Limits, Continuity, and the Derivative

Transcription

1 Unit #2 : Limits, Continuity, and the Derivative Goals: Study and define continuity Review limits Introduce the derivative as the limit of a difference quotient Discuss the derivative as a rate of change

2 Intro to Continuity - 1 We spent the previous unit discussing various families of functions, and how we could transform them to obtain functions that suit a particular purpose. In this unit we will study properties and tools for all functions: continuity and limits, and the idea of the derivative. Continuity Consider this statement: You were once exactly meters tall. A. True B. False

3 Intro to Continuity - 2 Below is a graph for drawing your height over time. Put your birth height, and your current height on the graph. Put the line h = meters on the graph. Why must the graph of your height cross the h = 1 line?

4 Intro to Continuity - 3 The more formal way to state the property we used is through the Intermediate Value Theorem. Intermediate Value Theorem Suppose f is continuous on a closed interval [a, b]. If k is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k. This introduces another question, though: how do we define a continuous function? What characterizes the graph of a continuous function?

5 Intro to Continuity - 4 Give examples of functions that are continuous everywhere. Give example of functions with discontinuities. Is the function y = x continuous at x = 0? A. Yes B. No

6 Continuity and Formulas - 1 Continuity and Formulas The continuity of a function (or lack thereof) is usually obvious if we have a graph of a function. It can be less clear if we just have a formula. Example: Where must the function y = sin(x) be continuous? Where might x it be discontinuous?

7 Continuity and Formulas - 2 Use sample points to sketch the graph of y = sin(x) x x [ 1, 1]. on the interval What is the value of y at x = 0?

8 Continuity and Formulas - 3 What value does y approach as x approaches 0? How do we write the last question mathematically?

9 Intuition about Limits - 1 Intuitive Definition of the Limit The limit lim x c f(x) equals the number L if we can make f(x) as close to L as we want, for all the values of x close enough to c. It is possible that such a number L may not exist; if that is the case, we would say the limit does not exist. Note: The limit never depends on the value of the function at the limiting point, or even if that point is defined or not.

10 Intuition about Limits - 2 Example: y Explain why the limit lim x 2 f(x) is not 6 for the graph below x

11 Intuition about Limits - 3 y x Describe the real value and limits for x = 2 of the graph of f(x) above.

12 Intuition about Limits - 4 Example: For each of the graphs below, explain why the limit as x 0 does not exist. f(x) x g(x) x

13 The Definition of Continuity - 1 We can use our definition of the limit to help us define continuity. Definition of Continuity A function is continuous at a point x = c if f(c) is defined lim x c f(x) is defined both these values are equal A function is continuous on an interval x [a, b] if it is continuous at all the points on the interval.

14 The Definition of Continuity - 2 Use this definition to state why y = sin(x) x is not continuous at x = 0. Use the consequences of this definition to compute lim x 4 (x 2 + x 3).

15 The Definition of Continuity - 3 Use the terms limit and continuous to describe the situation around x = 2 on the graph below. y x

16 Computing Limits - 1 Computing limits Example: Consider the function g(x) = x2 4 x + 2. Find the points where the function is discontinuous. Evaluate the limit of g(x) as you approach the discontinuity.

17 Computing Limits - 2 Example: Consider the function h(x) = 7 ex x 3. Find the points where the function is discontinuous. Evaluate the limit of h(x) as you approach the discontinuity.

18 One- and Two-Sided Limits - 1 One- and Two-Sided Limits Example: Consider the piecewise function { x + 1 x 1 h(x) = x 2 1 x > 1 On what intervals is the function clearly continuous? At what point(s) might the function be discontinuous?

19 One- and Two-Sided Limits - 2 Deciding whether a function is continuous or not, we realize that in our previous limit questions, we (implicitly or explicitly) considered points on both sides of the limiting point. In this new case, we have different behaviours on either side, so we need a way to handle this special case. Two-Sided Definition of the Limit The limit lim x c f(x) exists and equals L if and only if the right-hand limit lim x c + exists and equals L, and the left-hand limit lim x c also exists and equals L.

20 One- and Two-Sided Limits - 3 h(x) = { x + 1 x 1 x 2 1 x > 1 Does the limit lim x 1 h(x) exist, and if so what is its value?

21 One- and Two-Sided Limits - 4 h(x) = { x + 1 x 1 x 2 1 x > 1 Analyze the continuity of h(x) at x = 1.

22 One- and Two-Sided Limits - 5 Example: Sketch functions where both one-sided limits exist, but the overall limit does not. Example: exist. Sketch functions where one or both of the one sided limits do not

23 Limits at Infinity - 1 Limits at Infinity Aside from studying questions of continuity, limits can help to analyze the behaviour of the functions when x is very large. We say that the limit of f(x) as x approaches infinity is equal to L (written lim x f(x) = L) if f(x) becomes arbitrarily close to L when x is arbitrarily large. We make the similar definition for x when x is negative. Use limits at infinity to express the horizontal asymptote of e x. Find lim t 15 10e.04t

24 Limits at Infinity - 2 Determine lim x x x 2 1

25 Average Rates and the Derivative - 1 Average Rates and The Derivative Consider the graph below, and assume it represents the position of a person over time; t is in seconds, and f(t) is in meters. f(t) t Compute the average speed of the person between t = 1 and t = 4 seconds.

26 Average Rates and the Derivative - 2 f(t) t How can you show the value of the average speed by using the points t = 1 and t = 4 on the graph?

27 Average Rates and the Derivative - 3 f(t) t On the graph, sketch the line that would reflect the instantaneous speed at t = 1.

28 Average Rates and the Derivative - 4 Based on the earlier question about average speed, how would you compute the instantaneous speed? What is the difficulty with that approach?

29 Average Rates and the Derivative - 5 Our earlier work with limits was not tied to any particular function or any particular application. Now we use limits as a tool to help answer a completely separate question. If we cannot use rise-over-run or distance-over-time directly to compute the instantaneous speed, write how we could use limits to help us.

30 Average Rates and the Derivative - 6 Sketch on the graph what lim f(t) 2.0 t 0 f(1 + t) f(1) t represents t

31 Definition of the Derivative at a Point - 1 Definition of the Derivative at a Point By formalizing the intuitive idea of an instantaneous slope, or speed, or rate of change, we have created a formal way to compute slopes at points. We call the value of the slope at a point the derivative of the function at that point.

32 Definition of the Derivative at a Point - 2 The Derivative The derivative of a function f(x) at a point x = c is defined as lim h 0 f(c + h) f(c) h Sometimes this is written with x instead of h, to make the relationship to the graph more intuitive: lim x 0 f(c + x) f(c) x A third way is to write as if we were moving two separate points together, rather than using one reference point (c) and a distance ( x or h) lim x c f(x) f(c) x c Note that all these forms share the common rise over run ratio, and limit as denominator goes to zero. You may use any of the forms, as they are all equivalent.

33 Definition of the Derivative at a Point - 3 Example: Find the slope of the function f(x) = x 2 at the point x = 2, using the limit definition of the derivative y x

34 Definition of the Derivative at a Point - 4 The graph of x 2 around x = 2 is shown below in more detail. Confirm your answer graphically by sketching the tangent line with the slope you just computed. 10 y x