AP Calculus Review

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AP Calculus Review
Many slides will send you to websites for
additional examples. A very good website is
Visual Calculus at http//archives.math.utk.edu/
visual.calculus/index.html. Click anywhere in
this box to go there.
By Jeff Willets
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Table of Contents

• Mean Value Theorem
• Max/MinsPoints Of Inflection
• Fundamental Theorems of Calculus
• Reimann Sums
• Trapezoid Rule
• Linear Approximations
• Motion Problemsposition, velocity, acceleration
• Solids of Rotation
• List of Websites

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The Mean Value Theorem
If f(x) is a differentiable function on (a, b)
, then there exists a value c on (a, b) such that
ExamplesClick on graphs
In other words, there must be a point somewhere
on the curve with a tangent line parallel to the
line connecting the endpoints.
The Mean Value Theorem If f(x) is a
differentiable function on (a, b) , then there
exists a value c on (a, b) such that
In other words, there must be a
point somewhere on the curve with a tangent line
parallel to the line connecting the endpoints.
?
In other words, there must be a point somewhere
on the curve with a tangent line parallel to the
line connecting the endpoints.
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First Derivatives

• Consider a function f(x).
• Whenever f(x)gt0, f(x) is increasing
• Whenever f(x)lt0, f(x) is decreasing
• If f(c) is defined and f(c) 0 or f(c) is
  undefined, c is called a critical value.
• (Caution There are cases where f(c) is
  undefined but so is f(c). This would not be a
  critical value. An example of this would be the
  function y 1/x.)

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First Derivatives

• f(c) is the slope of the tangent line at c.
• Assuming f(c) is defined, if
• f(c)0, there is a horizontal tangent line
  (Points A and B)
• f(c) is undefined, there is a vertical tangent
  line (C and D)

D
C
B
A
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Max/Mins
If c is a critical value, then f(c) could be a
relative maximum, relative minimum, or neither.
We can determine this with a first derivative
test. If f(x) is negative to the left and
positive to the right of c, then f(c) is a
relative minimum. If f(x) is positive to the
left and negative to the right of c, then f(c) is
a relative maximum. If f(x) is positive on both
sides of c or negative on both sides of c, then
f(c) is neither a minimum or a maximum.
---
c
---
c

c
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Max/Mins
Endpoints on a closed interval must always be
considered for absolute max/mins. To determine
absolute max/mins, first you must make sure they
exist. If any limits are /- infinity (at
asymptotes or extremes) appropriate absolutes
will not exist. If absolutes do exist, they will
be the endpoint or relative max/mins with the
greatest (or least) y-values.
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Second Derivatives/Points of Inflection

• The second derivative tells you the concavity of
  the function.
• If f(x)gt0, the function is concave up.
• If f(x)lt0, the function is concave down.
• A point of inflection is
• where the function
• switches concavity
• (A, B, and C)

C
A
B
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Second Derivatives/Points of Inflection

• If f(c) exists and f(c)0 or f(c) is
  undefined, then (c, f(c)) is a possible point of
  inflection.
• If f(x) switches from positive to negative or
  negative to positive at c, then (c, f(c)) is a
  point of inflection.
• If f(x) does not switch signs at c, then (c,
  f(c)) is not a point of inflection.

---
---
c
c

--- ---
c
c
Table of Contents
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Second Derivatives/Tests for Max/Mins

• The second derivative test can also be used to
  determine if critical points are maximums or
  minimums.
• So if f(c) is defined and f(c)0 or f(c) is
  undefined and
• f(c)gt0, then (c, f(c)) is a relative minimum
  (since the function is concave up at that point)
• f(c)lt0, then (c, f(c)) is a relative maximum
  (since the function is concave down at that
  point)
• f(c)0, then the 2nd derivative test is
  inconclusive. A first derivative test must be
  used in this case.

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Fundamental Theorem

• f is a continuous function on a, b and Ff.
• Then
• There is also the Second Fundamental Theorem.
• It states that

Examples
Examples (at Bottom of site)
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Riemann Sums
Here are the three standard types of Riemann
Sums, each broken into ten rectangles
Left-handed Riemann SumNotation L(10)
Midpoint Riemann SumNotation M(10)
Right-handed Riemann SumNotation R(10)
Visit Website with more explanations And examples.
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Trapezoid Rule
n is the number of trapezoids, ?x is the width of
each trapezoid (which can be determined by
(b-a)/n.) Note that there will always be one
more term in the parentheses than there are
trapezoids.
T(10).
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Linear Approximations
Linear Approximation is a method to approximate a
value by using a value along the tangent line
close to the point of tangency. If (a, f(a)) is
the point of tangency of the line to the function
f(x) below, then for x values near a,
This
is merely the point-slope equation of line.
(x,f(x))
(a,f(a))
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Motion Problems
Let s(t) be the position of an object at time t.
(Sometimes it might be called x(t) or
y(t).) Then v(t) s(t), the velocity at time
t. a(t) v(t) s(t), the acceleration at
time t. The sign of the velocity tells the
direction it is moving. Positive usually means
right or up. The sign of the acceleration tells
the direction it is accelerating.
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Motion Problems
Speedthe difference between speed and velocity
is that velocity has direction, and speed does
not. Whenever the velocity and acceleration have
the same sign, speed is increasing. When they
have different signs, the speed is decreasing.
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Motion Problems
Finding Position Often you will be given a
velocity function v(x) and an initial position
s(0). You can find a position s(t) by
This follows directly from the fundamental
theorem of calculus.
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Motion Problems
Total Distance Traveled vs. Displacement Displacem
ent is the net change in position (final position
starting position) It is found as follows
Total distance traveled is found as follows
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Solids of Rotation
There are basically two types of rotations, the
disc/washer method and the shell method. The
main difference is that the disc/washer method
has the rectangles sliced perpendicular to the
axis of rotation, and the shell method has them
sliced parallel to the axis of rotation.
Whenever the rectangles are vertical, the
variable will be x (and dx) and the limits of
integration will be the x limits. When the
rectangles are horizontal, the variable will be y
(and dy) and the limits of integration will be
the y limits.
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Disc Method
Visit Website with more explanations And examples.
In this example, we are going to rotate the
region bounded by y x2, y1, and the y axis
about the y-axis. Slicing it horizontally, we
get rectangles perpendicular to the axis of
rotation, which is what is needed for the
disc/washer method. We will use the formula
R is the distance from the axis of rotation to
the function (note that we must convert
everything to be in terms of y.) R y1/2
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Washer Method
Visit Website with more explanations And examples.
In this example, we are going to rotate the
region bounded by y x and y x4 about the line
y 2. Slicing it vertically, we get rectangles
perpendicular to the axis of rotation, which is
what is needed for the disc/washer method. We
will use the formula
R and r are the distances from the axis of
rotation to the functions (with R being the
bigger one.) In this example, R 2-x4 and r
2-x.
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Shell Method
Visit Website with more explanations And examples.
In this example, we are going to rotate the
region bounded by y x and y x4 about the line
x 1. Slicing it vertically, we get rectangles
parallel to the axis of rotation, which is what
is needed for the shell method. We will use the
formula
r is the distance from the axis of rotation to
the rectangle, and h is the height of the
rectangle. In this example, r 1-x and h x-x4
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Known Cross Sections
With known cross sections, you will be given a
base and told that the base will be the base of
certain kind of shapes. In this example, the
base will be the area between y x4 and y x,
and each rectangle will be the base of a square.
We will always use the formula
A is the area of the square, which will be h2, or
(x x4)2.
Visit Website with more explanations And examples.
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List of Websites

• archives.math.utk.edu/visual.calculus/
• http//mathdemos.gcsu.edu/mathdemos/solids/index.h
  tml
• www.sosmath.com/calculus/calculus.html
• www.plu.edu/heathdj/java/
• people.hofstra.edu/faculty/Stefan_Waner/RealWorld/
  tccalcp.html
• apcentral.collegeboard.com/
• Calculus The Musical
• I Will Derive