of the Inverse Function -- The Derivative of a Parametrized Curve -- First Derivative, Tangent Line, and Growth -- The Mean Value Theorems -- L'Hôpital's Rule -- Snell's Law -- The Brachistochrone -- Exercises -- 5 The Integral -- Measure and Integral -- The Fundamental Theorem of Calculus -- A Pause for Comments -- Buffon's Needle -- Irrationality of π -- Improper Integrals -- Integration and Sums: Linearity of the Integral -- Uniform Convergence-The Weierstrass M-Test -- Gregory's Series -- Integration and Products: Integration by Parts -- Stirling's Formula. |
Integration and Composition: Integration by Substitution -- A Note on Notation -- Length of Curves. The Catenary -- Area Enclosed by a Simple Closed Curve -- Exercises -- 6 More Derivatives -- Second Derivative, Best-Fitting Parabola, and Curvature -- The Taylor Polynomial of Order Two -- Curvature -- Random Walk and the Gauss Curve -- The Taylor Series -- Exercises -- 7 Convexity and the Isoperimetric Inequality -- The Arithmetic-Geometric Inequality -- Convexity -- Young, Hölder, Jensen, Cauchy-Schwarz… -- The Isoperimetric Inequality -- Exercises -- 8 More Integrals -- Volume -- Double Integrals -- The Basel Problem -- Solids of Revolution -- Integration of e -- Density Functions, Barycenter, and Expectation -- Center of Mass or Barycenter -- Pappus' Theorem -- The Method -- Surface Area -- Normal Distribution. Gauss, Laplace, and Stirling -- Exercises -- 9 The Gamma Function -- The Gamma Function -- Weierstrass' Formula -- Growth of the Harmonic Series, Again -- Exercises -- Bibliography -- Index. |