Do you know the other form?
The one that is less commonly “used”?
d x
f
(
t
)
dt

f
(
x
)

a
dx
What is the general integral for
computing volume by slicing?
(Assume we are revolving f(x)
about the x-axis)
  ( f ( x)) dx
2
What if we revolve f(x)
around y=a ?
  ( f ( x)  a) dx
2
What if we revolve the area
between 2 functions: f(x) and
g(x) around the x-axis?
  ( f ( x))  ( g ( x)) dx
2
2
Be sure to square the radii
separately!!!
(and put the larger function first)
1. How do you compute displacement?
(distance between starting & ending points)
2. How do you compute total distance
traveled?
displacement:

tk

tk
t0
total distance:
t0
v(t )dt
| v (t ) | dt
Yea!!! That’s all
folks!
(Be sure to check out the
other calculus power point
drill and practices)