Math1220 Extra Final Review
Kelly MacArthur
1. Evaluate these integrals.
dx
(a) ∫
2
 x 1 x 
2
1
dx
(b)
∫  x−1
2
0
4
x
dx
(c) ∫ 2
0  x 9
(d) ∫ x sin x dx
2. Find the area of the region enclosed by
3. Solve
dy
4y=8 given
dt
4. What is the sum of
1
r =1sin  .
y=4 w h e n t=0 .
2
3
  
  ... ?
4 16 64
5. Find the area of the region outside
r =22 cos  and inside r =2 .
6. (a) Find the first four non-zero terms of
(b) Use that answer (from (a)) to estimate
f  x =e x
1
2
∫ e x dx
4
4
in its Maclaurin series.
.
0
7. Evaluate these integrals.
x2
dx
(a) ∫ 3
x 1
(b) ∫ y e−2y dy

(c)
∫ sin cos   d 
0
(d)
(e)
∫ x 2 e x dx
x1
∫ x  x−1 x 2 1 dx
3
8. For a new radioactive substance, it is found that after 10 years, 5% of the
substance has decayed. Find the half life of the substance.
1
9. Find the convergence set for the following power series.
∞
 x−3n
(a) ∑
n!
n=1
∞
( x−3)n
(b) ∑ n
n=0 2 +1
10. Does
−1n n 2 3n1
converge absolutely, conditionally or diverge?
∑
e 2n
n=1
∞
11. Find a power series for
convergence.
F  x =∫ x e x dx and state its radius of
3
dy
for each function. (Do not bother simplifying your answers! And
dx
make sure your answers are given as y = some function of x.)
ln (3x+1)
y=
(a)
cos( √ x)
(b)
y=( x 2 +2)1+x
12. Find
13. Solve this differential equation for y.
dy x 3−4y
=
dx
x
14. Evaluate these integrals.
√ 4x 2+9 dx
(a) ∫
x4
4
dx
(b) ∫
2 √ x −2
π/3
(c)
∫ x cos (3x )dx
0
(d)
(e)
(f)
dx
∫ 2x+3
x 3 +x
∫ cos3 x dx
0
∫ 8x 2 e−x dx
−∞
3
1
4y
dy
2
0 √ y +6
(h) ∫ e x sin (e x ) dx
x+9
dx
(i) ∫ 3
x +9x
cos x (sin x+cos x)
dx
(j) ∫
sin x
(g)
∫
2
4
x
dx
2
0 √ 9+x
7
2x
dx
(l) ∫
3 √ x −3
(m) ∫ x 2 ln x dx
(k)
∫
2
(n)
∫ x 3dxln x
√
1
2
15. Find the limit, if it exists.
sin (5x)
(a) lim
x → 0 tan (4x)
(b)
lim (2x)
1
3x
x →∞
(c)
lim (1+sin x )
x →0
(d)
lim
x →0
2
x
sin x− tan x
x 2 sin x
5n
, write out the first three terms of
√ 5n 2−3
the sequence. And, determine if the sequence converges or diverges. If it
converges, find its limit.
16. For the sequence given by
a n=
17. Determine if each series is absolutely convergent, conditionally convergent
or divergent. (Show your work to convince me of your result and state which
tests you're using to get to your answer.)
∞
√ 3n
(a) ∑ 3
n=1 n +5
∞
n−2
(b) ∑
n=1 4n+1
∞
n 4 (−3)n
(c) ∑
n=1 (n+2) !
n 2
∞
(−3) n
(d) ∑
n=1 (2n )!
∞
2n +7
(e) ∑ 4 4
n=1 √ 4n +5n +1
∞
3n 3+2n
(f) ∑
3
n=1 1+n
3
1
+e−3x . (Write out the
1−2x
3
terms through x .) State its radius of convergence.
18. Find a power series that represents
f ( x )=
1
3+ x
(a) Find the Taylor polynomial of order 3 centered about a = 2.
(b) Approximate f(2.3) using the Taylor polynomial in part (a).
(c) Find a bound for the error in your approximation.
19. For
f ( x )=
20. For Cartesian coordinates (−3 √ 3 , 3) , find three different ways to
represent this point in polar coordinates.
21. Find the slope of the tangent line to the graph r=2+3 sin θ at the point
π
where θ= 4 .
22. For the rectangular (Cartesian) equation
x 2 +( y−2)2 =4
(a) Find the equivalent polar equation for this curve.
(b) Using the polar equation in part (a), find the area of the region bounded by
the curve. (Hint: You may want to graph the function to ensure you get the
correct integration bounds.)
23. Find the derivative.
D x ( x 1+ x )
24. For f ( x )= x5 +2x3 +4x , determine if it's an invertible function. If it is
invertible, find ( f −1 )' (7) .
25. Find the equation of the tangent line to
26. Solve the differential equation
cos x
f ( x )= ( 1+sin x )
at
x= π .
2
dy
+2xy−2x =0 if it goes through (0, 3).
dx
1
1
a n= 3 + n
, write out the first three terms of
√ n √3
the sequence. And, determine if the sequence converges or diverges. If it
converges, find its limit.
27. For the sequence given by
2
x −1
(a) Find the Taylor polynomial of order 4 centered about a = 2.
(b) Approximate f(1.5) using the Taylor polynomial in part (a).
(c) Find a bound for the error in your approximation.
28. For
f ( x )=
4
29. Convert the point (−1 ,
coordinates.
5π
) from polar coordinates to rectangular
4
30. For the function (in polar coordinates) given by r 2 −6rcos θ−4r sin θ+9=0
(a) convert this function to rectangular coordinates, and
(b) describe the shape.
31. Find a power series that represents
convergence.
f ( x )=
3x 2
4− x3
and state its radius of
5
Answers:
−2
C
1 x
(b) diverges
(c) 2
(d) sin x− x cos xC
1. (a)
2.
3
2
3.
y=2
4.
5.
4
4−
8−
2
4t
e
1 8 1 12 1 16
4
1 x  x  x  x ...
2
6
24
(b) ~0.50636007
6. (a)
1
3
ln ∣x 1∣C
3
−1 −2x 1 −2x
x e − e C
2
4
7. (a)
(b)
(c) 2
(d)
(e)
1 x
e C
3
x−1
ln
−arctan x C
x
3
∣ ∣
8. ~135 years
9. (a) x ∈ℝ
(b) (1, 5)
10. converges absolutely
∞
11.
∑
n= 0
12. (a)
(b)
x 3n 2
n ! 3n2
dy
=
dx
cos   x 
dy
= x 2x 1x
dx



3
ln 3x1sin   x 

3x1
2 x
cos   x 
2x 1x 
ln  x 2 x 
2
x 2
2

6
13.
1 3 c
y= x  4
7
x
14. (a)
(b)
(c)
2 2
−2
9
3
− 4x2 9 2
C
27x 3
3
2
3 ln ∣x∣− ln  x 12 arctan xC
2
1 3
(e) sin x− sin xC
3
(f) diverges
(g) 4   7− 6
(h) −cos(e x )+C
1
1
x
2
+C
(i) ln∣x∣− ln ( x +9)+ arctan
2
3
3
(j) sin ( x)+cos ( x)+ln ∣csc ( x)−cot ( x )∣+C
(k) 2
104
(l)
3
1 3
1
x (ln x− )+C
(m)
3
3
(n) 0
5
15. (a)
4
(b) 1
(c) e2
1
(d) −
2
(d)
()
16.
a 1=
5
10
15
, a 2=
, a 3=
, converges to  5
2
17
 42
17. (a) converges absolutely
(b) diverges
(c) converges absolutely
(d) converges absolutely
(e) diverges
(f) diverges
18.
2−x 
17 2 7 3 155 4
x  x 
x ... , radius of convergence is
2
2
8
1
2
7
1 1
1
1
2
3
−  x−2
 x −2 −
 x −2 ...
5 25
125
625
(b) 0.1886768
(c) ∣R4  2.3∣≤0.000002592
19. (a)
5
−
−7
 , −6,
 , 6,

6
6
6
20.
6,
21.
−2−3  2
2
22. (a) r =4 sin 
(b) 4 
23.
x1+ x
(
1+ x
+ln x
x
)
1
15
π
25. y=(−ln 2) x +1+ 2 ln 2
26. y=1+2 e− x
3
3
4
3 √ 4+2 √ 3
2
2 √9
27. a 1= , a 2 =
; converges to 1
, a 3= 3
or
3
6
3
√3
28. (a) f ( x )≈2−2( x−2)+2 ( x−2)2 −2 ( x−2)3 +2 ( x−2)4
31
(b)
8
(c) 4
√2 , √2
29.
2
2
30. (a) ( x −3)2 +( y−2)2=4
(b) it's a circle with radius of 2 and center at (3, 2)
∞
3 x 3n+ 2
31. ∑ n+1 ; radius of convergence = √3 4
n=0 4
24.
2
(
(
)
)
8