Lecture 14: Antiderivatives
14.1 Antiderivatives
Denition Suppose F and f are dened on an open interval (a, b) with F (x) = f (x)
for all x in (a, b). Then we call F an antiderivative of f on (a, b).
The following result follows from our earlier work with
Lecture 13: Examples of Extreme Value Problems
13.1 Example: Continuous function on a closed interval
Example Suppose a farmer wishes to enclose a rectangular eld using 1000 yards of
fencing in such a way that the area of the eld is maximized. Let x and y
Lecture 15: Sums
15.1 Summation notation
It will be useful in our work to use the following notation: If a1 , a2 , . . . , an are numbers,
then
n
ai = a1 + a2 + + an .
i=1
Example
If a1 = 1, a2 = 3, a3 = 1, and a4 = 6, then
4
ai = a1 + a2 + a3 + a4 = 1 +
Lecture 11: Limits at Innity
11.1 Limits at innity
Denition We say the limit of f (x) as x approaches innity is L, denoted lim f (x) = L,
x
if for every > 0 there exists a number N such that
|f (x) L| <
whenever x > N . We say the limit of f (x) as x appr
Lecture 9: Increasing and Decreasing Functions
9.1 Increasing and decreasing functions Denition We say a function f is increasing on an interval I if for all x and y in I, x < y implies f (x) < f (y). We say f is decreasing on an interval I if for all x a
Lecture 12: Curve Sketching
12.1 Final example of curve sketching
Example
Let f (x) =
x2
. Then
1 x2
lim f (x) = lim
x 1
x2
x
1
= 1
1
and
lim f (x) = lim
x 1
x2
x
1
= 1.
1
Hence the line y = 1 is a horizontal asymptote for the graph of f . Also,
lim f (x)
Lecture 10: Concavity
10.1 Concave upward and concave downward
Example Note that both f (x) = x2 and g(x) = x are increasing on the interval [0, ),
but their graphs look signicantly dierent. This is explained by the fact that f (x) = 2x,
1
and so is an in
Lecture 7: Maximum and Minimum Values
7.1 Maximum and minimum values
Denition Let D be the domain of the function f . If f (c) f (x) for all x in D, then
we say f has an absolute maximum at c and we call f (c) the maximum value of f on D. If
f (c) f (x) f
Lecture 8: The Mean Value Theorem
8.1 Rolles Theorem
Suppose f is continuous on [a, b] and dierentiable on (a, b) with f (a) = f (b). Then either
f is a constant function, in which case f (x) = 0 for every x in (a, b), or f has an extreme
value at some po
Lecture 1: Trigonometric Functions: Denitions
1.1 The sine, cosine, and tangent functions
We say triangles ABC and DEF are similar if A = D, B = E, and C = F .
An important fact about similar triangles is that the ratios of corresponding sides are
equal.
Lecture 6: Trigonometric Functions: Final Examples
6.1 Implicit dierentiation
Example
Suppose we wish to nd an equation of the line tangent to the curve
8 sin(x) + 2 cos(2y) = 1
at 0, . Now
6
d
d
(8 sin(x) + 2 cos(2y) =
1,
dx
dx
so
8 cos(x) 4 sin(2y)
Henc
Lecture 5: More Calculus of Trigonometric Functions
5.1 Some limit calculations
We may use the limits
lim
x0
sin(x)
=1
x
and
1 cos(x)
=0
x0
x
lim
to help evaluate other limits.
Example
sin(3x)
= lim
x0
x0
x
Example
sin(2x)
2
= lim
x0 sin(3x)
3 x0
Example
Lecture 4: The Calculus of Trigonometric Functions
4.1 Continuity
As a consequence of the continuity of the unit circle, we have the following basic result.
Proposition
The functions f (x) = sin(x) and g(x) = cos(x) are continuous on (, ).
Example The fun