AP CALCULUS PERIODIC REVIEW

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AP CALCULUS PERIODIC REVIEW
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1 Limits and Continuity
A function y f(x) is continuous at x a if
i) f(a) is defined (it exists)
ii)
iii)
Otherwise, f is discontinuous at x a.
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2 Intermediate Value Theorem
A function y f(x) that is continuous on a
closed interval a,b takes on every value
between f(a) and f(b).
f(b)
f(a)
b
a
Note If f is continuous on a,b and f(a) and
f(b) differ in sign, then the equation f(x) 0
has at least one solution in the open interval
(a,b).
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3 Limits of Rational Functions as x???
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3 Limits of Rational Functions as x???
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3 Limits of Rational Functions as x???
Note The limit will be the ratio of the leading
coefficient of f(x) to g(x).
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4 Horizontal and Vertical Asymptotes
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4 Horizontal and Vertical Asymptotes
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5 Average Rate vs. Instantaneous Rate of Change
Average Rate of Change If (a, f(a)) and (b,
f(b)) are points on the graph of yf(x), then the
average rate of change of y with respect to x
over the interval a, b is
f(b)
f(a)
b
a
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5 Average Rate vs. Instantaneous Rate of Change
Instantaneous Rate of Change If (x0, y0) is a
point on the graph of yf(x), then the
instantaneous rate of change of y with respect to
x at x0 is f(x0).
f(b)
f(a)
b
a
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6 Limit Definition of a Derivative
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6 Limit Definition of a Derivative
AKA Difference Quotient
Geometrically, the derivative of a function at a
point is the slope of the tangent line to the
graph of the function at that point.
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7 The Number e is actually a limit
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8 Rolles Theorem
If f is continuous on a,b and differentiable on
(a, b) such that f(a) f(b), then there is at
least one number c in the open interval (a, b)
such that f(c) 0.
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9 Mean Value Theorem
If f is continuous on a,b and differentiable on
(a, b), then there is at least one number c in
the open interval (a, b) such that
f(b)
f(a)
b
a
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10 Extreme Value Theorem
If f is continuous on a closed interval a,b,
then f(x) has both a maximum and a minimum on
a,b.
CONSIDER
b
a
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11 Max / Min of Functions
To find maximum and minimum values of a function
y f(x), locate
1. the point(s) where f(x) changes sign. To
find the candidates first find where f(x) 0 or
is infinite or does not exist.
2. the end points, if any, on the domain of f(x).
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12 Increasing and Decreasing Intervals
If f(x) gt 0 for every x in (a, b), then f is
increasing on a, b.
If f(x) lt 0 for every x in (a, b), then f is
decreasing on a, b.
b
a
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13 Concavity and POI
If f(x) gt 0 for every x in (a, b), then f is
concave up a, b.
If f(x) lt 0 for every x in (a, b), then f is
concave down a, b.
b
a
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To locate the points of inflection of y f(x),
find the points where f(x) 0 or where f(x)
fails to exist. These are the only candidates
where f(x) may have a POI.
Then test these points to make sure that f(x) lt
0 on one side and f(x) gt 0 on the other
(changes sign).
b
a
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14 Differentiability and Continuity
Differentiability implies continuity If a
function is differentiable at a point x a, it
is continuous at that point.
The converse is false, that is, continuity does
NOT imply differentiability.
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15 Linear Approximation
The linear approximation of f(x) near x xo is
given by
1
1.1
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16 Comparing Rates of Change
The exponential function y ex grows rapidly as
x?? while the logarithmic function y ln x grows
very slowly as x??.
3x
x3
x2
ln x
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Exponential functions like y 2x or y ex grow
more rapidly as x?? than any positive power of x.
The function y ln x grows slower as x?? than
any nonconstant polynomial.
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Another way to look at this, as x??
1. f(x) grows faster than g(x) if
If f(x) grows faster than g(x) as x??, then g(x)
grows slower than f(x) as x??.
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Another way to look at this, as x??
2. f(x) and g(x) grow at the same rate as x?? if
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17 Inverse Functions
1. If f and g are two functions such that
f(g(x)) x for every x in the domain of g, and,
g(f(x)) x, for every x in the domain of f,
then, f and g are inverse functions of each other.
ex
f(x) ex
g(x) ln x
f(g(x)) eln x x
ln x
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17 Inverse Functions
4. If f is differentiable at every point on an
interval I, and f(x) ? 0 on I, then g f-1(x)
is differentiable at every point of the interior
of the interval f(I) and
f(x) ex
g(x) ln x
f(g(x)) eln x x
f(g(x)) g(x) 1
f(g(x)) g(x) 1/(f(g(x)))
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18 Properties of ex
1. The exponential function y ex is the
inverse function of y ln x.
2. The domain of y ex is the set of all real
numbers and the range is the set of all positive
numbers, ygt0.
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18 Properties of ex
3.
4. y ex is continuous, increasing, and concave
up for all x.
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18 Properties of ex
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19 Properties of ln x
1. The domain of y ln x is the set of all
positive numbers, x gt x.
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19 Properties of ln x
3. y ln x is continuous and increasing
everywhere on its domain.
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19 Properties of ln x
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20 Trapezoidal Rule
If a function, f, is continuous on the closed
interval a, b where a, b has been partitioned
into n subintervals of equal length, each
length (b a) / n, then
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21 Properties of the Definite Integral
If f(x) and g(x) are continuous on a, b
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21 Properties of the Definite Integral
If f(x) and g(x) are continuous on a, b
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21 Properties of the Definite Integral
If f(x) and g(x) are continuous on a, b
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21 Properties of the Definite Integral
If f(x) is an even function, then
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21 Properties of the Definite Integral
If f(x) is an odd function, then
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21 Properties of the Definite Integral
If f(x) ? 0 on a, b, then
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21 Properties of the Definite Integral
If g(x) ? f(x) on a, b, then
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22 Definition of a Definite Integral as the
Limit of a Sum
Suppose that a function f(x) is continuous on the
closed interval a, b. Divide the interval
into n equal subintervals, of length
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22 Definition of a Definite Integral as the
Limit of a Sum
Choose one number in each subinterval, i.e., x1
in the first, x2 in the second, ., xi in the ith
,., and xn in the nth. Then
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23 First Fundamental Theorem of Calculus
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23 Second Fundamental Theorem of Calculus
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24 PVA
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24 PVA
The velocity of an object tells how fast it is
going and in which direction. Velocity is an
instantaneous rate of change.
The speed of an object is the absolute value of
the velocity, ?v(t)?. It tells how fast it is
going disregarding its direction.
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24 PVA
The acceleration is the instantaneous rate of
change of velocityit is the derivative of the
velocitythat is, a(t) v(t).
Negative acceleration (deceleration) means that
the velocity is decreasing.
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24 PVA
The average velocity of a particle over the time
interval t0 to another time t, is
Where s(t) is the position of the particle at
time t.
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25 Average Value
The average value of f(x) on a, b is
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26 Area Between Curves
If f and g are continuous functions such that
g(x) ? f(x) on a, b, then the area
between the curves is
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27 Volume of Solids of Revolution
For volumes of solids rotated around the x (or y)
axis, volume
a
b
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27 Volume of Solids of Revolution
For washer method, volume
where f(x) is the large radius, and g(x) is the
small radius.
a
b
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27 Volume of Solids of Revolution
For cylinder method, volume
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28 Volume of Solids with Known Cross Sections
1. For cross sections of area A(x), taken
perpendicular to the x-axis, volume
2. For cross sections of area A(x), taken
perpendicular to the y-axis, volume
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28 Volume of Solids with Known Cross Sections
Some examples of these volumes are shown in the
next four slides
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29 Solving Differential Equations Graphically
and Numerically (Slope Fields)
At every point (x, y) a differential equation of
the form dy/dx f(x, y), gives the slope of the
member of the family of solutions that contains
that point.
At each point in the plane, a short segment is
drawn whose slope is equal to the value of the
derivative at that point. These segments are
tangent to the solutions graph at the point.
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29 Solving Differential Equations Graphically
and Numerically (Slope Fields)
y
You may be given an initial condition
x
O
This tells you exactly which of the possible
solutions is the answer.
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30 Solving Differential Equations by Separating
the Variables
Example of a differential equation
1. Rewrite the equation as an equivalent
equation with all the x and the dx terms on one
side and all the y and dy terms on the other.
2. Antidifferentiate both sides to obtain an
equation without dx or dy, but with one constant
of integration.
3. Use the initial condition (given) to evaluate
this constant.
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f
f
f
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