Calculus II

(1)

Calculus II Midterm Practice problems Chap 4: Sec. 4.10.

1. Evaluate the given integral (a) ^∫_−∞^∞ _(1+e^e^−x_−x₎2 dx (b) ^∫_−∞^∞ x³dx

(c) limR→∞∫_R

−Rx³dx

2. Determine whether the integral converges or diverges:

(a) ^∫₀¹x^−1/3dx (b) ^∫₀¹x^−4/3dx (c) ^∫₁^∞x^−1/3dx (d) ^∫₋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. (i) At time t, a particle has position x(t) = 1− cos t, y(t) = t − sin t Find the total distance traveled from t = 0 to t = 2π. Find the speed of the particle at t = π.

(ii) Find the area of the surface generated by revolving the curve y = cosh x, x∈ [0, ln 2] about the x-axis.

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. Solve the IVP, explicitly, if possible y⁰= ^x_y⁻¹2 , y(0) = 2.

• Sec. 7.1:

Find the limit of a sequence. Determine the convergence of a sequence.

Examples:2-12. Practice Problems:11,31,43,53,65.

• Sec. 7.2:

Convergence and divergence of a series; geometric series; p-series; k-th term test for divergence.

ExamplesExamples: 1-7. Practice Problems:1, 7, 17, 37, 39, 41.

• Sec. 7.3:

Integral Test; Comparison Test; Limit Comparison Test.

Examples: 1-2,5-9. Practice Problems:1, 11, 37, 41, 45, 57.

• Sec. 7.4:

Alternating Series Test;

Examples: 1-4. Practice Problems:1, 11, 41, 43.

• Sec. 7.5:

Absolute Convergence and Conditional Convergence; Ratio Test.

Examples: 1-7. Practice Problems: 7, 13, 25, 35, (40).

(2)

• Sec. 7.6:

Interval and Radius of Convergence; Term-by-term diﬀerentiation and integration.

Examples: 1-6. Practice Problems: 1, 3, 11, 21, 39.

• Sec. 7.7:

Taylor’s Theorem; Derive a Taylor series or polynomial; Find new Taylor series from old ones.

Examples: 1-3,8. Practice Problems: 1, 5, 33, 41, (47).

• Sec. 7.8:

Use Taylor polynomials to approximate a function, to ﬁnd the limit and to approximate an integral.

Examples: 1-5. Practice Problems: 7, 11, 13, 15.