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2.1 Rates of Change & Limits. Average Speed. Average Speed. Since d = rt ,. Example: Suppose you drive 200 miles in 4 hours. What is your average speed?. = 50 mph. Instantaneous Speed. The moment you look at your speedometer, you see your instantaneous speed. Example
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Average Speed Average Speed Since d = rt, Example: Suppose you drive 200 miles in 4 hours. What is your average speed? = 50 mph
Instantaneous Speed • The moment you look at your speedometer, you see your instantaneous speed. Example A rock breaks loose from the top of a tall cliff. What is the speed of the rock at 2 seconds? We can calculate the average speed of the rock from 2 seconds to a time slightly later than 2 seconds (t = 2 + Δt, where Δt is a slight change in time.)
Instantaneous Speed Example A rock breaks loose from the top of a tall cliff. What is the speed of the rock at 2 seconds? Free fall equation: y = 16t2 We cannot use this formula to calculate the speed at the exact instant t = 2because that would require letting Δt= 0, and that would give 0/0. However, we can look at what is happening when Δt is close to 0.
Instantaneous Speed What is happening? As Δt gets smaller, the rock’s average speed gets closer to 64 ft/sec.
Instantaneous Speed Algebraically:
Instantaneous Speed Algebraically: Now, when Δt is 0, our average speed is 64 ft/sec
Limits • Let f be a function defined on a open interval containing a, except possibly at a itself. Then, there exists a such that WHAT THE CRAP??????
Limits • The function f has a limit L as x approaches c if any positive number (ε), there is a positive number σ such that Still, WHAT THE CRAP?????? We read, “The limit as x approaches c of a function is L.”
The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1
Properties of Limits Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. If L, M, c, and k are real numbers and and then 1.) Sum Rule: 2.) Difference Rule: 3.) Product Rule:
Properties of Limits 4.) Constant Multiple Rule: 5.) Quotient Rule: 6.) Power Rule:
Example • Use and and the properties of limits to find the following limits: a.) b.)
Evaluating Limits • If fis a continuous function on an open interval containing the number a, then (In other words, you can many times substitute the number x is approaching into the function to find the limit.) Techniques for Evaluating Limits: 1.) Substituting Directly Ex: Find the limit:
Limiting Techniques: 2.) Using Properties of Limits (product rule) Ex: Find the limit:
Limiting Techniques: 3.) Factoring & Simplifying What happens if we just substitute in the limit? HOLY COWCULUS!!! Ex: Find the limit: When something like this happens, we need to see if we can factor & simplify!
Limiting Techniques 4.) Using the conjugate What happens if we just substitute in the limit? We must simplify again. Ex: Find the limit:
Limiting Techniques 5.) Use a table or graph What happens if we just substitute in the limit? Ex: Find the limit: As x approaches 0, you can see that the graph of f(x) approaches 3. Therefore the limit is 3. (You can also see this in your table.)
6. Sandwich (Squeeze) Theorem • If f, g, and h are functions defined on some open interval containing a such that g(x) ≤ f(x) ≤ h(x)for all x in the interval except at possibly at a itself, and h(x) • then, f(x) g(x)
Sandwich (Squeeze) Theorem sin oscillates between -1 and 1, so Now, let’s get the problem to look like the one given. Ex: Find the limit:
Sandwich (Squeeze) Theorem Therefore, by the Sandwich Theorem, Ex: Find the limit:
Existence of a Limit • In order for a limit to exist, the limit from the left must approach the same value as the limit from the right. If then and are called one-sided limits
does not exist because the left and right hand limits do not match! left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=1:
because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=2:
because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=3:
Suggested HW Probs: • Section 2.1 (#7, 11, 15, 19, 21, 23, 27, 31-36, 37, 43, 49, 63)