chapter5 - MATH1131 Calculus 5.1 Title MATH1131 Calculus 5.2 Introduction We have seen in Chapter 3 that continuous functions have various important

# chapter5 - MATH1131 Calculus 5.1 Title MATH1131 Calculus...

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MATH1131 Calculus 5.1: Title
MATH1131 Calculus 5.2: Introduction We have seen in Chapter 3 that continuous functions have various important properties (the Intermediate Value Theorem, existence of maximum and minimum points). Differentiable functions have these properties (because they are continuous), and further useful properties besides. As in Chapter 3, I suggest studying the results of this chapter in the following way: remember and understand the diagram; understand how the diagram is “translated” into words and mathematical symbolism; try to understand why each assumption in the theorem is necessary, by imagining what could happen to the diagram if the assumption were not true; make sure that you are able to apply the theorem to specific problems.
MATH1131 Calculus 5.3: MVT Theorem . The Mean Value Theorem . Let f be continuous on the interval [ a, b ] and differentiable on ( a, b ). Then there exists a value of c in ( a, b ) such that f ( b ) f ( a ) b a = f ( c ) . Example . Note that the quotient on the left hand side of the equation is the average rate of change of f over the whole interval [ a, b ], while f ( c ) is the instantaneous rate of change at c . Thus, the Mean Value Theorem says that for any (suitable) function, there must be a point at which the instantaneous rate of change is equal to the average (mean) rate of change over a whole interval. 0 x y a b c 1 c 2 f ( a ) f ( b ) Geometrically, there is a point (in this example, two points) where the tangent to the curve is parallel to the secant over the whole interval. Physical interpretation . A train travels exactly 100 km in exactly 1 hour. Assuming that it travels continuously and smoothly (that is, there are no jerks and no sudden starts or stops), there must be some point at which the speed is exactly 100 km/h.
MATH1131 Calculus 5.4: Problems The result of the Mean Value Theorem need not hold if f is not continuous on the closed interval [ a, b ], or if f is not differentiable at every point in ( a, b ). 0 x y a b f ( a ) f ( b ) 0 x y a b x 0 f ( a ) f ( b ) Note, however, that the conditions of the Mean Value Theorem do not require the function to be differentiable at the endpoints a and b . You can see from the next example that if, for example, f is not differentiable at b it makes no difference to the conclusion of the Theorem. 0 x y a b
MATH1131 Calculus 5.5: Find c Example.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.
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