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All section numbers refer to
APEX Calculus, Version 4.0.

Be sure to check the
Errata for corrections to the text.

Be sure to check back, because this page will be updated often during the semester.

u-substitution

- Watch:
- Welcome to Calc II! (Echo360)
- Overview of Calc II Content (Echo360)

- Submit:
- Pre-Semester Questionnaire, if you haven't already done so
- Submit: "The One Minute Quiz for first day" at onCourse

- Skim: Section 5.4 The Fundamental Theorem of Calculus, for review
- Watch: The Fundamental Theorem of Calculus (Echo360), for review

- Skim: Section 6.1 Substitution, for review
- Watch:
- Introduction to u-substition (Khan Academy), for review
- Watch: How to Integrate Using U-Substitution (NancyPi), for some more examples

- Section 2.7 Derivatives of Inverse Functions
- Section 6.2 Integration by Parts

- The Inverse Sine and Tangent Functions (Echo360)
- Integration by Parts Intro (Khan Academy)
- Integration by parts: An Example (Khan Academy)
- Integration by Parts. . .How? (NancyPi), if you would like some more examples

- Find an antiderivative of \( f(x) = \displaystyle \frac{3x^2}{ 1 + x^6}\)
- Integration by parts attempts to undo one of the techniques of differentiation. Which one is it?
- Use integration by parts to find an antiderivative of \(f(x) = 2x e^{x}\)

- Section 5.5 Numerical Integration

Focus on the intuitive ideas behind L_{n}, R_{n}, T_{n}, and S_{n}and the statement of Theorem 5.5.1 that gives error bounds for T_{n}, and S_{n}. We'll use technology to calculate the approximations. - Section 7.2 Volume by Cross-Sectional Area; Disk and Washer

- Numeric integration (Echo360)
- Volume with cross sections: intro (Khan Academy)
- Disc method around x-axis (Khan Academy)

- Why would you ever want to numerically approximate an integral?
- Let \( \mathcal{I} = \displaystyle\int_0^{\pi} \sin(x^2) dx\).
- Which would you expect to be MOST accurate in approximating \( \mathcal{I} \) : a Right Hand approximation \( R_n\), a Trapezoidal approximation \(T_n\), or a Simpson's approximation \(S_n\)? Why?
- Which would you expect to be LEAST accurate in approximating \( \mathcal{I} \) : a Right Hand approximation \( R_n\), a Trapezoidal approximation \(T_n\), or a Simpson's approximation \(S_n\)? Why?

- Let R be the rectangle formed by the x-axis, the y-axis, and the lines y=1 and x=4. Describe the shape of the solid formed when R is rotated about the x-axis.

- Section 7.4 Arc Length and Surface Area

Focus on pp 378-381 - Section 6.8 Improper Integration

- Deriving the Formula for Arc Length (Phil Clark)
- Introduction to improper integrals (Khan Academy)
- What makes an integral improper? (Krista King)

- Set up the integral that gives the length of the curve \( y=\sin(2x)\) from \(x=0\) to \( x=2\pi\).
- Explain why \( \displaystyle\int_1^{\infty} \frac{1}{x^2} dx \) is improper.
- Explain why \( \displaystyle\int_0^1 \frac{1}{x^2} dx \) is improper.

- Section 8.1 Sequences
- Section 8.2 Infinite Series

- Introduction to Sequences (Thinkwell)
- Introduction to Infinite Series (Echo 360)
- Geometric Series Are Your Friends (Echo 360)

- Does the following sequence converge or diverge? Explain.

\[ 1, 3, 5, 7, 9, 11, 13, \ldots \] - There are two sequences associated with every series. What are they?
- Does the geometric series \( \displaystyle \sum_{n=0}^{\infty} \left( \frac{1}{4}\right)^n\) converge or diverge? Why?
- What assignments are due this week? When are they due?

- Section 8.3 Integral and Comparison Tests

- The Integral Test for Series (Echo360)
- The Direct Comparison Test for Series (Echo360)

- What does the n
^{th}-Term Theorem tell you about the series \( \displaystyle \sum 2^n \)? - What does the n
^{th}-Term Theorem tell you about the series \( \displaystyle \sum \frac{1}{n} \)? - What does the Integral Test tell you about the series \( \displaystyle \sum \frac{1}{n^3} \)?
- What does the Integral Test tell you about the series \( \displaystyle \sum \frac{1}{\sqrt{n}} \)?

- Section 8.5 Alternating Series
- Section 8.6 Power Series

- Alternating Series (The Organic Chemistry Tutor)

The first 8 minutes or so is most important, and the rest contains more examples. Overall I think this is a good resource, but I cringe when they write \( \frac{1}{\infty} \). Please don't do this! - Power Series (Khan Academy)

- Explain why the alternating series \( \displaystyle\sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n}\) converges.
- How closely does \(\displaystyle S_{50}\), the 50th partial sum, approximate the value of the series \(\displaystyle \sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n}\)? Why?
- How do power series differ from other series we have looked at up to this point?

- Section 8.7 Taylor Polynomials
- Section 8.8 Taylor Series

- Taylor Polynomials (Echo360)
- A catalog of Taylor Series (Echo360)

- What is the difference between a Taylor polynomial and a Taylor series?
- What is the difference between a Taylor series and a Maclaurin series?
- Why would you ever want to compute a Taylor series for a function like f(x)=sin(x)?

- Section 12.1 Introduction to Multivariable Functions
- Section 12.3 Partial Derivatives

- Introduction to Multivariable Functions (Khan Academy)

We'll focus on functions f(x,y) with one output - Introduction to 3d Graphs (Khan Academy)
- Contour Plots (Khan Academy)
- Introduction to Partial Derivatives (Khan Academy)
- Partial Derivatives and Graphs (Khan Academy)

- Describe the level curves of the function \(f(x,y)= x^2 + y^2\) for c= 4, 0, and -1.
- If \( g(x,y)= x^2-y^2\), what is \( g_x(x,y) \), the partial derivative of \( g \) with respect to \( x \)?
- If \( g(x,y)= x^2-y^2\), what is \( g_x(2,1)? \)? What geometric information does this give you?

- Section 10.2 An Introdution to Vectors
- Sectxion 10.3 The Dot Product

- Unit vectors (Khan Academy)
- The dot product (Echo360)

- Give the unit vector in the same direction as \( \vec{\,v_1} \)
- What is \( \vec{\,v_1} \cdot \vec{\,v_2}\ \)? What geometric information does this give you about the vectors?

- Section 12.6 Directional Derivatives

- Directional Derivatives (Echo360)

- What does the directional derivative \( D_{\vec{\,u}} f(a,b)\) measure?
- If \(f(x,y) = 3xy^2 + 2x-4y^2\), what is \(\nabla f(x,y)\) ?

- Section 12.8 Extreme Values

- Multivariable Optimization Overview (Echo360)
- Multivariable Optimization Example (Echo360)

- Where can the local extrema of a function f(x,y) occur?
- In Example 12.8.3, why does it make sense that the critical point (1,2) is called a "saddle point"?

- Section 13.1 Iterated Integrals and Area
- Section 13.2 Double Integration and Volume

- Double Integration Overview (Echo360)
- Double Integration Examples (Echo360)

- Why would you want to switch the order of integration in an iterated integral?
- If \(f(x,y)\) is a function of two variables and \(R\) is a rectangle in the xy-plane, what does \( \iint_R f(x,y)\, dA\) measure?
- Explain the idea of Fubini's Theorem in a couple of sentences in your own words.