INTRO TO LIMITS & CALCULUS MR. VELAZQUEZ AP CALCULUS

Transcription

1 INTRO TO LIMITS & CALCULUS MR. VELAZQUEZ AP CALCULUS

2 WHAT IS CALCULUS? Simply put, Calculus is the mathematics of change. Since all things change often and in many ways, we can expect to understand a wide variety of problems in the sciences using Calculus. These are precisely the sort of problems Isaac Newton ran into when studying the motion of objects in space for his book, Philosophiæ Naturalis Principia Mathematica. Newton is credited with inventing Calculus, although Gottfried Liebniz made many significant contributions at around the time, working independently from Newton. Isaac Newton Gottfried Leibniz

3 Without Calculus With Differential Calculus

4 Without Calculus With Integral Calculus

5 THE LIMIT OF A FUNCTION The entire study of calculus is founded on the concept of the limit of a function, so it is here that we must begin. For a function f(x), we say that lim x a f(x) = L if the values of f(x) get closer and closer to L as the values of x get closer and closer to a. Example: For f x = x3 1 x 1 find lim x 1 f(x). IF YOU HAVE A CALCULATOR: Notice that we can plug any value other than x = 1 into this function and get a real answer. So to get an idea of what the limit might be, try plugging in values that are closer and closer to 1, and tracking the results.

6 THE LIMIT OF A FUNCTION Example: For f x = x3 1 x 1 find lim x 1 f(x). f x = x 3 1 x 1 The point x = 1 is missing from the graph, but we can clearly see that the values near it seem to be approaching a value of 3.

7 TRY THESE EASY ONES 3 if x = 1 If f x = ቊ, find lim x if x 1 f(x) x 1 2 if x 0 If g x = ቊ, find lim 1 if x > 0 g(x) x 0 If h x = x2 x x, find lim h x x 0

8 ONE-SIDED LIMITS We say that a function f(x) has a limit L as x approaches the number a from the right if we can make every value of f(x) as close to L as we want by choosing x sufficiently close to a, such that x > a. We call this the right-hand limit of f(x), and can write it as lim f(x) = L x a+ Similarly, we say that a function f(x) has a limit L as x approaches the number a from the left if we can make every value of f(x) as close to L as we want by choosing x sufficiently close to a, such that x < a. We call this the left-hand limit of f(x), and can write it as lim f(x) = L x a

9 ONE-SIDED LIMITS 2 if x 0 Take the previous example, g x = ቊ 1 if x > 0 Notice that: lim g x = 1 x 0 + However lim g x = 2 x 0 When this happens (limits differ on both sides) we say that the limit does not exist

10 ALTERNATE DEFINITION OF LIMITS lim x a f x = L if and only if lim f x = L and lim f x = L x a + x a In other words, for a limit to exist, the limits from either side of a must each exist and approach the same real number value L. Examples: For the function f x shown: lim f(x) = x 4 lim f(x) = x 1 lim f(x) = x 6

11 UNBOUNDED BEHAVIOR Let f be defined on both sides of a, but possibly not at a. Then f x = if the values of f(x) can be made as large as we lim x a want by choosing x sufficiently close to a, such that x a. Also lim x a f x = if the values of f x < 0 can be made as large in absolute value as we want by choosing x sufficiently close to a, such that x a. NOTE: We can make similar definitions for approaching a from the left or right.

12 UNBOUNDED BEHAVIOR 1 Example: Find lim x 0 x 2 The limit on either side of zero increases without bound. We can therefore conclude that: lim x 0 1 x 2 =

13 VERTICAL ASYMPTOTES The vertical line x = a is called a vertical asymptote of the function f(x) if one of the following conditions is satisfied: lim x a f x = lim x a f x = lim f x = x a + lim f x = x a + lim f x = x a lim f x = x a lim x a + lim x a + f x = and lim f x = x a f x = and lim f x = x a

14 VERTICAL ASYMPTOTES Examples: Find vertical asymptotes for the following functions f x = x 5 x 1 g x = x2 1 x 1

15 VERTICAL ASYMPTOTES In the theory of relativity, the mass of a particle with velocity v is: m = m 0 1 vτc 2 where m 0 is the mass of the particle at rest and c is the speed of light. Describe what happens to the mass of the particle as v c, and sketch the region of the graph of m near c.

16 CLASSWORK & HOMEWORK CLASSWORK: Graphical Limits Use the graph of h(x) shown below to determine the given limits Homework: Pg , #1-32 (DUE 9/7)