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MATH1131 Calculus5.1: Title
MATH1131 Calculus5.2: IntroductionWe have seen in Chapter 3 that continuous functions havevarious important properties (the Intermediate Value Theorem,existence of maximum and minimum points). Differentiablefunctions have these properties (because they are continuous),and further useful properties besides.As in Chapter 3, I suggest studying the results of this chapterin the following way:•remember and understand the diagram;•understand how the diagram is “translated” into wordsand mathematical symbolism;•try to understand why each assumption in the theoremis necessary, by imagining what could happen to thediagram if the assumption were not true;•make sure that you are able to apply the theorem tospecific problems.
MATH1131 Calculus5.3: MVTTheorem.The Mean Value Theorem. Letfbe continuouson the interval [a, b] and differentiable on (a, b). Then thereexists a value ofcin (a, b) such thatf(b)−f(a)b−a=f′(c).Example.Note that the quotient on the left hand sideof the equation is theaveragerate of change offover thewhole interval [a, b], whilef′(c) is theinstantaneousrate ofchange atc. Thus, the Mean Value Theorem says that forany (suitable) function, there must be a point at which theinstantaneous rate of change is equal to the average (mean)rate of change over a whole interval.0xyabc1c2f(a)f(b)Geometrically, there is a point (in this example, two points)where the tangent to the curve is parallel to the secant overthe whole interval.Physical interpretation. A train travels exactly 100 km inexactly 1 hour. Assuming that it travels continuously andsmoothly (that is, there are no jerks and no sudden starts orstops), there must be some point at which the speed isexactly100 km/h.
MATH1131 Calculus5.4: ProblemsThe result of the Mean Value Theorem need not hold iffis not continuous on the closed interval [a, b], or iffis notdifferentiable at every point in (a, b).0xyabf(a)f(b)0xyabx0f(a)f(b)Note, however, that the conditions of the Mean ValueTheorem do not require the function to be differentiable at theendpointsaandb. You can see from the next example that if,for example,fis not differentiable atbit makes no differenceto the conclusion of the Theorem.0xyab
MATH1131 Calculus5.5: Find cExample.Does the Mean Value Theorem apply whena=−1,b= 4 andf(x) =x5? If so, find the value(s) ofcwhich satisfy the conclusion of the Theorem.