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 decreasing on I
4
2
-5
5
-2
-4
-6
Example of a graph t
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)
3rd derivative
f ( x)
4th derivative
f
nth derivative
f
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 exponential function
f ( x) = e x
f ( x) = lim
h0
f ( x + h) f
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
f ( x)dx = F (b) F (a )
where F is any function such t
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)
Marginal Profit - P (x)
Example 1
Find the marginal c
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 the derivative of y = 4 ln( x)
y = 4 ln( x)
y = 4
1 4
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 x 2 + 5) 4 (6 x)
f ( x) = (30 x)(3 x 2 + 5) 4
Example 2
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 ( x) = (2 x 4 )( x 2 3 x) + (2 x 3)( x 2 4 x) = 2 x 3 6
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
Example 2
Give the intervals where the function increa
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 ( x) = 3 x 2 + 5
Example 2
Given f ( x) = 2 x 3 4 x 2 5 x
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 x1
x+hx
h
f ( x) = slope = lim
h 0
f ( x + h) f ( x )
h
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 = 7.39
e 2 = 7.39
ln(7.39) = 2
3) 10 4 = 10000
10 4 = 100
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 . Evaluate each function.
a) Evaluate f (4)
f (4) = 3 4 = 81
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 profit.
P 48x 2 0.02 x3
P 48 x 2 0.02 x 3
P 96 x .06 x 2
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 integral of 3 x 2 4 x would be x 3 2x 2
The power of x rul