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# Calculus and Analysis

### What is the value of given endowment in today's dollars?

A wealthy philanthropist has established the following endowment for a hospital. The details are as follows: a cash deposit of \$ 8 M one year from now; an annual cash deposit of \$3M per year for the next five years. The first \$3M will start today; at the end of 5 years, the hospital will also receive a lump sum payment of \$18M.

### Calculus - Finding the Area of a Specified Region

Please help with the following questions: 1. Use the limit process to find the area of the region between the graph of the function and the axis over the given interval. Sketch the region. y = 4 - x^2, [-2,2] 2. Find the area of the region bounded by the graphs of the equations. y = x^3 + x, x = 2, y = 0

### Set up linear equations for the word problem.

A gourmet coffee shop has a weekly budget for 2 imported coffee beans. Sixty dollars per week is allotted for Italian beans and Kenyan beans. Italian beans cost \$10/lb and Kenyan beans cost \$15/lb. Write a formula for the number of pounds of Kenyan beans, K, the gourmet coffee shop can buy as a function of the number of Ital

### Business Calculus and Average Cost Function

1. Average Cost. A company manufacturing snowboards has fixed cost of \$200 per day and total cost of \$3800 per day at a daily outputs of 20 boards. a) Assuming that the total cost per day c(x) is linearly related to the total output per day x, write an equation for the cost function. b) The average Cost per board for an out

### Forming a Basis in R4

Find a basis for each of the following subspaces of R4. a) Find all vectors that satisfy the equation x1 - 3x2 + 6x3 - 4x4 = 0. b) Find all vectors whose entries satisfy both equations x1 + x2 + x3 + x4 = 0 and x1 - x3 = 0.

### Interpretation.

a) Find a unit vector perpendicular to the surface x2 + y2 + z2 = 3 at the point (1,1,1). b) Derive the equation of the plane tangent to the surface at (1,1,1). I know this, but I forget how to get the answer....please help me..

### Pre-calc Questions Concerning dB & Doubling Period

Please help me with these 2 pre-calculus questions. 1) The drilling of a jackhammer was measured at 132 dB. The sound of whispering was measured at 28 dB. Find the ratio of the intensity of the drilling to that of the whispering. 2) A culture contains 10,000 bacteria initially. After an hour the bacteria count is 25,0

### Phase portrait of a linear system

Please solve the following ordinary differential equations problem. For what values of the parameters a and b does the linear system x' = Ax have a sink at the origin, given the matrix of the system to be: A = [ a -b ] [ b 2 ]

### Calculating Investment Amounts and Finding Real Solutions

Please look at the attachment. It includes two pre-calculus questions. Please show your work and explain the answer in the response. 28) If \$3,000 is invested at an interest rate of 9% per year, find the amount of the investment at the end of 5 years, for the following compounding methods: semiannual, weekly, hourly, contin

### Use Implicit Differentiation and Find Open Intervals

1. Find the derivative of the function: h(x) = [ (x-3)/(x^2) ]^2 2. Use implicit differentiation to find dy/dx. x^2 + 3xy + y^3 = 10. 3. Find the critical numbers (if any) and the open intervals on which the function is increasing or decreasing. f(x) = (x-1)^2(x-3)

### Ships Traveling South

At noon, ship A is 100 kilometers due east of ship B. Ship A is sailing west at 12 kilometers per hour, and ship B is sailing south at 10 kilometers per hour. At what time will the ships be nearest each other, and what will this distance be?

### find out the stationary point, saddle point, min and max

1) Let f(x,y)=25e^(-1/5x^2)-y^5+5y+3 a) Find all stationary points of the function f(x,y) and enter their coordinates by "" with at least 3 decimal places. b)Let (xs,ys) be the saddle point of the function f(x,y). Calculate the following expression: f(xs,ys)-(xs+ys) and enter the value with at least 3 dp. c) what is(are) t

### Temperature rate of change

Let y represent the temperature in fahrenheit of an object in a room whose temperature is kept at a constant 60 degrees. If the object cools from 100 degrees to 90 degrees in 10 minute, how long will it take for its temperature to decrease to 80 degrees?

### Projectile motion with linear drag

A baseball is hit when is 3ft above a ground. it leaves that bat with initial of 104ft/sec, making an angle 18 degree with the horizontal. Assume there is linear drag with a drag coefficient k=.22 the acceleration due to gravity is g=32ft/sec^2 answer the questions as following a) How would you find a vector form for th

### Concentration of a Certain Drug

The following figure shows the devastating effect the opening of a new discount department store had on an established department store in a small town. The revenue of the discount store at time t (in months) is given by f(t) million dollars, whereas the revenue of the established department store at time t is given by g(t) mill

### Applied Mathematics Domain and Range of a Function

1. Find the values which satisfy the given relations. (a) |x-4| = 6 (b) |2y + 3| > 5 (c) |4u-1|<= 3 2. Give the domain and range of each function (a) f(x) = (9 - x^2)^.5 (b) f(x) = 2/x^2 (c) f(x) = ((2x-5)/(x-3))^.5 3. Sketch the graph of f(x) = 2=x. Using this as a starting point, and using translations, plot

### Lines and Planes Constants

A) If a and b are constants that are not both zero then an equation of the form ax+by+c=0 represents a line in R^2 with normal nbar=(a,b) B) If a, b and c are constants that are not all zero, then an equation of the form ax+by+cz+d=0 represents a plane in R^3 with normal nbar=(a,b,c) Please demonstrate how to prove t

### Calculus - Intervals

Please help me with the following calculus problem Determine the intervals on which the following functions are concave up and concave down, a) h(x) = x^4 + 4x^3 - 144 x^2 + 324x b) f(x) = (x-5) e^x Determine the intervals on which the following functions are increasing and decreasing and classify each of the critical po

### Vectors and Logarithsm: Parametric Equation

1) Determine the parametric equations for the line that passes through the two points A = (5, 1, -3) and B = (4, 5, -1). At what point does this line intersect the x-y plane? Q2) Solve for x: Round to 2 decimal. 1) log7(x+4)+log7(x-2)=1 here is log basis 7 (x+4) +Log basis7 (x-2) =1 2) 2^(x+2)=3^(2x+1)

### Proofs of differential calculus

1) Sum Rule- If f and g are differentiable at x_0, then f+g is also differentiable at x_0, and (f+g)'(x_0)=f '(x_0) + g '(x_0) 2) Product Rule- If f and g are differentiable at x_0, then fg is also differentiable at x_0, and (fg)' (x_0)= f '(x_0)g(x_0)+f(x_0)g '(x_0) 3) If g is differentiable at x_0, and g is non-zero on X

### The height of a projectile at a particular time

The height of a projectile at a particular time (measured in seconds) is given by h(t) = 10t(1-t)m, Find the rate of change of height from first principles (using he definition of the rate of change).

### Sketching Blood Pressure Over Two Heart Beats

At rest, the human heart beats once every second. At the strongest part of the beat, a person's blood pressure peaks at 120mmHg. At the most relaxed part of the beat, a person's blood pressure drops to 80mmHh. Supposing that t=0 corresponds to the relaxed part of the beat, write down a trigonometric function that captures this b

### A petroleum refinery is being constructed on the west bank of a conveniently straight river which is 1 km wide.

Please show all work. A petroleum refinery is being constructed on the west bank of a conveniently straight river which is 1 km wide. Storage tanks are located on the opposite bank of the river, y = 2 km further south, and need to be connected to the refinery by pipelines. It costs 1 Megadollars/km to lay a pipeline alone the

### Population of Southberg Town

The population of the town of Southberg can be modeled by the function: P(t) = t^4 - 18t^3 + 77t^2 + 69t + 6405 for 0 < t < 20 where t is in years and t = 0 corresponds to the year 1991. a) What was the population of the town in the year 2002? b) What was the population of town in year 2011? c) What was the instantaneous r

### Differential Equations: Springs

A force of 64N stretches a spring 4 meters. The mass of 4kg is attached to the spring and set into motion in a medium that offer a damped force equal 16 times the velocity. If the mass is pulled down 2 centimeters from the equilibrium position and released. Find the equation for the position. Find the time(s) at which the mass p

### Solve the differential equations of LRC

Find the charge on the capacitor in an LRC series at t=0.02 sec when L=0.05h, R=2 Ohm, C=0.01 farad. The initial conditions are q(0)=5C, i(0)=0 Ampere. Determine the first time at which the charge on the capacitor is equal to zero. Also find the current at any time t.

### Solving Linear Differential Equations

Use the formulas for 'solving a linear differential equation'. 1) Write the equation in standard form y' = P(x)y = Q(x) 2) Find the integrating factor u(x) = e^( ? P(x) dx) 3) Evaluate the integral to find the general solution y = (1/u(x) ? Q(x) u(x) dx 4) Satisfy whatever initial condition is included. 1) y' + (2x

### Description of Business Calculus - Limit of Population

Complete the following problem explaining -all the formulas used and why those formulas were used -a clear explanation of all steps -all graphs -explain manipulation used to solve the problem -use proper punctuation, and complete sentences The population for a rural region is increasing due to the construction of a new

### Integrals for differential equations

Use initial condition to find the particular solution of the differential equations. 1) (sq root of x) + (Sq Root of y) dy/dx = 0 and initial condition is y = 4 when x = 1 2) dy/dx = (x^2)(1 + y) and initial condition is y = 3 when x = 0 3) dy/dx = 2xysin(x^2) and initial condition is y = 1 when x = 0

### Lagrange multiplier

f(x,y,z)=x^2+y^2+z^2, x-y=1, y^2-z^2=1 Please see the attachment.