Exercise 1.8

1. Your long distance phone service has a base monthly charge and a per-minute charge. When you used 350 minutes in a month the total cost was $32.50. When you used 400 minutes in a month the total cost was $36.50. You want an equation that will allow you to calculate your phone bill.

Please provide:

a. the definition of x, including the units

b. the definition of y, including the units

c. the equation in the form y = mx + b

d. the units of the slope

e. the units of b and an interpretation of b.

f. How much will your phone bill be if you talked for 711 minutes?

2. The dosage for a medicine is linear with the weight of the patient. There is a minimum dosage onto which is added a per pound dosage. You find that your dosage, at 110 pounds is 48 milliliters (ml). Your brother’s dosage, at 170 pounds, is 66 ml. You would like an equation that will relate the dosage to the weight of the patient. Please provide:

a. the definition of x, including the units

b. the definition of y, including the units

c. the equation in the form y = mx + b

d. the units of the slope

e. the units of b and an interpretation of b.

f. what is the dosage for a 165 lb patient?

For each of the problems,6 –10, cost information at a certain level of production for a manufacturing process is given. The revenue per unit is also given. Assume a linear relationship between the number of units produced and cost. For each problem please find:a. The cost,revenue and profit functions and the units of the slope and of b.b. The variable cost per unit, also known as the marginal cost, and the fixed cost. c. The cost, revenue and profit when z units are produced. (z will be specified in each problem)d. The break-even point.e. The average cost per unit of producing w units and the equation of the average cost per unit function. (w will be specified in each problem)

6. The cost of producing 1,250 units is $45,000. The cost of producing 1,500 units is $50,000. Each unit produced can be sold for $30. z = 3,000; w = 2,500.
7. The cost of producing 23,000 units is $225,000. The cost of producing 30,000 units is $260,000. Each unit produced can be sold for $12. z = 45,000; w = 68,000.

3.5. Each of the following measurements is a rounded value. We have no way of knowing the exact value that was rounded to obtain these rounded values. For each, i) state the range of possible exact values; ii) stating the absolute value of the maximum possible measurement rounding error that may have resulted from the rounding; and iii) state the minimum and maximum possible relative measurement error as a percent to two significant digits.

a. 0.02 ft b. 0.07 ft c. 2.634E+02 km d. 9.167E+02 km

a 600gal tank initially containing 75lb of salt. Brine containing 1lb of salt per gallon enters the tank at a rate of 5gal/s, and the well mixed brine in the tank flows out at a rate of 3gal/s. How much salt will the tank contain when it is full of brine?

A​ third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular solution satisfying the given initial conditions.

y⁽³⁾+2y''-y'-2y=0; y(0)=1, y'(0)=2, y''(0)=0,

y_{1=e^x ,} y₂=e^-x, y₃=e^-2x

Find the particular solution.

Let L be the line parametrically by~r(t) = [1 + 2t,4 +t,2 + 3t] and M be the line through the points P= (−5,2,−3) and Q=(1,2,−6).

a) The lines L and M intersect; find the point of intersection.

b) How many planes contain both lines?

c) Give a parametric equation for a plane Π that contains both lines

prove that every integer greater than 2 can be broken into sum of multiples of 2 and 3

A tank initially contains 80gal of pure water. Brine containing 2lb of salt per gallon enters the tank at 2gal/min, and the (perfectly mixed) solution leaves the tank at 3 gal/min. Thus the tank is empty after exactly 80 min.

(A) Find the amount of salt in the tank after t minutes.

(b) What is the maximum amount of salt ever in the tank?

Let F be a vector field. Find the flux of F through the given surface. Assume the surface S is oriented upward. F = eyi + exj + 24yk; S that portion of the plane x + y + z = 6 in the first octant.

1. Let v1, . . . , vn be nonzero vectors such that each vi+1 has more leading 0s than vi . Show that vectors v1, . . . , vn are linearly independent.

Consider the solution of the differential equation y′=−3yy′=−3y passing through y(0)=0.5y(0)=0.5.

A. Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,0.5).

B. Use Euler's method with step size Δx=0.2Δx=0.2 to estimate the solution at x=0.2,0.4,…,1x=0.2,0.4,…,1, using these to fill in the following table. (Be sure not to round your answers at each step!)

C. Plot your estimated solution on your slope field. Compare the solution and the slope field. Is the estimated solution an over or under estimate for the actual solution?
A. over
B. under

D. Check that y=0.5e−3xy=0.5e−3x is a solution to y′=−3yy′=−3y with y(0)=0.5y(0)=0.5.

Calculate the first three terms in the power series solutions of the following differential equations taken about x=0.

x^2y''+sinx y'-cosx y=0

Please solve the recurrence relation by finding the explicit formula of each problem. Show all work, thanks!

A. c_k = 6c_k-1 - 9c_k-2      k≥2,   c₀ =1, c₁=3

B. d_k = 2d_k-1 +k                       k≥1,   d₀ =1, d₁=3

C. a_k = 3a_k-1 + 2                         k≥1,   a₀ =3

D. b_k = -b_k-1 + 7b_k-2    k≥2,   b₀ =1, b₁=4