Florida Institute of Technology Compute the Values Calculus Questions

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MTH 1002 Practice Final Exam for SS22
Name _________________________
1) Use the Disk Method to find the volume of the solid when the area bounded by y = 2x 2 and y = 2 , is
rotated about the x-axis.
(10 points)
2) Use the Shell Method to find the volume of the solid when the area bounded by y = x 3 and y = 4x is
rotated about the x-axis.
(10 points)
3) Evaluate ∫ x sin(2x)dx . Use integration by parts. (12 points)
xdx
4) Make a trigonometric substitution to evaluate ∫
x 2 +1
.
(14 points)
3x 2 − 3x +12
dx . Hint: use partial fractions. (18 points)
5) Evaluate ∫
(x − 2)(x 2 + 5)
+∞
dx
dx
3
x(ln(x))
1
6) Evaluate ∫
(10 points)
+∞
(x + 3)n
. (17 points) Start with the Ratio Test.
n
2
n
n=1
dy
8) Use the method of integrating factors to find the general solution:
− 2xy = x , y(0) = 5 . ( 12 points)
dx
dy
x3
=
9) Solve the initial value problem by separation of variables:
. (10 points)
dx cos(y)
10) First, make the sketch in Cartesian θ r – coordinates of the polar curves: r = 1+ cos(θ ) . Then, make the
sketch in polar coordinates. Show the units and label two polar coordinates for the curve.
(12 points)
11) Find the set-up definite integral of area that is inside the circle r = 2 outside the cardioid r = 2 − 2cosθ .
7) Find the radius and interval of convergence for: ∑
(8 points ) Do not evaluate.
2
1
-4
-3
-2
1
-1
2
-1
-2
12) Give the Maclaurin series for the following functions in sigma (∑ ) form: (6 pts)
tan −1 (x) =
cosh(2x) =
2
⎛ dr ⎞
13. Given ds = r + ⎜ ⎟ dθ for r = 2sin(θ ). Find the arc length if 0 ≤ θ ≤ π / 2.
(8 points)
⎝ do ⎠
14. Write out the first two terms of the sequence, then determine if it converges or diverges by finding the limit
+∞
⎧ 2n 3 − n ⎫
of the sequence. Do not simplify. (8 pts) ⎨ 2

3
⎩ n + 3n ⎭n=1
2
Tests for Convergence or Divergence
+∞
1. The Divergence Test: If lim an ≠ 0 , then the series ∑ an diverges.
n→+∞
n=1
+∞
2. Geometric Series: The geometric series ∑ ar n−1 = a + ar + ar 2 +… + ar n−1 +… =
n=1
a
converges if r 1 and diverges if 0 0 and finite, then both of the series behave the same.
bn


n=1
n=1
7. The Comparison Test: (i) ∑ bn converges ⇒ ∑ an also converges if an ≤ bn , for n ∈ Ν = {1, 2, 3, 4,…}


n=1
n=1
(ii) ∑ an diverges ⇒ ∑ bn also diverges if an ≤ bn , for n ∈ Ν = {1, 2, 3, 4,…} .
8. The Alternating Series Test (AST): If (i) bn ≥ bn+1, for n ∈ Ν and (ii) lim bn = 0 , then the alternating
n→∞

series ∑ (−1)n−1 bn converges. (If lim bn ≠ 0 , then the series diverges by the Divergence Test.)
n−>∞
n=1
+∞
+∞
9. Absolute Convergence: ∑ a n is absolutely convergent if ∑ a n converges.
n=1
n=1
+∞
+∞
n=1
n=1
10. Conditional Convergence: ∑ a n is conditionally convergent if it converges and ∑ a n diverges.

an+1
a
= L 1 , then
n→+∞ a
n→+∞ a
n=1
n
n
the series diverges. If L = 1 , then the test is inconclusive.
11. The Ratio Test: If lim

12. The Root Test: If lim n an = L 1, then
n→+∞
n=1
the series diverges. If L = 1 , then the test is inconclusive.
n→+∞
MTH 1002 Final Exam for SS22 Name ____________________ 5 bonus points _______
1) Use the Disk Method to find the volume of the solid when the area bounded by y = 4x 2 , x = 3 , y = 0 is
rotated about the x-axis.
(10 points)
y
2) Use the Shell Method to find the volume of the solid when the area bounded by x = y and x = is
3
rotated about the x-axis.
(10 points)
3) Evaluate ∫ xe x dx . Use integration by parts.
(12 points)
4) Make a trigonometric substitution to evaluate ∫ x x 2 −1 dx . (14 points)
1
dx . Hint: factor the denominator and use partial fractions. (18 points)
5) Evaluate ∫ 3
x + 4x
+∞
ln(x)
dx
6) Evaluate ∫
(10 points)
x
e
+∞
(x −1)n
7) Find the radius and interval of convergence for: ∑
. (17 points) Start with the Ratio Test.
2
n=1 2n
dy
8) Use the method of integrating factors to find the general solution:
+ 2xy = x , y(0) = 2 . ( 12 points)
dx
cos(y) dy
, sin(y) > 0 . (10 points)
9) Solve the initial value problem by separation of variables: 1 = 2
x sin(y) dx
10) First, make the sketch in Cartesian θ r – coordinates of the polar curves: r = 1+ cos(θ ) . Then, make the
sketch in polar coordinates. Show the units and label two polar coordinates for the curve.
(12 points)
11) Find the set-up definite integral of area that is outside the circle r = 2 inside the cardioid r = 2 − 2cosθ .
(8 points ) Do not evaluate.
2
1
-4
-3
-2
1
-1
2
-1
-2
12) Give the Maclaurin series for the following functions in sigma (∑ ) form: (6 pts)
ex =
ln(1+ x) =
2
⎛ dr ⎞
13. Given ds = r + ⎜ ⎟ dθ for r = 4cos(θ ). Find the arc length if 0 ≤ θ ≤ π .
(8 points)
⎝ do ⎠
14. Write out the first two terms of the sequence, then determine if it converges or diverges by finding the limit
2
+∞
⎧ n3 − n ⎫
of the sequence. Do not simplify. (8 pts) ⎨ 2

⎩ n + 2 ⎭n=1
Tests for Convergence or Divergence
+∞
1. The Divergence Test: If lim an ≠ 0 , then the series ∑ an diverges.
n→+∞
n=1
+∞
2. Geometric Series: The geometric series ∑ ar n−1 = a + ar + ar 2 +… + ar n−1 +… =
n=1
a
converges if r 1 and diverges if 0 0 and finite, then both of the series behave the same.
bn


n=1
n=1
7. The Comparison Test: (i) ∑ bn converges ⇒ ∑ an also converges if an ≤ bn , for n ∈ Ν = {1, 2, 3, 4,…}


n=1
n=1
(ii) ∑ an diverges ⇒ ∑ bn also diverges if an ≤ bn , for n ∈ Ν = {1, 2, 3, 4,…} .
8. The Alternating Series Test (AST): If (i) bn ≥ bn+1, for n ∈ Ν and (ii) lim bn = 0 , then the alternating
n→∞

series ∑ (−1)n−1 bn converges. (If lim bn ≠ 0 , then the series diverges by the Divergence Test.)
n−>∞
n=1
+∞
+∞
9. Absolute Convergence: ∑ a n is absolutely convergent if ∑ a n converges.
n=1
n=1
+∞
+∞
n=1
n=1
10. Conditional Convergence: ∑ a n is conditionally convergent if it converges and ∑ a n diverges.

an+1
a
= L 1 , then
n→+∞ a
n→+∞ a
n=1
n
n
the series diverges. If L = 1 , then the test is inconclusive.
11. The Ratio Test: If lim

12. The Root Test: If lim n an = L 1, then
n→+∞
n=1
the series diverges. If L = 1 , then the test is inconclusive.
n→+∞

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