# 3 5 pts pretend f is a magical function that has the

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______________________________________________________________________ 3. (5 pts.) Pretend f is a magical function that has the property that at x = 3 the tangent line f is actually defined by the equation y = -2(x - 1) + 5. Obtain (a) f (3) = (b) f (3) =

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TEST2/MAC2311 Page 2 of 4 ______________________________________________________________________ 4. (25 pts.) Compute the first derivatives of the following functions. You may use any of the rules of differentiation that are at your disposal. Do not attempt to simplify the algebra in your answers. (a) f ( x ) 4 x 6 7 x 12 8tan( x ) f ( x ) (b) g ( x ) (4 x 2 2 x 1 )sec( x ) g ( x ) (c) h ( t ) 5 t 10 1 sin( t ) 2 h ( t ) (d) y cot 5 (2 θ 1) dy d θ (e) L ( z ) sin(4 z 8 ) 4 csc ( π 6 ) 4cos( z 2 ) dL dz ( z )
TEST2/MAC2311 Page 3 of 4 ______________________________________________________________________ 5. (10 pts.) What are the x- and y - intercepts of the tangent line to the graph of y = 1/x 2 at the point (2,1/4)? ______________________________________________________________________ 6. (10 pts.) (a) Find all values in the interval [0,2 π ] 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 .

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