p1812tut04 - x showing any local maxima local minima points...

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School of Mathematics and Statistics The University of Sydney PHAR1812 Calculus Tutorial 4 1. ( i ) Sketch a smooth curve whose slope is everywhere negative and increasing gradually (i.e. becoming less negative). ( ii ) Sketch a smooth curve whose second derivative is everywhere negative but whose first derivative is everywhere positive. ( iii ) Sketch a smooth curve whose first and second derivatives are everywhere negative. 2. For each of the following functions y = f ( x ) find y 0 , y 00 and sketch the curve y =
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Unformatted text preview: x ), showing any local maxima, local minima, points of inflections and asymptotes : ( i ) y = 1 1 + x 2 ( ii ) y = 2 x 3 + 3 x 2-36 x + 5 ( iii ) y = x x 2 + 1 ( iv ) y = x 10-10 x ( v ) y = x-ln x (for x > 0) ( vi ) y = x 2 + 2 ( x + 2) 2 3. Find a cubic polynomial P ( x ) with the following properties. ( i ) P (0) = 4 ( ii ) P (2) = 0 ( iii ) P has a critical point at x =-2 and P (-2) = 0. What is the nature of the critical point at x =-2?...
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