111F08Exam1Sol

# 2 is equal to the slope of the graph of g at the

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(2)) is equal to the slope of the graph of g at the point x = f (2) = 6. This slope is equal to the increment of g on the interval (1 , 6) divided by 6 - 1 = 5. Thus, g 0 ( f (2)) = 1 5 . Consequently, M 0 (2) = 1. Question 6. (15 points) The function y = f ( x ) has the following graph. 4. The function y = f ( x ) is defined by the following graph. y = f(x) Sketch a graph of f Â ( x ) . FOR PRACTICE {include a blank graph with axes and coordinate lines}

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Math 1110 (Fall 2008) Prelim 1 (9/30/2008) 3 Sketch the graph of f 0 ( x ). The derivative of f ( x ) must be negative where f ( x ) decreases and positive where it increases. On the intervals where the graph of f ( x ) is convex, the derivative must be decreasing. On the interval where the graph of f ( x ) is a straight line, the derivative must be constant. At the cusp on the left of the graph of f ( x ), the left limit of f 0 ( x ) must be -∞ , and the right limit must be . All these facts suggest the following graph of f 0 ( x ): Question 7. (15 points) A car drives back and forth along a straight road leading from its home (position 0 km) to its work (position 100 km). Its position on the road at time t is given by R ( t ). Some values for R ( t ) are known: t R(t) 0 0 km 3 hrs 50 km 4 hrs 100 km 6 hrs 40 km 7 hrs 70 km
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