Technical Calculus

(NOTE: Each chapter contains a Summary and a Review section.)

1. Analytic Geometry.


Functions. Graphing Equations. The Straight Line. Parallel and Perpendicular Lines. The Distance and Midpoint Formulas. The Circle. The Parabola. The Ellipse. The Hyperbola. Translation of Axes. The General Second-Degree Equation. Systems of Quadratic Equations. Polar Coordinates. Graphs in Polar Coordinates.



2. The Derivative.


Motion. The Limit. The Slope of a Tangent Line to a Curve. The Derivative. Differentiation of Polynomials. Derivatives of Products and Quotients. The Derivative of a Power. Implicit Differentiation. Proofs of Derivative Formulas. Higher Derivatives.



3. Applications of the Derivative.


Curve Sketching. Using Derivatives in Curve Sketching. More on Curve Sketching. Newton's Method for Improving Estimated Solutions. Maximum and Minimum Problems. Related Rates. Differentials and Linear Approximations.



4. Derivatives of Transcendental Functions.


The Trigonometric Functions. Derivatives of Sine and Cosine Functions. Derivatives of Other Trigonometric Functions. Derivatives of Inverse Trigonometric Functions. Derivatives of Logarithmic Functions. Derivatives of Exponential Functions. L'Hospital's Rule. Applications.



5. The Integral.


The Indefinite Integral. The Constant of Integration. Area Under a Curve. The Definite Integral.



6. Applications of Integrations.


Area Between Curves. Volumes of Revolution: Disk Method. Volumes of Revolution: Shell Method. Center of Mass of a System of Particles. Center of Mass of Continuous Mass Distributions. Moments of Inertia. Work, Fluid Pressure, and Average Value.



7. Methods of Integration.


The General Power Formula. Logarithmic and Exponential Forms. Basic Trigonometric Forms. Other Trigonometric Forms. Inverse Trigonometric Forms. Integration Using Partial Fractions. Integration by Parts. Integration Using Tables. Integration by Trigonometric Substitution. Integration Using Tables. Numerical Methods of Integration. Areas in Polar Coordinates. Improper Integrals.



8. Three-Space: Partial Derivatives and Double Integrals.


Functions in Three-Space. Partial Derivatives. Applications in Partial Derivatives. Double Integrals.



9. Progressions and the Binomial Theorem.


Arithmetic Progressions. Geometric Progressions. The Binomial Theorem.



10. Series.


Series and Convergence. Ratio and Integral Tests. Alternating Series and Conditional Convergence. Power Series. Maclaurin Series. Operations with Series. Taylor Series. Computational Approximations. Fourier Series.



11. First-Order Differential Equations.


Solving Differential Equations. Separation of Variables. Use of Exact Differentials. Linear Equations of First Order. Applications of First-Order Differential Equations.



12. Second-Order Differential Equations.


Higher-Order Homogenous Differential Equations. Repeated Roots and Complex Roots. Nonhomogenous Equations. Applications of Second-Order Differential Equations. The Laplace Transform. Solutions by Method of Laplace Transforms.



Appendix A: U.S. Weights and Measures.