Section 12.3 Exam Question

The figure below is a contour diagram of the monthly payment on a 5 year car loan as a function of the interest rate and the amount you borrow.

PASTE  FIGURE HERE (Fig 12.55 pg 583 in Text book)

Suppose that the interest rate is 13% and that you decide to borrow $6000.

a. What is your monthly payment?

_______________________ [1pt]

ANSWER: Approximately $137

b. If interest rates drop to 11%, how much more can you borrow without increasing your monthly payment?

_______________________ [1pt]

ANSWER: Approx. $250

c. Complete the table below for how much you can borrow, without 
    increasing your monthly payment in part a), as a function of the 
    interest  rate. [8 pts]

({{Interest   Rate (%), 0, 5, 10, 15}, {Loan  Amount,                ,                ,                   ,                   }})

ANSWER: ({{Interest  Rate (%), 0, 5, 10, 15}, {Loan  Amount, 8200, 7200, 6350, 5800}})

Section 14.7 Exam Question

f (x, y) = (Sqrt[x + 2  y + 1])

a. Find the linear, L(x,y), and the quadratic, Q(x,y), Taylor polynomial of second degree valid near (0,0). [6 pts]

L(0,0) =_______________________

Q(0,0) =_______________________

ANSWER

L (0, 0) = 1 + (1/2) x + y

Q (0, 0) = 
      1 + (1/2) x + 
        y - (1/8) (x^2) - (1/2) xy - (1/2) (y^2)

b. Compare the values of the approximations L(- 0.1,0.1) and Q(- 0.1,0.1) with the exact value of the function f(- 0.1,0.1). [3 pts]

({{Point, Linear, Quadratic, True}, {(x, y), L (x, y), Q (x, y), f (x, y)}, {(-0.1, 0.1),      ,     ,     }})

Which approximation gives the closest values ? [1 pt]

ANSWER

({{Point, Linear, Quadratic, True}, {(x, y), L (x, y), Q (x, y), f (x, y)}, {(0, 0), 1, 1, 1}, {(-0.1, 0.1), 1.05, 1.04875, 1.048809}})

Quadratic approximations are closer to the true values for f(x,y)