Math 1A PDP Worksheet Rob Bayer October 23 2008 You should work on the following problems in groups of 3 Try to get through as many as you can but you aren t expected to finish everything Instead you should make sure everyone in your group knows how to solve all the problems and not just the answers Max Min Values 1 Suppose we have a function whose domain is each point on the surface of the earth and the value at each point is the altitude of that point a Where are the absolute maximum and minimum values attained b What do local max mins represent 2 Find the critical numbers of each of the following functions a 4x3 9x2 12x 3 b 2x 3 c x4 5 x 4 2 3 Find the extreme values and the places they are achieved of each function for the given intervals a e x e 2x 0 1 b 3x2 12x 5 0 3 c x ln x 12 2 d x x2 4 0 3 4 Find the absolute maximum and minimum values of f x 9 x2 5 A plane flying a constant speed of 500mi h at an altitude of 1 mile flies directly over a radar station at time t 0 and disappears over the horizon 30 minutes later a At what time during the interval while the plane is visible to the radar tower is the distance between the tower and the plane changing the fastest Slowest b At what time is the angle between the ground the radar station and the plane changing the fastest slowest 6 During takeoff the velocity of a certain rocket obeys the equation v t 0013t3 09t2 23t 3 If it takes 120 seconds to reach orbit what is the maximum acceleration experienced by the rocket during takeoff 7 True False For those that are true explain why For those that are false provide a counterexample a Every function has at least one local maximum or local minimum b Every function has an absolute maximum c Every function has an absolute minimum on every closed interval d Every continuous function has an absolute maximum on every closed interval e If c is a local extremum then c is a critical point f If c is a critical point then c is a local extremum 8 Prove that the function x207 x63 x3 9x 10 has no local extrema Mean Value Theorem 1 Consider the function f x 1 x Show that f 1 f 1 but that there is no c such that f 0 c 0 Why does this not contradict Rolle s Theorem 2 a If a function has at most two roots could it have just one root How about no roots at all Could it have three roots b If a function has at least two roots could it have just one root How about two roots Three c If a function has at most two roots and at least two roots how many roots does it have 3 Show that the equation x5 3x3 10x 1 has exactly one real solution 4 Show that the equation x5 6x c 0 where c is a constant has at most one real root in the interval 1 1 5 Suppose f 1 3 and 2 f 0 x 6 for 1 x 5 Assuming f is continuously differentiable on 1 5 what are the possible values for f 5 6 Show that sin 1 tanh x tan 1 sinh x Hint it s not enough to show that they have the same derivative What else do you need 7 It turns out that if x 0 1 x 1 21 x Let s prove it a Start by letting f x 1 12 x 1 x What are f 0 and f 1 b Prove that if b 0 f b 6 0 c Use a and b to conclude the original inequality 8 Generalize the method from the above problem to prove the following very useful theorem If f g are continous on a b differentiable on a b f a g a and f 0 x g 0 x for all x in a b then f x g x for all x in a b 9 a Use the fact that a linear function has at most one root to show that a quadratic polynomial has at most two roots b Now show a cubic has at most three roots 10 A number a is called a fixed point of f if f a a Show that if f 0 x 6 1 then f has at most 1 fixed point
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