Calculus II Midterm Practice problems Time: 4/30(M), 3:10-5:00; Place:

Calculus II Midterm Practice problems Time: 4/30(M), 3:10-5:00; Place:

Chap 4: Sec. 4.10.

1. Evaluate the given integral

i)^∫_−∞^∞ _(1+e^e−x_−x)2dx ^∫_−∞^∞ x³dx iii) lim_R→∞^∫_−R^R x³dx 2. Determine whether the integral converges or diverges:

i)^∫₀¹x^−1/3dx ii) ^∫₀¹x^−4/3dx

iii)^∫₁^∞x^−1/3dx iv)^∫₋₁¹ x^−1/3dx

Chap 5: Sec. 5.3-Sec. 5.4.

1. Set up a definite integral for the arc length of an ellipse x²+ 4y² = 4.

2. Set up the integral for the surface area of the surface of revolution. y = e^x, 0 ≤ x ≤ 1, revolved about x-axis.

3. An object is released from a height of 10m with an upward velocity of 5m/s. Let y(t) be the height of the object. Identify the initial conditions y(0) and y⁰(0). Find y(t).

Chap 6: Sec. 6.1-Sec. 6.3.

1. Two years ago, there were 4 grams of a radioactive substance . Now there are 3 grams. How much was there 10 years ago?

2. Find the size of permanent endowment needed to generate an annual $2,000 forever at 10% (annual) interest compounded continuously.

3. Select the differential equation which corresponds to the direction field below.

(a) y⁰ =−_2y¹ (b) y⁰ =−^2y_x,

(c) y⁰ =−^2x_y (d) y⁰ =−_y¹2

4. Select the differential equation which corresponds to the direction field below.

(a) y⁰ =−xy (b) y⁰ = y + x²,

(c) y⁰ = y²+ x³ (d) y⁰ =−x − y²

5. Solve the IVP:

i) y⁰ = ^x_y⁻¹2 , y(0) = 2; ii) y⁰ = ^x−1_y , y(0) =−2.

6. Solve the initial value problem

i) y⁰(t) = y(t), y(0) = 0; ii) y⁰(t) = (y(t)− 4), y(0) = 4.

Chap 7. Sec. 7.1-Sec. 7.8.

1. Determine the convergence of a sequence. If it converges, find the limit of the sequence.

A) a_n= ³ⁿ_4n²⁺ⁿ2−n⁻² B) an= ^{cos n}_n2

C) a_n= 1 D) a_n= (−1)ⁿ

2. Determine if the series converges or diverges. If it converges, find the sum of the series.

i)^∑∞_k=0(¹₃)^k ii) ^∑∞_{k=0 k}^k+12−4

iii)^∑∞_{k=2 k(k−1)}¹

3. Determine if the series converges or diverges.

i)^∑∞_k=0 √3³

k ii) ^∑∞_{k=2 k(k−1)}¹

iii)^∑∞_k=0((−1)^k− (¹₃)^k) iv)^∑∞_{k=0 k}^k+13−4

v)^∑∞_k=2^{cos k+1}_k2

4. Use the Integral Test to test the convergence of the series ^∑∞_{k=2k(ln k)}¹ 3

5. If a_k= {

1/k if k is odd

1/k² if k is even , determine the convergence of the series,^∑∞_k=1(−1)^ka_k. 6. Determine if the series is absolutely convergent, conditionally convergent or divergent.

i)^∑∞_k=1(−1)^k_k3^k+1 ii) ^∑∞_k=1(−1)^k_k+1^√k iii)^∑∞_k=1(−1)^{k 3}_(2k)!^k iv)^∑∞_k=1(−1)^k_(2k+1)¹ 7. Determine the radius and interval of convergence of the power series.

i)^∑∞_k=12^kk(x− 2)^k ii) ^∑∞_k=1(x− 3)^k iii)^∑∞_k=1k!(x− 2)^k iv)^∑∞_k=1 ²_k!^k(x− 3)^k 8. For f (x) = e^x, find the Taylor polynomial of degree 2 expanded about c = 1.

9. Find the Taylor series of e^x, e^−x2 and x²e^−x2 about c = 0. Determine the corresponding radius and interval of convergence.

10. Given that _1+x¹ =^∑∞_k=0(−1)^kx^k, for − 1 < x < 1, find the Taylor series of

i) ln (1 + x) ii) _1+x¹ 2.

iii) x ln (1 + x) iv) _1+x^x22.

Determine the corresponding radius and interval of convergence.

11. ^∑∞_k=0(−1)^k_k+1¹ =?