Let f(x) = a(7-x^2) where a is not equal 0(a) find, in terms of a, the equation of the line tangent to the curve at x = -1 (use point slope)(b) find, in terms of a, the y intercept of the tangent line at x = -1(c) find the x intercept of the tangent line at x=-1
Let f(t)=5 cos^2( t),; with domain frac-{pi}{1} to frac{pi}{1}.(A) Find all (open) intervals on which f(t) is increasing and concave up. Use I for infty, -I for -infty, and U for the union symbol.B) Find all (open) intervals on which f(t) is decreasing and concave up.C)Find all (open) intervals on which f(t) is increasing and concave down(D) Find all (open) intervals on which f(t) is decreasing and concave down.Decreasing and concave down: (E) Find the vertical asymptote. Enter “DNE” if there are no vertical asymptotes.x = (F) Find the horizontal asymptote. Enter “DNE” if there are no horizontal asymptotes.y = (G) List all the t values where f(t) has inflection points, in increasing order, separated by commas. Enter “DNE” if there are no inflection points. (H) List the values of the function f(t) corresponding to the inflection points, in respective order, separated by commas.(I) List the values of the slopes of the tangent lines at the inflection points, in respective order, separated by commas.
Let P denote the set of points for a Hilbert plane. Suppose that f : P -> P is an isometry ofthis plane. Recall that by definition, this means that for all points A and B in P, the segments ABand f(A)f(B) are congruent. Show that the isometry f also preserves angles: i.e. if A, B, and Care any three non-collinear points, and if D = f(A),E = f(B), F = f(C), then angle ABC is congruent to angle DEF.(Hint: Use triangle congruences
Let p denote the probability that a particular item A appears in a simple random sample (SRS). Suppose we collect 5 independent simple random samples, i.e., each SRS is obtained by drawing from the entire population. Let X denote the random variable for the total number of times that A appears in these 5 samples. What is the expected value of X, i.e., E[X]? Your answer should be in terms of p. What is V ar(X)? Again, your answer should be in terms of p.
Let P be the plane in R3 whose coordinates satisfy the equation x + 3y -2z = 5, find a parametric representation for P. Find the parametric representation for a line in P which passes through the point [7 0 1].
Let p = 47 and q = 59, n = p . q, and e = 157.
a. Compute a multiplicative inverse of d, modulo φ(n).
b. Every two-letter string (including A-Z and spaces) can be converted to a number-message between 0 and 2626, by replacing a space by 00, ‘A’ by 01, ‘B’ by 02, etc.. For example. ‘ME’ becomes 1305. Encrypt the two-letter string ‘HI’ by computing its number-message m, and the ciphertext me mod n.
c. Decrypt the sequence of ciphertexts 0802, 2179, 2276, 1024 to find a message.
Please help me prove the following question about the space of all bounded, continuous functions?
Let ORR) denote the space of all bounded, continuous functions f : R —> (C. Let COOK)denote the set of continuous functions f : 1R —) (C for which lim f(a:) = 0. msdzoo a) Prove that every f E ODOR) is bounded. b) Prove that ODOR) is closed in CAR) (equipped with the uniform metric d(f, g) :=5111336112 If (99) – g($)l-).
Let O = {x ∈ Z | x is odd} be the set of odd integers and E = {x ∈ Z | x is even} be the set of even integers. (a) Explain whether {O, E} is a partition of Z. (b) Explain whether {O × O, E × E} is a partition of Z × Z. If the answer is no for either question, can you extend the collection so that it becomes a partition of the given set.
Let {N1 (t), t ≥ 0} and {N2 (t), t ≥ 0} be two independent, time homogeneous Poisson processes with constant rate λ1 and λ2 respectively. Then, for a fixed time point t, the distribution of the random variable N1 (t) + N2 (t) is what?
Is it is a poisson process, if yes then with what parameters?
Let MUA = z = 10 x and MUB = z = 21 2y, where z is marginal utility per dollar measured in utils, x is the amount spent on product A, and y is the amount spent on product B. Assume that the consumer has $10 to spend on A and Bthat is, x + y = 10. How is the $10 best allocated between A and B? How much utility will the marginal dollar yield?(Campbell, McConnell. Economics, 19th Edition. McGraw-Hill Learning Solutions, 01/2011. p. 132).
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