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NAME_______________________________________
I___II___III___IV___V___ VI___VII___VIII___IX___X___XI___XII___ TOTAL _______
April 10 Mathematics 106 Mr. Haines
2003 Calculus II
Final Examination
(4) I. Circle the correct word for these approximations to
/ 2
0
sin( )
π x dx :
LEFT(12) is an overestimate an underestimate
RIGHT(40) is an overestimate an underestimate
MID(100) is an overestimate an underestimate
TRAP(5) is an overestimate an underestimate
(20) II. Evaluate these five integrals:
A.
x(4 - x) dx
24
B. ∫
dw 4 3 w
\
C. dw w
2
(5) III. Determine if this integral converges. If it converges, give the number to which it converges.
If it diverges, say why.
dz z
∞
1
4 / 3
(5) IV. Give an integral for the volume of the solid obtained when you rotate about the x-axis
the region bounded by the line y = 7 x and the parabola
2 y = x. You do not need to
evaluate the integral.
(8)V. The density function, p(x) , has equation:
p(x) = 0 if 0 > x
p(x) =
3 4 x if 1 ≥ x ≥ 0
p(x) = 0 if x > 1
A. Evaluate ∫
∞
−∞
p ( x ) dx.
B. If P is the cumulative distribution function for p(x), sketch a graph of P.
(10) VII. Taylor polynomials and series.
A. Give the twenty-fifth degree Taylor polynomial for ( ) 43 1492 1
25 3 f x = x + x − x +
at x = 0.
B. Give the Taylor Series for x
sin x about the point a = 0.
(12) VIII. Test each of these series and tell whether it converges or diverges. Give reasons for
your answers.
A. ∑
∞
n n
n
B. ∑
∞
= 1
n n n
C. ∑
∞
n
n
n
(5) XI Find that solution to the differential equation
− 4 A = 0
dt
dA which passes through the point (0, 10).
(12) XII. A population of sparrows has a continuous annual birth rate of 80% and a continuous
annual death rate of 90%. Scientists are importing sparrows at a rate of 100 per year to
try to keep the population from dying out.
A. Write a differential equation which has a solution P(t) , the population expressed as a function of t years.
B. Find all of the equilibrium solutions.
C. Solve your differential equation.
