2-1: RATES OF CHANGE AND LIMITS Objectives: To evaluate limits numerically, graphically, and analytically. To use properties of limits.

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Presentation transcript:

2-1: RATES OF CHANGE AND LIMITS Objectives: To evaluate limits numerically, graphically, and analytically. To use properties of limits

Average Speed or Rate of Change Distance Covered Elapsed time A rock breaks free from the top of a cliff. What is the average speed during the first 2 seconds?? (y=16t 2 )

What if I wanted to know the speed at EXACTLY 2 seconds (use same function)? Let us use t = 2 and t= 2 + h.

Definition of a Limit Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement Means “The values f(x) of the function f approach or equal L as the values of x approach (but do not necessarily equal) c.

In other words… If the values of f(x) approach the number L as x approaches a from both the left and the right, we say that the limit L as x approaches a exists and **Please note..a limit describes how the outputs of a function behave as the inputs approach some particular value. It is NOT necessarily the value of the function at that x value.

Evaluating numerically…using a table of values. Evaluate Try: x f(x)

One-Sided Limits RIGHT-HAND LIMIT (RHL) (The limit of f as x approaches c from the right) LEFT-HAND LIMIT(LHL) (The limit of f as x approaches c from the left)

IN ORDER FOR A LIMIT TO EXIST, THE FUNCTION HAS TO BE APPROACHING THE SAME VALUE FROM BOTH THE LEFT AND THE RIGHT (LHL and RHL must exist and be equal) IF = THEN

a.) Graph the function b.) Determine the LHL and the RHL c.) Does the limit exist? Explain.

Properties of Limits: If L, M, c and k are real numbers and and then: 1. Sum and Difference rule: 2. Product Rule: 3. Constant Multiple Rule: 4. Quotient Rule: 5. Power Rule:

Evaluating Algebraically Theorems: Polynomial and Rational Functions 1. If f(x) = a n x n + a n-1 x n-1 +…+a 0 is any polynomial function and c is a real number, then SUBSTITUTE!!!!!! 2.If f(x) and g(x) are polynomials and c is a real number, then SUBSTITUTE!!!

EXAMPLES

PRIZE ROUND Factor: 1. x x t 2 +5t x 2 -64

To evaluate limits algebraically: 1. Try substitution. (c has to be in the domain). If you get 0/0, there is something you can do!! 2. If substitution doesn’t work, factor if possible, simplify, then try to evaluate 3. Conjugate Multiplication: If function contains a square root and no other method works, multiply numerator and denominator by conjugate. Simplify and evaluate. 4. Use table or graph to reinforce your conclustion.

Examples: Evaluate the limit.

Evaluate the limit:

Evaluate.

Sandwich (Squeeze) Theorem If g(x) < f(x) < h(x) when x is near c (except possibly at c) and THEN

Show

Useful Limits to Know!! Evaluate…

Evaluate