MATH102: CALCULUS I

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Course Prerequisite(s)

About Course

Involves a study of limits, continuity, derivatives and integrals; computations of derivatives—sum, product, and quotient formulas, chain rule, implicit differentiation, applications of derivatives to optimization problems and related rate problems; mean-value theorem; definite integrals and fundamental theorem of calculus; application of definite integrals to computations of areas (length, surface) and volumes.

What Will You Learn?

  • 1. Calculate limits, derivatives, and indefinite integrals of various algebraic and trigonometric functions of a single
  • variable
  • 2. Apply the definition of continuity to pure and applied mathematics problems
  • 3. Utilize the definition of the derivative to differentiate various algebraic and trigonometric functions of a single
  • variable
  • 4. Use the properties of limits and the derivative to analyze graphs of various functions of a single variable
  • including transcendental functions
  • 5. Employ the principles of the differential calculus to solve optimization problems, related rates exercises, and
  • other applications
  • 6. Calculate the area of regions in the plane with elementary Riemann sums
  • 7. Utilize the Fundamental Theorem of Calculus and the techniques of integration, including u-substitution, to
  • calculate the area of regions in the plane and the volume and surface area of solids of revolution

Course Content

Equations and inequalities
Solving linear and quadratic equations, linear inequalities. Division of polynomials, synthetic division. Roots of a polynomial, rational roots; Viete Relations. Descartes rule of signs. Solutions of equations with absolute value sign. Solution of linear and non-linear inequalities with absolute value sign.

Functions and graphs
Domain and range of a function. Examples: polynomial, rational, piecewise defined functions, absolute value functions, and evaluation of such functions. Operations with functions: sum, product, quotient and composition. Graphs of functions: linear, quadratic, piecewise defined functions.

Lines and systems of equations
Equation of a straight line, slope and intercept of a line, parallel and perpendicular lines. Systems of linear equations, solution of system of linear equations. Nonlinear systems: at least one quadratic equation.

Limits and continuity
Functions, limit of a function. Graphical approach. Properties of limits. Theorems of limits. Limits of polynomials, rational and transcendental functions. Limits at infinity, infinite limits, one-sided limits. Continuity.

Derivatives
Definition, techniques of differentiation. Derivatives of polynomials and rational, exponential, logarithmic and trigonometric functions. The chain rule. Implicit differentiation. Rates of change in natural and social sciences. Related rates. Linear approximations and differentials. Higher derivatives, Leibnitz's theorem

Applications of derivatives
Increasing and decreasing functions. Relative extrema and optimization. First derivative test for relative extrema. Convexity and point of inflection. The second derivative test for extrema. Curve sketching. Mean value theorems. Indeterminate forms and L'Hopitals rule. Inverse functions and their derivatives.

Integration
Anti derivatives and integrals. Riemann sums and the definite integral. Properties of Integral. The fundamental theorem of calculus. The substitution rule.

APPLICATIONS OF INTEGRATION
Area B. The Mean Value Theorem for Integrals* C. Volume 1. Disk and washer method 2. Shell method 3. Slice method* D. Arc length* E. Work* F. Liquid pressure and force* G. Center of mass* H. Centroid*