Search & Filtering
Let a, b ∈ R with a < b. Let Ta,b ⊆ Q be the subset defined by Ta,b := n r ∈ Q : a < r < bo . Prove that the set Ta,b is infinite.
(Density property) if a, b ∈ R with a < b, then there exists a rational number r ∈ Q such that a < r < b.
If x is an arbitrary real number, show that there exists a unique n ∈ Z such that n ≤ x < n + 1. (This is called the greatest integer in x and denoted by [x].)
Exercise 2. If x > 0, show that there exists n ∈ N such that 1/n < x.
Exercise 1. If x is a real number, prove that there are integers p, q ∈ Z such that p < x < q.
If f(*)=$(1/*^2 +2* -3)d* find f(*) given that f(2) =2
Let f be a function such that each point (x,y) on the graph of f, the slope given by dy/dx = y^2-x. The graph of f passes through the point (1,2) and is concave down on the interval 1<x<1.5. Let k be the approximation of f (1.2) found by using the locally linear approximation of f at x=1. Which of the following statements about k is true?
a) k=5.6 and is an overestimate of f(1.2)
b) k=5.6 and is an underestimate of f(1.2)
c) k-2.6 and is an overestimate of f(1.2)
d) k=2.6 and is an underestimate of f(1.2)
A poster must have 32 square inches of printed matter with margins of 4 inches at the top and bottom, and 2 inches at each side. Find the dimensions of the whole poster if its area is minimum.
Sketch the graph of
y=1/π+r2
by finding the domain, symmetries, critical points, inflection points,
intercept points, asymptotes, extremas, intervals on which the function is increasing or decreasing,
concave up or down.
Find dy/dx and d²y/dx² without eliminating the parameter.
a.) x= e^(2t) , y= 1+cos(t)
b.) x= acosh(t) , y= bsinh(t)
