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Dartmouth Mathematics Placement System

## Syllabus: Math 8

The following is a standard syllabus for Math 8. If you wish to review some or all of the material, we provide three methods.

First, the syllabus closely follows Stewart's Calculus textbook chapters 6, 8, and 12-15, which are marked on the syllabus below.

Second, we provide links to Khan Academy (KA) videos relevant to the material on that part of the syllabus (Note that, at this time, KA's coverage of differential multivariable calculus is not complete).

Third, as practice problems are always useful, in addition to the problems available in Stewart and at Khan Academy, we provide practice exams.

Note that MATH 8 is split into two, more or less independent, parts - the end of single variable calculus (part A below) and the beginning of multivariable calculus (part B below). If you are reviewing for the MATH 8 Credit exam, you need only review material from the Sample Math 3 Syllabus and from part A below - those are the only topics covered on the exam. If you are reviewing for the Multivariable Credit Exam, you should review part B below as well as topics from the Sample Math 13 syllabus.

## A. The remainder of BC Calculus

### 1. Applications of Integration

a. Areas between curves (Stewart 6.1)

b. Volumes of revolution (Stewart 6.2)

### 2. Techniques of Integration

a. Integration by parts (Stewart 8.1)

b. Trigonometric integrals (Stewart 8.2)

c. Trigonometric Substitution (Stewart 8.3)

d. Partial Fractions (Stewart 8.4)

e. Numerical Integration (Stewart 8.7)

f. Improper Integrals(Stewart 8.8)

### 3. Sequences and Series

a. Sequences and series of constants (Stewart 12.1)

b. Inetgral Test (Stewart 12.3)

c. Comparison Test (Stewart 12.4)

d. Alternating series (Stewart 12.5)

e. Ratio Test (Stewart 12.6)

f. Power series, Representation as power series (Stewart 12.8-12.9)

g. Taylor and Maclaurin series (Stewart 12.10)

## B. Differential Multivariable Calculus

### 4. The language of vectors

a. Coordinates and vectors in $\mathbb R^2,\ \mathbb R^3$ (Stewart 13.1,13.2)

b. Dot and cross product (Stewart 13.3,13.4)

c. Lines and planes in three-dimensional space (Stewart 13.5)

### 5. Calculus for vector valued functions of one variable

a. Vector functions, space curves, and their derivatives and integrals (Stewart 14.1, 14.2)

b. Arclength, velocity, and accelerations (Stewart 14.3, 14.4)

### 6. Differentiation for functions of more than one variable

a. Functions of several variables (Stewart 15.1)

b. Limits and continuity (Stewart 15.2)

c. Partial Derivatives (Stewart 15.3)

d. Tangent Planes and approximation (Stewart 15.4)

e. Chain Rule (Stewart 15.5)

f. Directional derivatives and the gradient (Stewart 15.6)

g. Maxima and minima (Stewart 15.7)

h. Lagrange Multipliers (Stewart 15.8)