Tugas 1
Kalkulus Multivariabel II
Petunjuk pengerjaan :
1. Tugas dikerjakan pada lembar folio bergaris dan ditulis rapi.
2. Tuliskan Nama, NIM dan Kelas di halaman depan di pojok kiri atas.
3. Tugas dikumpulkan pada saat sebelum ujian dimulai ke pengawas.
Soal.
1. Suppose that a company expects its annual profits t years from now to be f(t)
dollars and that interest is considered to be coumpounded continously at an annual
rate r. Then the present value of all future profitd can be shown to be

FP   e rt f (t ) dt
0
Find FP if r = 0,08 dan f(t) = 100.000.
2. The gamma probability density function is
 Cx 1e x , if x  0
f ( x)  
0 , if x  0

Where  and  are positive constants. (Both the gamma and the Weibull
distributions are used to model lifetimes of people, animals, and equipment.)
a. Find the value of C, depending on both  and , that makes f(x) a probability
density function.
b. For the value of C found in part (a), find the value of the mean .
c. For the value of C found in part (a), find the variance 2.
3. Suppose that the government pumps an extra $1 billion into the economy. Assume
that each business and individual saves 25% of its income and spends the rest, so of
the initial $1 billion, 75% is spent, and so forth. What is the total increase in
spending due to the government action ?
4. Evaluate each improper integral using gamma function or beta function.

a.
 cos
a
4
 d
0
5. Test for convergence or divergence.

n!
a. 
n 1 5  n
4n 3  3n

5
2
n  2 n  4n  1

1
c. 
4
n 2 (ln n)

b.
b.
y
a 2  y 2 dy
4
0

 ln n 
d.  

n 1  n 
2