Average Rate of Change
Section 1.3
Find the average rate of change
for
f ( x) = x x
2
as x goes from 1 to 3
First find f(1) and f(3)
f ( 1) = 1 1 = 0
2
f ( 3) = 3 3 = 6
2
Then find the average rate of change
f ( 3 ) f ( 1) 6 0 = 31 2
=3
Difference Quotien
Business Calculus Worksheet on Solving Exponential Equations 1) 53 x = 54 x 2 2) 1023 x = 105 x 6 3) 4) 5)
7 x =7 2 x +3
2
45 x x =46 x 2 =33x 1 9
2
2
6) 4 x = x 8 7)
9 x =27 x +3
x
2
2 9 8) = 4 3 9) 125 x = 5 1 10) = 4 2 11) 23 x = 8 12) 52 p +1 = 125 13
Worksheet for Limits at Infinity (1.1B) Find the following limits, if they do not exist say so using the most precise symbol or words. 1.
x
lim ( 7 x 3 + 2 x 11) lim ( 2 x 6 3 x 5 + 8 )
2.
x
3.
5 lim x x 2
3 x+2 lim x 5 x 3
x
4.
5.
lim ( x 2 )
2
( 2
Worksheet for Limits (1.2A) Find the following limits, if they do not exist say so. 1.
lim 3 x 2 2
x 5
(
)
2.
( x 2 2 x +8 ) x 2
lim 5x lim x 1 2 + x 2 3 x+2 lim x 10 5 x 3 lim ( x 2 ) ( 2 x 3 ) x 4
2 x 1
3.
4.
5.
6.
lim
3 x 2
7.
x 5
lim 25 x 2
8.
x 2 3
3.2 A Logarithm Properties Expand the following log expressions and simplify as much as possible. 1) ln 4x 2) ln 1 x
3) ln e2x 4) ln [(x+1)(x- 3)] 5) ln [(x 3 (x-6)] x+9
Condense the following log expressions to a single log with coefficient of 1. 6) ln x
3.2C Exponential Equations Solve the following exponential equations. Give the exact answer and its decimal approximation.
1. 2. 3. 4. 5. 6. 7.
e2 x = 5
e1 x = 4
2e 3 9 = 0
x
5 x+2 = 12 x
122 x = 150
7 x+1 + 2 = 20
5e x 3 = 4
Answers for Business Calculus Worksheet on Exponents Simplify 1) 2)
103x 1104 x = 102 x +3 (43x ) 2 y = 46 xy 5 x 3 =5 x 4 5 4x 43xz y = 3 yz 5 5 3x = 32 x 1 31 x
3z
3)
4)
5)
6)
(2 3 )
xyz
= 2 xz 3 yz
7)
75 x = 73x 1 2 x +1 7 1043x = 102+ 2 x 1025x 53x +
Answers for Business Calculus Worksheet on Solving Exponential Equations 1) 2) 3)
53x =54 x 2 3x = 4 x 2 x = 2 1023x =105 x 6 23x = 5x 6 x =1
7
x2
=7 2 x +3 x 2 =2 x +3 x 2 2 x 3 =0 x =1or x =3
2
4)
45 x x =46 5x x 2 =6 0 = x 2 5x 6 x =1or x = 6
2 9x
=33x
Business Calculus Worksheet on Exponents Simplify 1) 103 x 1104 x 2) (43 x ) 2 y 3) 5 x 3 5x4
3z
4x 4) y 5 5) 6) 7) 3x 31 x
(2 3 )
x
yz
75 x 7 2 x +1 1043 x 1025 x
8)
9) 53 x + 45 x5 10) 3x 27 x 3 22 x +5
1.6
1.6A
Differentiate using the Product Rule.
The Product Rule Let
F ( x ) = f ( x ) g( x ).
Then,
d F ( x ) = [ f ( x ) g( x )] dx d d F ( x ) = f ( x ) g( x ) + g( x ) f (x) dx dx
Example Find
d 4 3 2 x 2 x 7 3x 5 x . dx
(
)(
)
d 4 3 2 x 2 x 7 3x 5 x =
Practice with the Quotient Rule-Find the derivative of each function. Simplify the answer.
1)
y=
7x 1 2x + 3
2)
3 x2 6 x + 2 g ( x) = 2 x + 4x + 1
7t 3 3 f (t) = 3 t +1
3)
4)
5u 2 y= 4 u 1
Chapter 1 - Limits, Alternatives, Choices Worked Problems 1. Consider that you have an income of $80 a week. You decide to purchase two goods: tacos and movie tickets. Tacos (y-variable) cost $1 and movie tickets (x-variable) cost $8. a) What is the slope
1.4B & 1.5B
Where is a function not differentiable Equation of a Tangent line
Where a Function is Not Differentiable: 1) A function f(x) is not differentiable at a point x = a,
if there is a corner (or sharp point) at a.
Where a Function is Not Differenti
1.5A
Basic Rules for Derivatives
Example 1
Find the derivative with respect to x for
y =x x+ 2
2
Do not leave a negative exponent in the final answer.
y = x x+ 2
2
d d d 2 y = x [ x ] + 2 dx dx dx d 2 = x 1 + 0 dx 21 = ( 2 x ) 1 = 2 x 1
Example 2
Find the
Find the horizontal asymptote of each function. If none, say so. 1.
3x + 2 x 4 f ( x) = 2 x2 x + 1
2
2.
3x 4 g ( x) = x 2x + 3 h( x ) = 3 x 2x2 + 4 7 x2 j ( x) = x3
3.
4.
Find the open intervals over which the function is a) increasing and b) decreasing. 1) 2) 3) 4) 5) 6)
f ( x ) = x 2 6 x + 19
f ( x ) = 1 4x2
f ( x ) = 10 x x 2
f ( x) =
1 x
f ( x) = 3 x
f
( x) =
1 x2
Find the critical numbers in each of the following func
Infinite Limits
When f(x) increases without bound as x approaches c
we write
x c
lim f ( x ) =
Note that this means
x c
lim f
( x)
and
=
x c+
lim f
( x)
=
When f(x) decreases without bound as x approaches c
we write
x c
lim f ( x ) =
Note that this also
Limits at Infinity
for Polynomials
To find the limit at infinity of a polynomial, take the limit of the term of the highest degree.
x
lim ( 5 x 3 x + 2 ) = lim ( 5 x
4 2 x
4
)
Now apply the Principles of Limits
x
lim ( 5 x 3 x + 2 ) = lim ( 5 x
4 2 x x
Limits at Infinity For Rational Functions
1 = 0 lim x x
1 lim = 0 x x
k = lim k 1 lim x x x x
1 = k lim = k 0 x x
=0
For positive integer n,
k = 0 lim n x x
Use the previous limits to find the limit of a rational function as x increases without bo
Find the derivative. Do not leave a negative exponent in the final answer. 1)
y = x 5 +4 x
3 4 x3
2) 3)
f ( x) =
y=
4
x5
20 4 5 + 6x 2 xx
4) 5)
g( x ) =
y = 28 x
6)
d 1 + 3 dx x
7)
y = x 4 x
2 3
1
5
+ x 4
8)
f ( x) = 5
y =( 2 x )
4
5 x3
9)
10)
d 2 x 3
eTroy
MTH 2201 XTIA
Business Calculus
COURSE SYLLABUS
Term 1, 2011
August 15 October 16
NOTE: For course syllabus posted prior to the beginning of the term, the instructor reserves the
right to make minor changes prior to or during the term. The instructo