104003 – Differential and Integral Calculus 1
Introduction: real numbers; absolute value; open and closed intervals; upper and lower bounds of sets of real numbers; infimum (inf) and supremum (sup).
Infinite Sequences: convergence of sequences; the arithmetic of limits; the sandwich theorem; monotone sequences; convergence of bounded, monotonic sequences; subsequences, the Bolzano-Weierstrass theorem.
The Limit of a Function: Explanation and definition in terms of ? and ?; the arithmetic of limits; one-sided limits and limits at ?; infinite limits; inequalities for limits; the sandwich theorem; bounded functions and monotone functions; monotonicity and boundedness imply the existence of a limit.
Continuous Functions: Definition of continuity (including one-sided); the arithmetic of continuous functions; the continuity of the elementary functions; classification of discontinuities; a monotone function has only jump discontinuities; the intermediate value theorem; application to the roots of a polynomial of odd order; the boundedness and the existence of maximum and minimum for a bounded, continuous function on a closed bounded interval.
Derivatives: definition of the derivative (including one-sided); the physical meaning of derivative; derivatives of elementary functions; differentiability implies continuity; the arithmetic of derivatives; the best linear approximation; the chain rule; higher order derivatives.
Local extrema and the theorems of Fermat, Rolle and Lagrange: Local extrema; Fermat’s theorem – if is differentiable at a extremum, then vanishes there; algorithm for finding global extrema; Rolle’s theorem and Lagrange’s theorem; if and only if is constant; monotonicity and derivatives; the second derivative test; the distance from a point to a line.
Analyzing the behavior of a function: domain of definition; extrema; regions of increase and decrease; convexity and inflection points; sketching the graph of a function.
Approximation methods: Taylor’s remainder theorem; examples; L’H pital’s rule; orders of magnitude – polynomial, exponential, logarithmic; Newton’s method.
Indefinite integrals: Definition and immediate examples; integration by parts; substitution methods; rational functions; trigonometric functions.
Definite Integrals: Calculating area; definition of Riemann sums; the definition of integrability; theorems on integration – a monotone function is integrable, a continuous (or piecewise continuous) function is integrable; integrability and the value of the integral are not affected by changing the value of the function at a finite number of points; properties of the integral – linearity, additivity, comparison. upper and lower bounds in terms of the maximum and minimum of the function; the fundamental theorem of calculus; calculating the area between two curves; the definition of ; recovering the distance from the speed and the mass from the density; work; average value of a function on an interval; integration by parts and substitution for the definite integral; approximating definite integrals by Taylor’s theorem.
Improper Integrals: Types of improper integrals; examples; comparison; convergence and absolute convergence.
Infinite Series: Convergence; examples; linearity; necessary condition; series with positive terms; comparison to improper integral; absolute convergence; alternating series.
Power Series: Definition; radius of convergence; differentiation and integration; Taylor series; basic examples.
Homework – 10%
There will be between 10 and 12 homework sets assigned during the semester. They will be graded and returned to you. Your total grade on homework will be calculated by deleting the two lowest grades on homework sets, and then averaging the rest of the grades. The homework grade will constitute 10 percent of the final grade for the course. Each student must hand in his/her own homework.
Midterm – 30%
There will be one midterm. This midterm counts as 30 percent of the final grade.
Final Exam – 60%
The final exam will count as 60 percent of the final grade.
For specific details of fall 2012-13:
Contact Hours per Week
Lecture: 4 hours
Recitation: 2 hours