MATH 234, Calculus for BusinessLecture 6, Textbook Sections 3.2, 3.3Continuity and Rates of ChangeFebruary 6th, 2017Announcements (1) Quiz 3 is tomorrow in section. The coverage is Lectures 4, 5; GWs 4 and 5; HW2problems 7-10, HW3 problems 1- 6. (2) HW 3 is due Thursday, 2/9, 11:59pm; late HWs are not accepted.ReviewExample 1. Compute limx→−∞2x3+ x − 43x2− xSolution.Example 2. Compute limh→0(5 + h)2− 52hSolution.Section 3.2, ContinuityDefinition. A function f (x) is continuous at x === c if it satisfies all three conditions:(1) f(c) (which is to say, )(2)(3)If any of these conditions fail, then f (x) is .1Example 3. Is f(x) = x2continuous at x = 1?Solution.Example 4. Let f(x) =√x. Is f(x) continuous at x = −3?Solution.Example 5. Consider the piece-wise functionf(x) =x2− 2 if x < 2x − 2 if x ≥ 2Is f(x) continuous at x = 2?Solution.Example 6. Consider the piece-wise functionf(x) =x2if x 6= 13 if x = 1Is f(x) continuous at x = 1?Solution.2Theorem 1.Function type General Form Example Continuous for...Polynomial anxn+ an−1xn−1+ 3x4+ 5function ··· + a1x + a0Rational r(x) =p(x)q (x)wherex2− 3x − 2function p(x), q(x) are poly.Root√ax + b√3x + 7functionExponential ax15xfunction where a > 0Logarithmic loga(x) log15(x)function where a > 0, a 6= 1Graph of 3x4+ 5 is Graph ofx2− 3x − 2is Graph of√3x + 7 isGraph of 15xis Graph of log15(x) isDefinition. The open interval (a, b) is the set ofThe closed interval [a, b] is the set of(Difference is closed intervals )3Definition. A function f (x) is continuous on (a,b) ifDefinition. A function f (x) is continuous on [a,b] if all 3 conditions hold:(1)(2)(3)If any of the 3 conditions fail, the f (x) is .Example 7. Discuss the continuity of f (x) = 3x4+ 5 on intervals (2, 7) and [2, 7].Solution.Example 8. Discuss the continuity of f (x) = 15xon intervals (2, 7) and [2, 7].Solution.Example 9. Discuss the continuity of f (x) =x2− 3x − 2on intervals (2, 7) and [2, 7].Solution.Example 10. Discuss the continuity of f (x) = log15(x) on intervals (0, 7) and [0, 7].Solution.Example 11. Discuss the continuity of f (x) =√3x + 7 on intervals (−73, 7) and [−73, 7].Solution.4Example 12. Find the value of B that makes the functionf(x) =x2− 2 if x < 2x + B if x ≥ 2continuous for all x.Remark. The value B = −2 corresponds to Example 5.Solution. Step 1: Is f (x) continuous for x 6= 2?Step 2: Discuss continuity at x = 2.For B = −2 (Ex. 5) graph is, For B = 0, graph is For B = 0.5, graph isRemark. The graph of a continuous function .5Section 3.3, Rates of ChangeRemark. The slope of a line through points (x1, y1) and (x2, y2) isDefinition. The average rate of change of a function f (x) from x = a to x = b iswhich is also .Example 13. Let t = hours after 12pm. At 1pm, Alice starts driving from Urbana. She arrives at O’Hareairport at 3:30pm, which is 150 miles away. What is Alice’s average rate of change, or average speed?Solution.Example 14. Let f (x) = x2. Find average rate of change (a) from x = 1 to x = 2 (b) from x = 1 to x = 1.5.Solution.Part (a): average rate of change is Part (b): average rate of change isExample 15. Let f (x) = x2. Find average rate of change from x = 1 to x = 1 + h, where h can be any realnumber.Solution.Remark. For previous example, part (a) means .Also, part (b) means .6Example 16. Let f(x) = mx + b, where m and b are (fixed) real numbers.Find the average rate of change of f(x) from x = a to x = a + h for every value of a and h.Solution.... in hindsight, think of why should this answer should be intuitive.Food for thoughtQuestion 1. The average rate of change of f (x) from x = a to x = a + h isf(a + h) − f (a)(a + h) − a=f(a + h) − f (a)hWhat should the “instantaneous” rate of change of f (x) at x = a
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