Rob Bayer Math 1A PDP Worksheet October 23, 2008You should work on the following problems in groups of 3. Try to get through as many as you can, but you aren’texpected to finish everything. Instead, you should make sure everyone in your group knows how to solve all theproblems, and not just the answers.Max/Min Values1. Suppose we have a function whose domain is each point on the surface of the earth, and the value at each pointis 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)23. 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)xx2+4; [0, 3]4. Find the absolute maximum and minimum values of f(x) =√9 − x25. A plane flying a constant speed of 500mi/h at an altitude of 1 mile flies directly over a radar station at timet = 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 thetower 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 takes120 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 extremum8. Prove that the function x207+ x63+ x3+ 9x − 10 has no local extremaMean Value Theorem1. Consider the function f(x) = 1 −|x|. Show that f(−1) = f(1), but that there is no c such that f0(c) = 0. Whydoes 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 ithave 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 ≤ f0(x) ≤ 6 for 1 < x < 5. Assuming f is continuously differentiable on [1, 5], what arethe 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 +12x. Let’s prove it:(a) Start by letting f(x) = 1 +12x −√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 inequality8. Generalize the method from the above problem to prove the following very useful theorem: If f, g are continouson [a, b], differentiable on (a, b), f (a) = g(a) and f0(x) < g0(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 mosttwo 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 f0(x) 6= 1, then f has at most 1 fixed
View Full Document
Unlocking...