Credit: (4-2)5 

Content: Limits and derivatives. The Mean Value Theorem. Definite and indefinite integral. The logarithmic, exponential, inverse trigonometric and hyperbolic functions. L'Hospital rule. Techniques of integration. Numerical methods of integration. Applications to geometry and physics. Area in polar coordinates. Improper integrals. Sequences. Infinite series, power series and Taylor's series. 

Goals: Calculus was first discovered to meet the needs of the scientists of the sixteenth and seventeenth centuries. Diferential calculus deals with the
   problem of calculating rates of change. It enables us to define slope of curves, to calculate velocities, accelerations of moving bodies and to predict the  times when planets would be closest together or farthest apart. Integral calculus deals with the problem of determining a function from information about its  rate of change. It enables us to calculate the future location of a body from its present position, to find the areas of irregular regions in the plane, to measure the lengths of curves, and to find the volumes and masses of arbitrary solids. The goal of this course is to present a modern view of calculus enhanced by the use of technology. 

Course Outline: 
 

• (1 Week) Examples of Velocity, Growth Rate, and Area 
•    Limits of Functions 
•    The Formal Definition of Limit 
•    Limit at Infinity and Infinite Limits 
•    The Formal Definition of Limit 
•    (1 Week) Continuity 
•    Tangent Lines and Their Slopes 
•    The Derivative (up to Differentials) 
•    Using Derivatives  (only Average and Instantaneous Rates of Change) 
•    Differentiation Rules 
•    (1 Week) Derivatives of Trigonometric Functions 
•    The Chain Rule 
•    Higher-Order Derivatives 
•    Implicitly Differentiation 
•    Velocity and Acceleration (up to Falling Under Gravity) 
•    Related Rates 
•    Extreme Values (up to Finding Extreme Values) 
•    (1 Week) Extreme Values 
•    Extreme Value Problems 
•    Review 
•    (1 Week) The Mean Value Theorem 
•    Antiderivatives and Initial Value Problems 
•    (omit Differential Equations and Initial Value Problems) 
•    Velocity and Acceleration  (only Falling Under Gravity) 
•    The Derivative  (only Derivatives Have the Intermediate Value Property) 
•    Increasing and Decreasing Functions  (up to the First Derivative Test) 
•    (1 Week) Linear Approximations 
•    Using Derivatives (only Approximating Small Changes) 
•    The Derivative (only Differentials) 
•    Higher Order Approximations 
•    Finding Roots of Equations (omit Fixed Point Iteration) 
•    Inverse Functions 
•    The Natural Logarithm and Exponential 
•    Exponential and Logarithmic Functions 
•    (1 Week) Growth and Decay (omit Logistic Growth) 
•    The Inverse Trigonometric Functions 
•    Hyperbolic Functions 
•    Indeterminate Forms and l’Hopital’s Rule 
•    Increasing and Decreasing Functions 
•    Concavity and Inflections 
•    Sketching the Graph of a Function 
•    (1 Week) Sums and Sigma Notation 
•    Areas as Limits of Sums 
•    The Definite Integral 
•    Properties of the Definite Integral 
•    Review 
•    (1 Week) The Fundamental Theorem of Calculus 
•    The Method of Substitution 
•    Areas of Plane Regions 
•    Integration by Parts 
•    (1 Week) Inverse Substitions 
•    Integrals of Rational Functions 
•    Improper Integrals (include Limit Comparison Test and 
•    Absolute and conditional Convergence) 
•    (1 Week) The Trapezoid and Midpoint Rules 
•    Simpson’s Rule 
•    Volumes of Solids of Revolution 
•    Other Volumes by Slicing 
•    Arc Length and Surface Area 
•    (1 Week) Polar Coordinates and Polar Curves (omit Polar Conics) 
•    Slopes, Areas, and Arc Lengths for Polar Curves    (omit Tangent direction for a Polar Curve) 
•    Review 
•    (1 Week) Sequences and Convergence 
•    Infinite Series 
•    Convergence Tests for Positive Series 
•    (1 Week) Absolute and Conditional Convergence 
•    Estimating the Sum of a Series (omit Geometric Bounds) 
•    Power Series 
•    Taylor and Maclaurin Series  (up to Maclaurin Series for Some Elementary Functions) 
•    (1 Week) Taylor Polynomials and Taylor’s Formula 
•    Taylor and Maclaurin Series (only Other Maclaurin and Taylor Series) 
•    Applications of Taylor and Maclaurin Series 
•    The Binomial Theorem and Binomial Series 
•    Review 

Reference Books: 

• G.B. Thomas and R.Finney; Calculus and Analytic Geometry. Addison-Wesley Publishing Com. 1996.  
• R.Ellis and D. Gulick; Calculus with Analytic Geometry, 5 th Edition, SCP, 1994  
• M.Spivak; Calculus 3 rd Edition, Publish of Perish, Inc.