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Coffee is poured at a uniform rate of into a cup whose inside is shaped like a truncated cone (see the accompanying figure). If the upper and lower radii of the cup are 4 cm and 2 cm and the height of the cup is 6 cm, how fast will the coffee level be rising when the coffee is halfway up? [Hint: Extend the cup downward to form a cone.]
Suppose that a function f is differentiable at and that . Prove that there exists an open interval containing such that if and are any two points in this interval with , then .
Given that the means that the slope of the tangent line to the graph at that point is positive.
It means that the value of the function as increases also increases.
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