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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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