UMHS Calculus Honors -Summer Review
Summer Math assignments for current and incoming students
Calculus
All Calculus students are required to have a
TI 84-Plus Graphing Calculator for this course.
The calculator will be used throughout this course, and in future math courses.
Please keep your eyes open for sales and purchase one before school begins.
Show All Work
Part 1: POLYNOMIALS
You should be able to factor polynomials, simplify rational expressions, and divide polynomials.
Factoring the Sum/Difference of Cubes:
Factor completely.
Simplify each rational expression. SHOW YOUR WORK.
Simplify each rational expression. SHOW YOUR WORK.
Dividing Fractions
Use Generic Rectangles to divide the polynomials.
Divide the following polynomial functions using generic rectangles AND using long division.
Part 2: DOMAIN AND RANGE
You should be able to identify the domain and range of a function both algebraically (without a calculator) and graphically.
DOMAIN: The domain of a function is the set of x-values that can be inputted into a function,
so yield a y-value. To find the domain of a function, ask yourself what values CANNOT be used.
● The domain of all polynomial functions is ALL REAL NUMBERS, because you can raise any number to an whole number exponent and you can add, subtract, and multiply all numbers.
● You cannot take the square root of a negative number, so when dealing with a square root, you know that whatever is under the radical must be greater than or equal to zero.
● You are NEVER allowed to divide by zero, so we know that the denominator of a rational function cannot equal zero.
● You cannot take the log of a negative number, so we know that when we have a logarithmic function, whatever we take the log of must be greater than or equal to zero.
RANGE and INTERVAL and SET NOTATION
RANGE: The range is the resulting outputs or y-values. To find the range of a function, think about what the function looks like (general shape and starting point or shifts); also consider what is happening in the function and what that does to the problem (EX: when you square a positive or negative number, your answer always turns positive).
INTERVAL and SET NOTATION: When identifying your domain and range, you should write your answer in interval or set notation.
INTERVAL NOTATION: Interval notation uses parentheses () and brackets []. A parentheses indicates that the number next to it is NOT included while a bracket indicates that the number next to it IS part of the domain or range.
SET NOTATION: Set notation is good to use when your domain or range is a list of numbers or a set of numbers on an interval. For domain, it would look like: {x: x>5}. Range would look similar, except with y’s instead of x’s.
- EX: The range is all numbers between 3 and 7 (including 3, but not including 7).
Identify the domain and range of each function. When given an equation for the function, you MUST show work to justify your answer.
Part 3: EVALUATING FUNCTIONS and COMPOSITE FUNCTIONS
You can use your calculator to evaluate a function, but you should be able to evaluate a function by hand as well.
EVALUATING FUNCTIONS: f(x) is used to write a function in function notation (instead of y=). If f(x) = 3x - 2, then f(5) means evaluate f (x) for x=5.
COMPOSITE FUNCTIONS: With composite functions, you combine two functions by taking the output of one function and making it the input of the other function.
Evaluate each function. SHOW YOUR WORK.
Evaluate each function. SHOW YOUR WORK.
Part 4: TRIGONOMETRY
In this course, you will need to evaluate trigonometric functions and use trigonometric identities to simplify expressions or solve equations. You should be able to do all of those things WITHOUT a calculator; therefore, you are responsible for knowing the following:
UNIT CIRCLE: You should know (or be able to figure out) the value of sine, cosine, tangent, cosecant, secant, and cotangent of any angle on the unit circle ... angles will be written in RADIANS.
TRIG. IDENTITIES: You should know the following trigonometric identities and be able to apply them when necessary.