Room
|Room |Sections |
|Matheson 109 |011, 013, 016, 017 |
|CAT 61 |009, 010 |
Exam Coverage: 1.1-1.6
Date: Friday, April, 24th 8-8:50 am
Recap of section 1.1:
Interval Notation
|Inequality |Interval notation |Meaning |Graph |
| [pic] | [pic] |x is greater than or = to a | |
| [pic] | [pic] |x is greater than a | |
| [pic] | [pic] |x is less than or = to a | |
| [pic] | [pic] |x is less than a | |
|Inequality |Interval notation |Meaning |Graph |
|[pic] | [pic] |Include both a and b (closed | |
| | |interval) | |
| [pic] | [pic] |Include a but not b | |
| | |(half closed interval) | |
| [pic] | [pic] |Include b, but not a | |
| | |(half closed interval) | |
| [pic] | [pic] |Neither a nor b is included (open | |
| | |interval) | |
Slope and equations of Lines:
Slope
Formula for slope: [pic]
1. Slope of a horizontal line is 0
2. Slope of a vertical line is undefined
Equations of Lines:
1. Use slope-intercept form (y = mx + b) if you are given
The slope and the y-intercept
2. Use point-slope form ( [pic]) if you are given
- The slope (m) and any point on the line,
- You are given two points on the line only (find the slope first, using [pic], then pick a point and use
point-slope form)
3. Equation of Horizontal line through the point (a, b): y = b
4. Equation of Vertical line through the point (a, b): x = a
Section 1.2
Properties of Exponents
[pic]
1.3 Functions:
Domain and Range:
x = input value = independent variable; domain describes allowable values for x;
f(x) = y = output value = dependent variable; range describes possible output for f(x)
Vertical Line Test:
The Vertical Line Test:
A curve in the Cartesian plane is the graph of a function if and only if, if no vertical line intersects the curve at more than one point.
Types of Functions:
Linear Function: A linear function is a function that can be expressed in the form f(x) = mx +b with constants m and b. Its graph is a line with slope m and y-intercept b.
Quadratic Functions:
[pic]
[pic]
To find the vertex of quadratic functions:
Again, the vertex is the highest or lowest point of a parabola. To find the vertex of a parabola:
[pic]
Solving Quadratic Equations:
A value of x that solves an equation f(x) = 0 is called a root of the equation or the x-intercept of the graph of y = f(x)
There are 2 methods to solve a quadratic equation:
• Factoring
To solve a quadratic equation, you must
1. Have the equation set = 0
2. Factor
3. Set each factor = 0 and solve for the variable
• Using the Quadratic Formula[pic]
Applications:
Profit, Revenue, and Cost:
Revenue: Revenue is the amount of money a company takes in from producing x items.
Cost: The cost is how much it costs for the company to make the same x items.
Profit = Revenue – Cost
A company breaks even if Revevue = Cost
Section 1.4 Functions Continued…
A Polynomial Function is a function that can be written in the form:
[pic]
Where n is a non-negative integer and
are called coefficients.
The domain is all Real numbers.
The degree is the highest power of the variable
Solving Polynomials:
Recall, solving means to find the value for the variable that makes the equation true.
▪ Have the equation set = 0
▪ Factor
▪ Set each factor = 0 and solve for the variable
Rational Functions:
A rational function is a fraction consisting of one polynomial divided by another polynomial.
Domain: division by zero is undefined, so the domain of a rational function is all real numbers, except those which make the denominator 0.
Piecewise Linear Functions:
A piecewise linear function is a function which is given in several parts. For example
[pic]
Composite functions:
The composition of f with g evaluated at x is f(g(x))
[pic]
The Difference Quotient
[pic]
Section 1.5 Exponential Functions:
An exponential function is a function that has the variable in the exponent, for example:
[pic]
General Form of an Exponential Function:
[pic]
[pic]
[pic] [pic]
[pic] [pic]
Applications of Exponential Functions: Compound Interest:
To generalize:
For P dollars invested at an interest rate of r compounded m times per year, the value after t years, denoted by A is
[pic]
Depreciation by a Fixed Percentage
Depreciation by a fixed percentage, such as for a refrigerator, or a car, (say 10% of its value) is like compounding interest, only with a negative interest rate. So to compute this depreciation, we use m = 1 (since the item loses value only once per year) and with r being negative.
[pic]
Continuous Compounding
We looked at compounding annually, quarterly, and monthly. But you could also compound weekly, daily, or even minute by minute. Ultimately, you can compound continuously. The formula for continuous compounding is:
For $P invested at a rate r compound continuously for t years
[pic]
Section 1.6 Logarithmic Functions
The result of a log is the exponent
Common Logarithms
Logs with base 10 are called common logarithms.
Usually, we omit the base with the base 10 understood.
[pic]
General form of a logarithm:
[pic]
Natural Logarithm:
[pic]
Properties of Logs:
[pic]
Problems for Review:
1. Write each interval in set notation and graph it on the real line:
a. [0,4)
b. [-3,3]
2. Write each inequality in set notation:
a. [pic]
b. x > 3
3. Write an equation of the line satisfying the following conditions. If possible write your answer in the form y = mx + b
i. Slope -1 and passing through (4,3)
ii. Horizontal line passing through (1/2, ¾)
iii. Vertical line passing through (1/2, ¾)
iv. Passing through the points (3 ,-1) and (6, 0)
4. Simplify:
a. [pic]
b. [pic]
c. [pic]
5. Does this graph represent the graph of a function?
[pic]
6. Graph the following functions. Determine the domain and the range of each one.
a. [pic]
[pic]
b. [pic]
[pic]
c. [pic]
[pic]
7. A company manufactures bicycles at a cost of $55 each. If the company’s fixed costs are $900, express the company’s cost as a linear function of x, the number of bicycles produced.
8. Find both coordinates of the vertex of the given parabola: [pic]
9. A company that installs car alarm systems finds that if it installs x systems per week, its costs will be [pic] and its revenue will be [pic] (both in dollars).
i. Find the company’s break-even points.
ii. Find the number of installations that will maximize profit and the maximum profit.
10. For the following functions
i. evaluate the given expression
ii. Find the domain of the function
a. [pic]
b. [pic]
11. Solve
a. [pic]
b. [pic]
12. Find compositions f(g(x)) and g(f(x)) of the given functions: [pic]
13. For the function [pic] find and simplify [pic]
14. The Black-Scholes formula for pricing options involves continuous compounding. If an option is now worth $10,000 and its value grows at an interest rate of 6.1% compounded continuously, what will be its value in 5 years? (A= Pert).
15. In certain experiments the percentage of items remembered after t time units is [pic]. Such curves are called “forgetting” curves. Find the percentage remembered after:
a. 0 time units
b. 2 time units
16. Evaluate without using a calculator:
a. [pic]
b. [pic]
17. Use the properties of logarithms to simplify [pic]
18. The Dead Sea Scrolls, discovered in a cave near the Dead Sea, are among the earliest documents of Western civilization. Estimate the age of the Dead Sea Scrolls if the animal skins on which some were written contain 78% of their original carbon-14.
(Proportion of carbon-14 remaining after t years) = e-0.00012t.
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