Section 10.2
Concavity
Concave Up: If a function f ( x) on I is concave up, then f (x) is increasing on I
8
6
4
2
-10
-5
5
-2
-4
Concave Down: If a function f ( x) on I is concave down, then f ( x) is
Math 126
Section 12.1
The Indefinite Integral
The integral of function does the opposite of the derivative.
Review of the derivative:
The derivative of f ( x) = x 3 2 x 2 is f ( x) = 3 x 2 4 x
The int
Math 126
Section 9.8
Higher-Order Derivatives
First Derivative
Second Derivative
Third Derivative
f ( x) = x 4
f ( x) = 4 x 3
f ( x) = 12 x 2
f ( x) = 24 x
1st derivative
f (x)
2nd derivative
f ( x)
3
Math 126
Section 11.2
Derivatives of Exponential Functions
The exponential function
The definition of the number e
eh 1
e is the number such that lim
=1
h 0
h
4
2
-5
5
-2
The derivative of the exponen
Math 126
Section 12.2
Definite Integrals
Fundamental Theorem of Calculus
Let f be a continuous function on the closed interval [a, b] : then the definite integral of f exists
on this interval, and
b
a
Math 126
Section 9.9
Business Functions
Cost Function - C(x)
Revenue Function R(x)
Profit Function P(x)
P( x) = R( x) C ( x)
R ( x) = xp
Marginal Measures
Marginal Cost - C (x)
Marginal Revenue - R (x
Math 126
Section 11.1
Derivatives of logarithms
The derivative of the natural logarithm
d
(ln x ) = 1
dx
x
Example 1
Find the derivative of f ( x) = ln( x)
f ( x) = ln( x)
1
f ( x) =
x
Example 2
Find
Math 126
Section 9.6 Chain Rule
Example 1
(
Find the derivative of the function: f ( x) = 3 x 2 + 5
(
f ( x) = 3x 2 + 5
)
5
Let f ( x) = u 5 where u = 3 x 2 + 5 du = 6 x
f ( x) = 5u 4 du
f ( x) = 5(3
Math 126
Section 9.5
Product and Quotient Rule
The Product Rule
If f ( x) = F ( x) S ( x) , then f ( x) = F ( x) S ( x) + S ( x) F ( x)
Example 1
(
)
Given f ( x) = x 2 4 x ( x 2 3 x) , find f ( x)
f
Section 10.1 A
Increasing and Decreasing Functions
8
6
4
2
-5
5
Example 1
Give the intervals where the function is increasing and decreasing.
8
6
4
2
-5
5
-2
-4
3
Decreasing: ,
2
3
Increasing: ,
2
Math 126
Section 9.4
Derivatives and the slope of a tangent line
Review of the Power Rule
Given f ( x) = ax n , f ( x) = nax n1
Example 1
Given f ( x) = x 3 + 5 x, find f ( x)
f ( x) = 3 x 31 + 5
f (
Section 9.3
Derivatives
The derivative is the slope tangent line to the curve
Limit Definition of a Derivative
8
6
(x2,y2)
4
2
(x1,y1)
-5
5
-2
m=
y 2 y1
f ( x + h ) f ( x ) f ( x + h) f ( x )
=
=
x 2
Section 5.2
Logarithms
Common Logarithms
log b a = x b x = a
ln a = x e x = a
log10 a = x 10 x = a
Example 1
Write each expression as a logarithm expression
1) 3 4 = 81
3 4 = 81
log 3 81 = 4
2) e 2 =
Math 126
Section 5.1
Exponential Functions
Definition:
If a is a real number with a > 0 and a 1 then the function f ( x) = a x is an exponential function
with base a.
Example 1
Let f ( x) = 3 x . Eval
Section 10.3
Optimization (Business Applications)
Maximize Profit
Minimize Average Cost
Average Cost: C ( x)
C ( x)
x
Maximizing Revenue
Example 1
Find the number of units x that produces a maximum p
Section 5.2
The Number of Elements in a Finite Set
Let A be a set, then n(A) denotes the number of elements in the set A.
A = cfw_a, b, c, d ;
B = cfw_c, d , e ;
A B = cfw_a, b, c, d , e
Example:
Giv