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 Ans: (i) 1 (ii) DIV (iii) 0

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 Ans: (i) 3/2

(ii) DIV (iii) DIV (iv) 0 Chap 5: Sec. 5.3-Sec. 5.4.

1. Set up a definite integral for the arc length of an ellipse x²+ 4y² = 4. Ans: 4^∫₀²

√

1 + (− ^x

4√

1−x²/4)²dx or 4^∫₀^π/2√(−2 sin θ)²+ (cos θ)²dθ

2. Set up the integral for the surface area of the surface of revolution. y = e^x, 0 ≤ x ≤ 1, revolved about x-axis. Ans: S =^∫₀¹2πe^x√1 + (e^x)²dx

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). Ans: y(0) = 10;y⁰(0) = 5;y(t) =

−12gt²+ 5t + 10;

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?

Ans: ⁴₃⁵4

2. Find the size of permanent endowment needed to generate an annual $2,000 forever at 10% (annual) interest compounded continuously. Ans: 2000· e^−0.1

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.

Ans: (i) y = (³₂x²− 3x + 8)^1/3;(ii) y(t) =−√

x²− 2x + 4 6. Solve the initial value problem

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

Ans: (i) y(t0 = 0;(ii) y(t) = 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) a_n= ^{cos n}_n2

C) an= 1 D) an= (−1)ⁿ

Ans: A) ³₄; B) 0; C) 1; D) Div

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₎

Hint/Ans: i) Geometric Series; ii) DIV; Limit comparison Test (Compared with Harmonic Series); iii) Telescope Series; example 2.3.

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

Hint/Ans:i) Div (p-series) ;ii) Conv (Telescope Series); iii)Div (Thm 2.3); iv) Conv (Limit Comparison Test, p-series) ;v)Conv (Comparison Test)

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

Hint: Substitution s = ln x 5. If ak=

{ 1/k if k is odd

1/k² if k is even , determine the convergence of the series,^∑∞_k=1(−1)^kak. Hint: Div (Thm 2.3)

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

Hint/Ans: i)Abs Conv (Limit Comparison Test, p-series);ii) Conditionally Conv (Limit Comparison Test, p-series, Alternating Series Test);

iii) Abs Conv (Ratio Test);iv) Conditionally Conv (Limit Comparison Test, p-series, Alternating Series Test)

7. Determine the radius and interval of convergence of the power series.

i)^∑∞_k=12^k_k(x− 2)^k ii) ^∑∞_k=1(x− 3)^k iii)^∑∞_k=1k!(x− 2)^k iv)^∑∞_k=1 ²_k!^k(x− 3)^k Ans: i) (0, 4), r = 2;ii) (2, 4), r = 1;iii){2},r = 0;iv) (−∞, ∞),r = ∞

8. For f (x) = e^x, find the Taylor polynomial of degree 2 expanded about c = 1.

Ans: P2(x) = e +₁^e(x− 1) + ^e₂²(x− 1)²

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

Ans: e^x=^∑∞_{k=0 k!}¹x^k, for x∈ (−∞, ∞), r = ∞;

e^−x2 =^∑∞_k=0(−1)^{k 1}_k!x^2k, for x∈ (−∞, ∞), r = ∞;

x²e^−x2 =^∑∞_k=0(−1)^{k 1}_k!x^2k+2, for x∈ (−∞, ∞), r = ∞

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.

Ans: i)ln (1 + x) =^∑∞_k=0⁽⁻¹⁾_k+1^kx^k+1, for − 1 < x ≤ 1, r = 1;

ii)_1+x¹ 2 =^∑∞_k=0(−1)^kx^2k, for − 1 < x < 1,, r = 1;

iii)x ln (1 + x) =^∑∞_k=0 ⁽⁻¹⁾_k+1^kx^k+2, for − 1 < x ≤ 1, r = 1 iv)_1+x^x22 =^∑∞_k=0(−1)^kx^2k+2, for − 1 < x < 1,, r = 1;

11. ^∑∞_k=0(−1)^k_k+1¹ =?

Ans: ln 2