quiz1_solution - MATH235 Calculus 1 Quiz 1 Show all work to...

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MATH235 Calculus 1 Quiz 1 Show all work to receive full credit. Carefully write down your thought process. Your solution must not contain any logical errors. Good luck! 1. Solve the following inequality : 3 x 1 < 2 x + 1 . 3 x 1 2 x + 1 < 0 , 3( x + 1) 2( x 1) ( x 1)( x + 1) < 0 , x + 5 ( x 1)( x + 1) < 0 . Notice that there are two possible ways that the above inequality can hold: case 1. x + 5 > 0 and ( x 1)( x + 1) < 0 : 1 < x < 1 case 2. x + 5 < 0 and ( x 1)( x + 1) > 0 : x < 5 Therefore, the answer is ( −∞ , 5) ( 1 , 1). 2. Find the domain of the function f ( x ) = ln ( x + 1) ln ( x 2) . Restrictions on the domain : ln( x 2) = 0 , x = 3 . x + 1 > 0 , x > 1 . x 2 > 0 , x > 2 . Therefore the domain of f is (2 , 3) (3 , ). 3. Find the slope of the line tangent to the curve f ( x ) = x at the point (9 , 3) by using the limit definition of the instantaneous rate of change. Then find the equation of this tangent line. We will calculate the limit lim h 0 f ( x + h ) f (
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