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AP Calculus
Limits, Continuity and Derivatives
The limit of a function f ( x) as x approaches some number a is the value the function is getting
close to as x gets close to a. It is not necessarily equal to the value of the function when x = a. If
it does equal that value, then the function is said to be continuous at x = a.
Example 1:
lim f ( x)  3 but f (2)  1 :
y
x 2
f
f ( x) is not continuous at x = 2.
4
2
O
Example 2:
x 2
lim f ( x)  f (2)  3
y
f ( x) is continuous at x = 2.
4
2
4
x
2
4
x
f
2
O
The left limit of a function as x approaches a is the number the function approaches as x gets near a
while always remaining on the left of a, and the right limit is the number the function approaches as x
gets near a while always remaining on the right of a:
Example 3:
y
lim f ( x)  1
f
x2
4
f (2)  4
lim f ( x)  3
2
x  2
O
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2
4
x
A function is continuous on an interval if it is continuous at every value of x in that interval. You can
draw a continuous function without lifting your pen off the paper. A function is differentiable on an
interval if it is continuous on that interval and has no corners.
Example 4:
This function is continuous on [1, 3]
y
but not differentiable at x = 2:
O
1
2
3
x
1
2
3
x
y
Example 5:
This function is differentiable on [1, 3]:
O
A differentiable function is not only continuous, it is also “smooth.”
An important result about continuous functions is the Intermediate Value Theorem:
If a function is continuous on [a, b], then it takes on every value
between f (a) and f (b) .
Precisely,
If f ( x) is continuous on [a, b] and k is a number between f (a) and f (b)
then there is a number c between a and b for which f (c)  k :
y
f (a )
k
f (b)
O
a
c
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b
x
If a function f gets close to a certain number L when x gets larger and larger, then we say that the
limit as x goes to infinity is L and we write: lim f ( x)  L .
x 
Likewise, if f gets close to L when x gets smaller and smaller, then the limit as x goes to negative
infinity is L and we write: lim f ( x)  L .
x 
In both cases, the line y = L is a horizontal asymptote of f.
Example 6:
lim
x
1
1
 0 because when x is very large,
is close to 0. The x-axis is a horizontal
x
x
asymptote of the function f ( x) 
1
:
x
y
1
y 
x
x
If a function f ( x) gets larger and larger as x gets close to a number a, then it “goes to infinity” and
we write: lim f ( x)   . The line x = a is a vertical asymptote of f.
x a
Similarly, if f ( x) gets smaller and smaller as x gets close to a, then it “goes to negative infinity”
and we write: lim f ( x)   . Again, the line x = a is a vertical asymptote of f.
x a
An important result:
If lim | f ( x) |   then lim
x a
xa
1
 0 . This is because one over a very large
f ( x)
positive number and one over a huge negative number are both close to 0.
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The derivative of a smooth function f at a point (a, f (a) ) is the slope of the line tangent to the
graph of y  f ( x) at the point where x = a. It can be defined as the limiting slope of secant lines in
two ways:
y
f (a)  lim
h0
f (a  h)  f (a)
h
m 
O
a
f (a  h)  f (a)
h
x
a+h
y
f (a)  lim
ba
f (b)  f (a)
ba
m 
O
a
b
f (b)  f (a)
ba
x
The second derivative of a function f is derivative of the derivative of f:
f ( x)  lim
h0
f ( x  h)  f ( x)
h
In problems relating position (or distance) to time, velocity is the first derivative of the position with
respect to time, and acceleration is the derivative of the velocity with respect to time, which is the
second derivative of the position with respect to time.
Some important results about first derivatives can be deduced from graphs of smooth functions:
(1)
If f ( x) is positive on an interval, then f is increasing on that interval.
(2)
If f ( x) is negative on an interval, then f is decreasing on that interval.
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(3)
If f (a)  0 , and for some small positive number p and all h between p and 0,
f (a  h)  0 and f (a  h)  0 then the point (a, f (a) ) is a relative maximum of f
(that is, it is at the “top of the hill”).
(4)
If f (a)  0 , and for some small positive number p and all h between p and 0,
f (a  h)  0 and f (a  h)  0 then the point (a, f (a) ) is a relative minimum of f (that
is, it is at the “bottom of the hill”).
(5)
If f (a)  0 , and for some small positive number p and all h between p and 0, f (a  h)
and f (a  h) are both positive or both negative, then the point (a, f (a) ) is an “inflection
point” of f (that is, a point where the curvature of the graph changes shape).
Example 8:
In the graph below, f has a relative maximum when x = a and a relative minimum
when x = b.
relative
maximum
y
f  0
f  0
f  0
f  0
f  0
f  0
O
relative
minimum
b
a
x
Some important results about the second derivative can also be deduced from looking at graphs:
(6)
If f ( x)  0 on an interval, then f  is increasing, which means the graph of f is curving
upwards faster and faster as you go from left to right. We call this “concave upward”:
y
Example 9:
f is concave upward on [a, b]:
O
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a
b
x
(7)
If f ( x)  0 on an interval, then f  is decreasing, which means the graph of f is curving
upwards slower and slower as you go from left to right. We call this “concave downward”:
Example 10: f is concave downward on [a, b]:
y
O
(8)
a
b
x
A point at which f (a)  0 , and f (a  h) and f (a  h) are of opposite signs is an
inflection point, since the concavity of the curve changes on both sides of x = a.
To understand inflection points, think of driving a car. You take off with your foot on the
accelerator, so your acceleration (the second derivative of your position) is positive. Then you put
on the brakes, so your acceleration is negative. At the moment you move your foot from the
accelerator to the brakes, your acceleration is zero, and this is an inflection point of the graph of
position vs. time:
x
x  f (t )
inflection point
slowing down
f   0
speeding up
f   0
O
t
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