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# At which the graph of f has a horizontal tangent line

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] at which the graph of f has a horizontal tangent line when f ( x ) = x + 2 cos( x ). (b) The following limit represents f ( a ) for some function f and some number a . Using that information, evaluate the limit. lim x → π sin(3 x ) 0 x π ______________________________________________________________________ 7. (10 pts.) (a) Using complete sentences and appropriate notation, provide the precise mathematical definition for the derivative, f ( x ), of a function f ( x ). (b) Using only the definition of the derivative as a limit, show all steps of the computation of f ( x ) when f ( x ) = 1/ x . f ( x )

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TEST2/MAC2311 Page 4 of 4 ______________________________________________________________________ 8. (5 pts.) Determine whether the following function is differentiable at x = 1. ______________________________________________________________________ 9. (5 pts.) Compute f (x) when f( x ) sin(2 x 3 ). ______________________________________________________________________ 10. (10 pts.) A spherical balloon is to be deflated so that its radius decreases at a constant rate of 15 cm/min. At what rate must air be removed when the radius is 9 cm.? [ V = (4/3) π r 3 ??] ______________________________________________________________________ Silly 10 point Bonus Problem: Explain completely how to obtain the limit lim x 0 x sin 1 ( x ) 1 from lim x 0 sin( x ) x 1. Say where your work is, for it won’t fit here.
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