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Please solve the following with as much explanation of each step as possible.
Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Also, suppose that f(a) = f(b) and that c is a real number in the interval such that f'(c) = 0. Find an interval for the function g over which Rolle's theorem can be applied, and find the corresponding critical number of g (k is a constant).
(a) g(x) = f(x) + k
(b) g(x) = f(x-k)
(c) g(x) = f(kx)
Let f:R->R be uniformly continuous on R (the reals) and let f_n(x)=f(x+1/n) for x in the reals. Show that (f_n) converges uniformly on the reals to f.
What is the probability that the nut fits the bolt? (Note: the sample space consists of all pairs of nuts and bolts, and each pair is equally likely to be chosen.)
If F is a field, prove that the field of fractions of F[[x]] is the ring F((x)) of formal Laurent series. Show that the field of fractions of the power series ring Z[[x]] is properly contained in the field of Laurent series Q((x)).
In furnishing their new office space, a company allows each employee to select a desk, a chair and a bookcase or file cabinet based on personal preference. There are two desk models, 3 different chairs, 2 styles of file cabinet and 4 bookcase styl..
Find the slope of the line 2x + y = 4. Find the point of intersection of the lines x + y = 5 and 3x - y = 7. Graph the pair of lines.
the probability that she is either rich or beautiful is 0.5, what is the probability that she is both rich AND beautiful?
Describe how you would calculate the odds of winning the jackpot. Do you think it is worth it to buy a ticket? Why or why not?
Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve.
he distribution of ant mounds on a farm can be accurately modeled by a Poisson distribution.
Solving problems on venn diagram
Assume that 10% of the population is left-handed. If three people are chosen at random, what is the probability that at least one will be left-handed?
Determine the circumference of a circle circumscribed about a square, an equilateral triangle, a regular hexagon, a right triangle
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