Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval.

Where no interval is specified, use the real line.

a)?(?)=−8x^3 + 9x^2 + 6x - 5; [−2,0]

b) ?(?)= x^2 + 2/x ; (0 ,∞)

A tank has the shape of a sphere with a radius of 5 m. It is filled with water up to 9 m.


(i) Let x be the height in meters measured from the bottom of the tank. The weight in Newtons of a thin layer of water between x and x + Δx is approximately P (x) Δx. What is P (x)?
P (x) =
Express the answer by a formula. Recall that the density of water is ρ = 1000Kg / m3 and that the gravitational acceleration on the surface of the earth is g = 9.8m / s2.


(ii) The work in Joules required to pump this thin layer of water 2 meters above the tank is approximately w (x) Δx. What is w (x)?
w (x) =
Express the answer by a formula.


(iii) Using the previous results, determine, in Joules, the work required to pump all the water 2 m above the tank. Give its numerical value to 3 significant figures.
Answer: 

Find the Taylor series for f(x) centered at the given value of

a=1

fx=6x^-2

Danica and Darrelle have developed a lucrative business selling handmade silver bracelets. Darrelle makes the more complex bracelets and Danica makes the simpler ones. It takes Darrelle 4 hours to make 3 bracelets and Danica can make 5 bracelets every 2 hours.

a. Convert this scenario into linear equation(s); show both the standard form and the slope-intercept form of your equation(s).
c. Each bracelet Darrelle sells makes $10.50 in profit. Each bracelet Danica sells makes $3.50 in profit. If Darrelle and Danica work the same amount of time, explain each of the steps you will use and then calculate how many hours will it take for the whole business to make at least $500 profit?

There are three boxes in front of you. They are as follows: – BOX A, which has the following statement written on it: “Box C is empty.” – BOX B, which has the following statement written on it: “This box contains $100.” – BOX C, which has the following statement written on it: “Box B is empty.” One of the three contains $100 and the other two are empty. The box with the money has a true statement on it, while the empty boxes have false statements on them. Which box has the money?

Consider the following function. (If an answer does not exist, enter DNE.) f(x) = x2 + 5 − 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.

Let C be the boundary of the quarter circle with radius 1, oriented counterclockwise (Figure 1). Evaluate H C(ex + y2) dx + (ey + x2) dy

1) Find the following indefinite integrals.

    a) (4-3xsec^2 x)/x dx

    b) (5 sin^ 3 x ) / (1+cosx)(1-cosx) dx

2) A particle starts from rest and moves along the x-axis from the origin at t = 0 with acceleration

       a(t) =   6 - 2t (ms^-2) at time t. When and where will it come to rest.

       Remember dvdt = acceleration and dsdt = velocity

3) Use substitution to find the following integrals.

    a) (9x)/ sqrt of (3+x2) dx                         Let U = 3+x^2

    b) (dx) / (25- x^2 ) Let x = 5 sin U

4) Consider the integral 1∞ 1 / (2x+1)^3 dx.

    a) Explain why the integral is improper.

    b) Determine whether the integral is convergent or divergent and if convergent, evaluate he integral. (Show all working)

For the following pair of functions, ?(?) = ? 2 − 2, ?(?) = √? − 3 Find each of the following, and State its Domain: a) Find (f + g)(x)

b) Find (f – g)(x)

c) Find (f*g)(x)

d) Find (f*f)(x)

e) Find (f/g)(x)

Net income for the company for the year was $300,000, and 100,000 shares of common stock were outstanding during the year. The income tax rate is 30%. For each of the following potentially dilutive securities, perform the shortcut antidilution test to determine whether the security is dilutive. Assume that each of the securities was issued on or before January 1. Treat each security independently; in other words, when testing one security, assume that the others do not exist.      (10)
1. 10,000 convertible preferred shares (cumulative, 5%, $100 par). Each preferred share is convertible into three shares of common stock.
2. 500 convertible bonds ($1,000 face value, 10%). Each bond is convertible into 25 shares of common stock.
3. 20,000 convertible preferred shares (cumulative, 10%, $50 par). Each preferred share is convertible into two shares of common stock.
4. 2,000 convertible bonds ($1,000 face value, 8%). Each bond is convertible into 15 shares of common stock.
(b) Is It Possible For A Company To Have Positive Cash Flow But Still Be In Serious Financial Trouble?

A baseball pitcher throws a baseball with an initial speed of 91 feet per second at an angle of 5 degree to the horizontal. The ball leaves the pitcher's hand at a height of 15 feet. Write the parametric equations that describe the motion of the ball as a function of time.

a) Write a set of parametric equations that model the baseball.

b) What is the max height of the ball?

c) When does the ball hit the ground?

d) How far away from the pitcher does the ball hit the ground?

e) Will the batter hit the ball if the batter is 60 feet away and swings the bat at a height of 3 feet? Why or why not?

A business owner needs to run a gas line from his business to a gas main as shown in the accompanying diagram. The main is 30-ft down the 12-ft wide driveway and on the opposite side. A plumber charges $4 per foot alongside the driveway and $5 per foot for underneath the driveway.

a) What will be the cost if the plumber runs the gas line entirely under the driveway along the diagonal of the 30-ft by 12-ft rectangle? b) What will be the cost if the plumber runs the gas line 30-ft alongside the driveway and then 12-ft straight across? c) The plumber claims that he can do the job for $160 by going alongside the driveway for some distance and then going under the drive diagonally to the terminal. Find x, the distance alongside the driveway. d) Write the cost as a function of x and sketch the graph of the function. e) Use the minimum feature of a graphing calculator to find the approximate value for x that will minimize the cost. f) What is the minimum cost (to the nearest cent) for which the job can be done?

1. ʃ11-3x/(x^2+2x-3) dx

2. ʃ4x-16/(x^2-2x -3 )dx

3. ʃ(2x^2-9x-35)/(x+1)(x-2)(x+3) dx

Evaluate the following integral ∫ ∫ R 4x + y (3x − y) ln(3x − y) dA where R is the region bounded by the graphs of  y  =  3x − e7,  y  =  3x − e5,  y  =  −4x + 8,  and  y  =  −4x + 5. Use the change of variables  u  =  3x − y,  v  =  4x + y.

QA: (inspired by a problem from Stewart’s textbook) Suppose that there is some function f whose _derivative_ is f ’ =sin(x)/x, with f ‘(0)=1 by definition rather than DNE. Draw that, on the interval [-4pi,+4pi].

(i) On what intervals is the original f increasing? Decreasing? Indicate the intervals on the graph as well as writing them in interval notation like [0,pi]

(ii) At what x values does f have a local max? A local min? Indicate them on the graph as well as writing them out like: maxes at …. ; mins at ….

(iii) On what intervals is f CD? CU? Indicate the intervals on the graph as well as writing them in interval notation.

(iv) At what x values does f have an inflection point? Indicate them on the graph as well as writing them out like: IP at ...

(v) Sketch a graph of f, starting at f(0)=0.