Solve the following:
1. Find an equation of variation where y varies jointly, directly as the cube of x and cube root of z, and inversely as the square of w, and y = 4 when x = 2 and z = 8 and w = 4.
2. Perform the indicated operation and simplify. y2 + 3y/y2 -5y + 6 - y2 -y/(y-3)(y+2) / (y - 1 /y + 2)
3. Solve the equation: 3/x2 - 2/x = 1
4. For the functions f (x) = x2 - 9 and g (x) = x - 3 find:
a. (f/g) (x)
b. (fog)(x)
c. f (a - 1)
5. Two boats leave the same port at right angles. The first boat travels at a speed of 12 mph. In three hours the two boats are 60 miles apart. How fast is the second boat traveling?
6. Find an equation of the line containing the point (2, -1) and perpendicular to the line: 3y - 5x = 5.
7. For the function f (x) = - x2 + 4x + 3:
a. Find the coordinates of the vertex.
b. Determine whether there is a maximum or minimum function value and find that value.
c. Find the x-intercepts.
d. Determine the line of symmetry.
e. Graph the function.
8. Solve: 4x - 2y = 32
1/3 x - 1/5 y = 3
9. Solve. (Write set or interval notation.) x2-4x-5/x-1≥0
10. Which of the following are functions and have inverses that are functions? [Explain your choice(s)]
Reason(s):
11. The budget department of a certain yacht manufacturer has formulated the following revenue and cost functions. Find the number of yachts that must be produced to maximize profits and find that profit.
R(x) = 250,000x and C(x) = 625x2 + 175,000x + 165,000.
12. A rancher uses 1600 feet of fence to enclose a rectangular region and also to subdivide the region into three smaller rectangular regions by placing the fences parallel to one of the sides. Find the dimensions that produce the greatest enclosed area and find that area.
13. Find the inverse, f-1 (x), for the function: f(x) = 6 - 2x2
14. Graph: y = log3 x and find log315
15. Solve. 6 x + 1 + 9 = 45.
16. Solve: log6 (x+ 4) + log6 x = 2.
17. Gotham city had a population of 15,000 in 1987, and a population of 20,100 in 2005. Estimate the population in the year 2010. Discuss the validity (constraints) of your prediction. (P(t)= P0 e kt)
18. For the Polynomial Function: f (x) = x3 + 2x2 - x - 2, determine:
a. the zeros and the solutions for the Polynomial.
b. the graph.
19. An airplane flies 2550 miles with a headwind in 5 hours. Traveling in the reverse direction, with a tailwind, the plane covers 2655 miles in 4.5 hours. Find the speed of the plane and the speed of the wind.
20. How many years will it take an investment of $20,000 to triple if interest is compounded monthly at 6%?