Math1220
Spring, 2009
Final Quiz (#11)
1. Evaluate D x  x 1 x  .
(a) 1x  x x
(b) ln x  x 1x
1 x
(c)
ln x
x
1 x
(d) x1 x
ln x
x

x

x
2. Evaluate ∫ e sin e  dx .
(a) −cos xC
x
(b) −cos e C
(c) cos xC
(d) cose x C
3. For f  x= x52x 34x we can prove it's a monotonically increasing function and therefore has an
inverse function. Find (1) f −1 7 and (2)  f −1' 7 .
1
(a) 11 ; 2
12303
(b) 11 ; 212303
(c) 117521 ; 212303
1
(d) 11 ; 2
15
4. If $500 is put in the bank today at 9% interest compounded monthly, how much will it be worth in 10
years?
(a) $1183.68
(b) $538.79
(c) $1225.68
(d) $546.90
5. Find the equation of the tangent line to

ln 2
2
1

1
(b) y=ln
x −1 ln
2
2
2
(c) y =1

(d) y=−ln 2 x ln 2
2
(a) y =−ln 2 x1


y=sin x1
cos x
at x=

.
2
dy
2xy−2x=0 if it goes through (0, 3).
dx
6. Solve this differential equation
(a)
(b)
(c)
(d)
y =1
y=1c e−x
y =12 e −x
y=1−3x 2
2
2
7. Evaluate the integral.
x9
∫ x 39x dx

1
x
(a) ln∣x∣ arctan
C
3
3
(b) x ln x C
1
x
(c) ln∣x∣−x ln  x 29 arctan
C
3
3
1
x
(d) ln∣x∣ arctan
−ln  x 29C
3
3


cos x sin xcos x 
dx
sin x
(a) sin xln ∣sin x∣C
(b) sin xcos xln ∣csc x−cot x∣C
(c) sin xln ∣csc x−cot x∣C
(d) cos xln∣sin x∣C
8. Evaluate the integral.
∫
4
∫
9. Evaluate the integral.
0
(a)
(b)
(c)
(d)
2
1
4
0
7
10. Evaluate the integral.
(a)
(b)
(c)
(d)
52
3
64  7
−16  3
3
diverges
104
3
x
dx
 9 x 2
2x
dx
∫  x−3
3
2
11. Evaluate the integral. ∫ x ln x dx
(a) −x C
1
1
(b) x3 ln x− x 3C
3
3
1
1
(c) x 2 ln x− x 2C
2
4
1 3
1
(d) x  ln x− C
3
3
12. Find the limit.
lim 1sin x
2
x
x0
(a) 2
(b) e 2
(c) 1
(d) e−2
lim
13. Find the limit.
x0
1
2
(b) 0
sin x −tan x
2
x sin x
(a)
(c) −
1
2
(d) 1
2
14. Evaluate the integral.
(a) 0
(b) 2 ln 2
∫
1
2
dx
x ln x 
3
2
(c) 3 ln 2 3
(d) diverges
1
1
15. Given the sequence a n= 3  n
, does it converge or diverge? If it converges, what does it
 n 3
converge to?
(a) diverges
(b) converges to 1
(c) converges to 0
1
(d) converges to
3
(di)
−3n n2
∑  2n! .
n=1
∞
16. Determine the convergence of this series.
(a) converges absolutely
(b) converges conditionally
(c) diverges
∞
17. Determine the convergence of this series.
∑
n=1
(a) converges absolutely
(b) converges conditionally
(c) diverges
2n7
 4n 4 5n1
.
3n 32n
∑ 1n 3 .
n=1
∞
18. Determine the convergence of this series.
(a) converges absolutely
(b) converges conditionally
(c) diverges
(ci)
 x−3n
∑ n
n=0 2 1
∞
19. Find the convergence set for this power series.
(a) −∞ , ∞
(b) [ 1, 5 )
(c) 1,5
(d) [2, 4]
20. Find the Taylor polynomial of order 4 centered at x = 2 for
f  x =
2
.
x−1
(a) 21− x−2 x−22− x−23 x −24 
(b) −2 1x x 2 x 3x 4 
1
1
1
(c) 21− x−2  x−22−  x−23  x−24 
2
6
24
1
1
1
1
2
3
4
(d) 21−  x−2  x −2 −  x −2   x−2 
2
2
2
2
21. For
f  x =
in problem #20).
1
(a)
16
1
(b)
40
(c) 4
4
(d)
5
2
, find the error in computing f(1.5) using the 4th order Taylor polynomial (found
x−1
22. Find the polar coordinates for the rectangular coordinates −3,  3 .
−
(a) 2  3 ,

6
5
(b) 2  3 ,

6

(c) 2  3 , 
3
−
(d) −2  3 ,

3
23. Find the rectangular coordinates for the polar coordinates −1,
5

4
 
  
 
  
− 2 − 2
,
2
2
2 − 2
(b)
,
2
2
− 2 2
(c)
,
2
2
2 2
(d)
,
2 2
(a)
24. Find the Cartesian equation for the polar equation.
2
2
(a)  x−2  y −3 =4
(b)  x−32 y−22 =4
(c) y =x−1
 x−22  y−32
(d)

=1
9
4
2
r −6r cos −4 r sin 9=0
25. Find the power series and the radius of convergence for
∞
3n2
∑ 34xn1 ; 3 4
n=0
∞
3 x 3n2
(b) ∑
;1
4
n=0
∞
3 x n 2
(c) ∑
;4
n=0
4n
∞
3 x 2n
(d) ∑ n1 ; 1
n=0 4
(a)
f  x =
3x 2
.
4−x 3