MTH U141 Outline
Goal: Develop graphical, numerical, and algorithmic understanding of the basic ideas of differential calculus and in what situations they might be applicable. Develop ability to solve simple maxima and minima problems. Develop an understanding of the concept of mathematical modeling, using differential equations. Introduce antiderivative and indefinite integral. Students are encouraged to use a graphing calculator or equivalent aid as a tool to understanding.
Functions
Concept of function. Linear functions. Slope and average rate of change.
The Derivative
The derivative as instantaneous rate of change, and as slope at a point.. Simple rules for taking derivatives of polynomials and powers. Newton, Leibniz, and delta notation.
Rules for Taking Derivatives
Product rule, quotient rule, chain rule, extended power rule.
Sines and Cosines
Radian measure of angle. Sine and cosine functions. Graphs. Physical interpretations. Derivatives of sine and cosine.
Exponentials and Logarithms
Exponential functions and logarithms, and their derivatives. Use in modeling population growth and radioactive decay.
Curve Sketching
First derivative. Second derivative and concavity. Asymptotes. Quadratics. Cubics. Quartics. Functions involving exponentials. sines and cosines.
Finding local maxima and minimal; finding maxima and minima over an interval.
Functions involving polynomials, rational functions, exponentials, sines and cosines.
Use of calculus to understand and solve word problems: maxima and minima involving fence, box, motion, absorption of medicine, etc.
Antidifferentiation
Antiderivative. Indefinite integral. Word problems including motion.
Differential Equations
Concept. Slope fields. Examples of Growth and decay. NewtonÕs law of cooling. Modeling population growth and simple physical, biological processes.
Area function, Areas and integrals as sums, Integral as accumulation,
Fundamental theorem of calculus. Applications.