hw6 - numbers f ( x ) 6 = 1, then f has at most one fixed...

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University of Engineering and Technology Lahore Department of Electrical Engineering Calculus I Problem Set #6 Due date: December 23, 2010 R eading: Thomas’s Calculus Section 4.1-4.3. 1. A shuttle fires it rockets at t = 0 and all the fuel is burnt out at t = 126. During this phase the velocity of the shuttle is given by v ( t ) = 0 . 001302 t - 0 . 09029 t 2 + 23 . 61 t 3 - 3 . 083. Find the maximum acceleration during the time when the shuttle was being propelled by rockets. 2. Let f ( x ) = 1 - x 2 / 3 defined on [ - 1 , 1]. Show that while f ( - 1) = f (1), there is no point c such that f 0 ( c ) = 0. Does this contradict Rolle’s theorem? 3. Define f ( x ) = 1 /x for all x 6 = 0. Define g ( x ) to be the function g ( x ) = ( 1 x if x > 0 1 + 1 x if x < 0 Show that f 0 ( x ) = g 0 ( x ). Can you conclude that f ( x ) - g ( x ) = c where c is some constant? Why? 4. A number a is called a fixed point of a function f if f ( a ) = a . Prove that if for all real
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Unformatted text preview: numbers f ( x ) 6 = 1, then f has at most one fixed point. 5. Suppose the derivative of a function f is given by f ( x ) = ( x + 1) 2 ( x-3) 5 ( x-6) 4 . On what interval is f increasing? 6. Suppose f is differentiable on an interval I and f ( x ) &gt; 0 for all numbers x ∈ I , except for a single number. Prove that f is increasing on the entire interval. 7. Consider the function f defined by f ( x ) = ( if x = 0 x 4 sin 1 x if x 6 = 0 Show that 0 is a critical point. Can you use the first derivative test to figure out whether 0 is a local extrema? Why? 8. Use the mean-value theorem to deduce that ± ± sin x-sin y ± ± &lt; ± ± x-y ± ± . 1...
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