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Unformatted text preview: DERIVATIVES AND THE SHAPES OF CURVES Suppose that a car traveled 100 miles in 2 hours. Then, we can infer the speedometer must have read 50mph at lease once; for otherwise, the car had always been faster than 50mph or slower than 50mph, and in these cases, the car would have traveled more than or less than 100 miles, respectively. This suggests that there existed the time when the velocity was equal to the average velocity. Theorem 1 (The Mean Value Theorem) . If f is a differentiable function on the interval [ a,b ] , then there exists a number c between a and b such that f ( c ) = f ( b )- f ( a ) b- a or, equivalently, f ( b )- f ( a ) = f ( c )( b- a ) . This theorem is one of the most important one in calculus, since it enables us to prove various theorems. In section 2.9, we observed that a function with a positive derivative is increasing. We can not prove this from the mean value theorem. Theorem 2. If f ( x ) > on an interval, then f is increasing on that interval. If f ( x ) < on an interval, then f is decreasing on that interval. Proof. Let x 1 and x 2 be any two numbers in the interval and x 1 < x 2 . Because we have f ( x ) > 0, f is differentiable on the interval, and by the mean value theorem, there exists a number c between x 1 and x 2 such that f ( x 2 )- f ( x 1 ) = f ( c )( x 2- x 1 ) ....
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- Fall '09