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a) Let M>0, and let f:[a,b]-->R be a function which is continuous on [a,b] and differentiable on (a,b), and such that |f'(x)| <= M for all x belonging to (a,b) (derivative of f is bounded). Show that for any x,y belonging to [a,b] we have the inequality |f(x)-f(y)| <= M|x-y|. *apply mean value theorem.
b) Let f:R-->R be a differentiable function such that f' is bounded. Show that f is uniformly continuous. Use part a from above.
Prove De Morgan's laws for sets by using set theory arguments
Let G be a group (finite or infinite) and let a and b in G. Let o(a) represent the order of a. Suppose G is abelian and both a and b are of finite order. Show that ab is of finite order and o(ab) divides o(a)o(b).
The curve y = x^3 + x^2 - x has two horizontal tangents. Find the distance between these two tangents and draw a picture. Please show how to solve the following problem to the answer of 32/27.
Compute the integral of xdz (|z|=r) for the positive sense of the circle in two ways first by using parametrization and second by observing that x=(1/2)(z+z conjugate)=(1/2)(z+r^2/z) on the circle.
Find the total distance from the given particle - Evaluate the total distance traveled by the particle from t = 0 and t = 4
If A and B are normal subgroups of G such that G/A and G/B are abelian, prove that G/(A intersect B) is abelian
A college cafe receives a profit of $1.25 for each cup of house tea that the sell, $1.40 for each cup of the premium brand, and $1.50 for each cup of their special blend that they sell.
Application of Calculus-Optimization problem - Show that the shortage length of such a rope occurs when θ 1 = θ 2
Teachers also found that strength of class is maximum satisfying all above conditions. Determine the total no. chocolates got by student A of the first class.
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 whether the given information results in one triangle.
A client has an order for a foot soak prepared with 25% povidine iodine and the total volume of preparation is to be 1 liter. How many mls of povidine/iodine should be used?
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