MATH 265 Lecture 8: MATH 265-LECTURE 8-Continuity

124 CHAPTER 2 Limits and Derivatives that a function fis 10. Explain why each function is continous The temperature at a specific location 2.5 EXERCISES ific location a (b) The temperature at a specific time due west from New York City (e) The current in the circuit for the li fic time as distance due west from New York (c) The altitude above sea level as (a) an equation that expresses the fact that a function f is 1. Write continoous at the number 4. level as a funct 2 If f is continuous on (, »«), what can you say about its 3. (a) From the graph of f, state the numbers at which f is graph? i ride as a function of (d) The cost of a taxi discontinuous and explain why (b) For each of the numbers stated in part (a), determine function of time 11-14 Use the definition of continuity 11. x) -( +2x a 12.9(,) , a-2 whether f is continuous from the right, or from the left. or neither and th to show that the function is continuous at the + 5 2t + 1 a-2 14.f(x)一3r' _ 5x + :/r-4, 15-16 Use the definition of continuity and to show that the function is continuous on t 4. From the graph of g. state the intervals on which g is continuoS 16. g(x)(oc,-2) 3x + 6 17-22 Explain why the function is discon number a. Sketch the graph of the functios 17. f) +2 5-8 Sketch the graph of a function f that is continuous except for the stated discontinuity 5. Discontinuous, but continuous from the right, at2 6. Discontinuities at -1 and 4, but continuous from the left at 18, f(x)-{x + 2 if x 2 u from o le i19.1). 3i1 -I and from the right at 4 7. Removable discontinuity at 3, jump discontinuity at 5 8. Neither left nor right continuous at -2, continuous only from the left at 2 20. fx)x- 9. The toll T charged for driving on a certain stretch of a toll road 21. )-if x is $5 except during rush hours (between 7 AM and 10 AM and between 4 PM and 7 PM) when the toll is $7 cosx if x0

A useful remark: Since the derivative function is defined through limits, it is easy to show the following result: If f(s) and g(x) are equal on an open interval (a, b), and f?(c) exists for some c E (a, b), then g?(c) also exists and g(c) = f (c). this allows one to calculate the derivative of functions Defined piecewise, except at the x values where the definition changes from one formula to another. For example: if then h (x) = 4x on (- infinity , 5) (because it agrees with (2x^2 + 4)? there) and h?(x) = 2e^2x on (5, infinity) (because it agrees with (e^2x) there). Whether or not h?(5) exists requires more work and we will see how to answer this in the following two questions. (1) Consider the function (a) Show that f(s) is continuous at x = 0 by calculating the left and right hand limits of f(s) as it approaches 0 and showing they are both equal to f(0). (b) Show that f(s) is not differentiable at 0 by calculating and showing the two sided limit does not exist (note: calculate these limits explicitly, do not take shortcuts using derivative formulas). These one - sided limits are called the left and right - sided derivatives of f(x) at 0. By definition, f?(0) exists if and only if the two one - sided derivatives both exists and are equal (note: there is nothing special about 0; one sided derivatives are defined similarity for any s value). (c) Using the Power Rule Theorem from class and the remark preceding this question, calculate f(x) (note that by (b), 0 should not be in the domain of .f?(x)). Sketch a graph of f(s) and f(x) on the same XY - axis (you do not need to be precise, just sketch time shape).