e209pmk - MAC 2311 Exam II and Key PM Prof S Hudson 1[30...

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MAC 2311 PM Feb 27, 2009 Exam II and Key Prof. S. Hudson 1) [30 pts] Compute the derivative, f 0 ( x ) (shortcuts such as the Product Rule are OK for these, but show all work); a) f ( x ) = x sin( x 2 ) b) f ( x ) = tan - 1 (2 x ) c) f ( x ) = x +1 x d) f ( x ) = ln(ln( x )) e) e sec( x ) f) Compute the second derivative, f 00 ( x ), given that f ( x ) = sin(3 x +1). 2) [10 pts] Find the derivative using logarithmic differentiation: y = x x . 3) [10 pts] A 6-foot man is walking towards a 15-foot lamppost at 3 feet per second. How fast is the length of his shadow decreasing ? 4) [10 pts] Find the slope of the tangent line to y = x 2 at x = 4, using the definition of m tan . So, you will need to compute a limit. Do not use any methods beyond Ch 3.1, such as derivative shortcuts. 5) [15 pts] Answer True or False. You do not have to explain. For all x , ln( e 5 x ) = 5 x . If f is differentiable, then f 0 is continuous. [using notation from the LLA section] 4 y = dy dx dx . If f is continuous (on its domain), then f - 1 is continuous (on its domain).
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