Advanced Calculus and Function Analysis: Derivatives, Limits, and Continuity

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Advanced Calculus and Function Analysis: Derivatives, Limits, and Continuity 1) Find the composite function as ( ) ( ) ( ) ( ) ( ) ( ) 3 12 3 12 3 4 4 f g h x f g x x f f x x = + +   =     = + = + Hence, correct option is D . 2) Find the derivative as ( ) ( ) ( ) h c h h h h h h c h c h h 2 h h h c 2 h c 2 h Q Q de dQ dQ Q Q Q Q Q Q Q dQ dQ Q Q Q Q Q Q Q d d d   − =     − − − = − − = = Hence, correct option is A . 3) Find derivative of the function as 4 2 dy x dx = − When 4 2 2 x − = then 1 x = When 1 x = then y is 1 Equation of line having slope 2 and through point ( ) 1,1 is ( ) 1 2 1 1 2 2 2 1 y x y x y x − = − − = − = − Hence, correct option is A . 4) Find the composite function as ( ) ( ) 9 2 9 2 9 2 9 9 x g f x g x x x −   =     −   = +     = − + = Hence, correct option is A . 5) For 1 x  the graph is a line with slope 2 − passing through origin and for 1 x  the graph is a line with slope 1. ( ) 2 1 1 1 x x f x x x −   =  +   Hence, correct option is B . 6) Find the derivative as
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