Determine the intervals on which the function below is increasing or decreasing, concave or convex. Determine also relative maxima and minima, inflections points, symmetry, asymptotes, and intercepts if any. Then sketch the curve.

. ? = 3?+3/(3? − 3)^2

Let AB be a diameter of a circle and suppose that UV is a perpendicular chord. Show that cr(A,U,B,V)=2

(where cr = cross-ratio)

Construct expansions in a two-individual universe of discourse for the following sentence:

(x) [Fx --> (Gx v Hx)]

A circular puddle is expanding. How quickly is the radius changing when the area is 50 square centimeters and the area is increasing by 4 square centimeters per second?

Explain what the Gaussian elimination does, by column picture, to a linear system with 3 unknowns and 3 equations.

An open-top rectangular box has a volume of 250 cm 3. The width of the box is 5 cm. The cost is $2/ cm 2 for the base and $1/ cm 2 for the other sides. What is the minimum cost for making the box?

Consider the following function. (If an answer does not exist, enter DNE.) f(x) = ln(8 − ln(x))

(a) Find the vertical asymptote(s). (Enter your answers as a comma-separated list.)

x =

Find the horizontal asymptote(s). (Enter your answers as a comma-separated list.)
y =

(b) Find the interval where the function is increasing. (Enter your answer using interval notation.

Find the interval where the function is decreasing. (Enter your answer using interval notation.

(c) Find the local maximum and minimum values.

(d) Find the interval where the function is concave up. (Enter your answer using interval notation.

Find the interval where the function is concave down. (Enter your answer using interval notation.)

Find the inflection point.

(x, y) =

(e) Use the information from parts (a)-(d) to sketch the graph of f.

f (x) = -0.248226*cos (2 x) - 0.0184829*cos ((2+2)x) - 0.0594608*cos(x)*sin(x) + 0.123626*sin ((2+2)x).

The intervall is ]0, 3/2[

What is the local maximum and local minimum? Answer with 5 decimals

Suppose the response of a drug is modeled by the function ?(?) = ?²/ 3+?² where ? is the drug concentration. At what drug concentration does the inflection point occur? Show that there is a change of concavity at that point.

Consider rotations Rθ and Rα in R^2 . (a) From geometrical considerations, we know that Rα◦Rθ =Rα+θ. Verify the corresponding matrix equation. (b) Prove that [R−θ] T =[Rθ]

In a survey of 300 individual investors regarding subscriptions to the New York Times (NYT), Wall Street Journal (WSJ), and USA Today (UST), the following data were obtained.

121 subscribe to the NYT.
147 subscribe to the WSJ.
66 subscribe to the UST.
40 subscribe to the NYT and WSJ.
25 subscribe to the WSJ and UST.
22 subscribe to the NYT and UST.
42 do not subscribe to any of these newspapers.

(a) How many of the individual investors surveyed subscribe to all three newspapers?
investors

(b) How many subscribe to only one of these newspapers?
investors

a. *If y=(x^3+7)/(x^2/3) , then find dy/dx . Make sure your answer is fully simplified.

b. *If y=(5x-8)/(4x+3)   , then find dy/dx .

c. *If x=(x2-5x+3)(2x2+4) , then find f ‘(x).

Please neatly show your work.

Matilda makes specialized chocolates. Her two best-selling chocolates, the chocolate heart and the chocolate flower, are made by pouring milk chocolate into a mold. A heart requires 3.5 ounces of milk chocolate and earns a profit of 20 cents, while a flower requires 1.5 ounces of milk chocolate and earns a profit of 30 cents. If she has 663 ounces of milk chocolate available and she wants to make at least twice as many hearts as flowers, what is the maximum profit she can make?

Maximum profit in dollars:

Find the maximum area of a rectangle inscribed in a triangle of area A.(NOTE: the triangle need not necessarily be a right angled triangle).

Explain and state the formula of (a) Trapezoidal rule, (b) the Midpoint Rule, and (c) Simpson’s Rule and the error formula of each