chapter 1: numbers, functions, and graphs
1-1 introduction
1-2 the real line and coordinate plane: pythagoras
1-3 slopes and equations of straight lines
1-4 circles and parabolas: descartes and fermat
1-5 the concept of a function
1-6 graphs of functions
1-7 introductory trigonometry
1-8 the functions sin o and cos o
chapter 2: the derivative of a function
2-0 what is calculus ?
2-1 the problems of tangents
2-2 how to calculate the slope of the tangent
2-3 the definition of the derivative
2-4 velocity and rates of change: newton and leibriz
2-5 the concept of a limit: two trigonometric limits
2-6 continuous functions: the mean value theorem and other theorem
chapter 3: the computation of derivatives
3-1 derivatives of polynomials
3-2 the product and quotient rules
3-3 composite functions and the chain rule
3-4 some trigonometric derivatives
3-5 implicit functions and fractional exponents
3-6 derivatives of higher order
chapter 4: applications of derivatives
4-1 increasing and decreasing functions: maxima and minima
4-2 concavity and points of inflection
4-3 applied maximum and minimum problems
4-4 more maximum-minimum problems
4-5 related rates
4-6 newtons method for solving equations
4-7 applications to economics: marginal analysis
chapter 5: indefinite integrals and differential equations
5-1 introduction
5-2 differentials and tangent line approximations
5-3 indefinite integrals: integration by substitution
5-4 differential equations: separation of variables
5-5 motion under gravity: escape velocity and black holes
chapter 6: definite integrals
6-1 introduction
6-2 the problem of areas
6-3 the sigma notation and certain special sums
6-4 the area under a curve: definite integrals
6-5 the computation of areas as limits
6-6 the fundamental theorem of calculus
6-7 properties of definite integrals
chapter 7: applications of integration
7-1 introduction: the intuitive meaning of integration
7-2 the area between two curves
7-3 volumes: the disk method
7-4 volumes: the method of cylindrical shells
7-5 arc length
7-6 the area of a surface of revolution
7-7 work and energy
7-8 hydrostatic force
part ii
chapter 8: exponential and logarithm functions
8-1 introduction
8-2 review of exponents and logarithms
8-3 the number e and the function y = e <^>x
8-4 the natural logarithm function y = ln x
8-5 applications
population growth and radioactive decay
8-6 more applications
chapter 9: trigonometric functions
9-1 review of trigonometry
9-2 the derivatives of the sine and cosine
9-3 the integrals of the sine and cosine
9-4 the derivatives of the other four functions
9-5 the inverse trigonometric functions
9-6 simple harmonic motion
9-7 hyperbolic functions
chapter 10 : methods of integration
10-1 introduction
10-2 the method of substitution
10-3 certain trigonometric integrals
10-4 trigonometric substitutions
10-5 completing the square
10-6 the method of partial fractions
10-7 integration by parts
10-8 a mixed bag
10-9 numerical integration
chapter 11: further applications of integration
11-1 the center of mass of a discrete system
11-2 centroids
11-3 the theorems of pappus
11-4 moment of inertia
chapter 12: indeterminate forms and improper integrals
12-1 introduction. the mean value theorem revisited
12-2 the interminate form 0/0. l'hospital's rule
12-3 other interminate forms
12-4 improper integrals
12-5 the normal distribution
chapter 13: infinite series of constants
13-1 what is an infinite series ?
13-2 convergent sequences
13-3 convergent and divergent series
13-4 general properties of convergent series
13-5 series on non-negative terms: comparison tests
13-6 the integral test
13-7 the ratio test and root test
13-8 the alternating series test
chapter 14: power series
14-1 introduction
14-2 the interval of convergence
14-3 differentiation and integration of power series
14-4 taylor series and taylor's formula
14-5 computations using taylor's formula
14-6 applications to differential equations
14. 7 (optional) operations on power series
14. 8 (optional) complex numbers and euler's formula
part iii
chapter 15: conic sections
15-1 introduction
15-2 another look at circles and parabolas
15-3 ellipses
15-4 hyperbolas
15-5 the focus-directrix-eccentricity definitions
15-6 (optional) second degree equations
chapter 16: polar coordinates
16-1 the polar coordinate system
16-2 more graphs of polar equations

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