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Math 150

Calculus I (Differentiation)

Text Book: Calculus, early transcendental functions (A Custom publication to KSU, Third Edition).

Authors:  Robert T. Smith, and Roland R. Minton.

الخطة الدراسية لمقرر 150 ريض

 جامعة الملك سعود السنة التحضيرية قسم مهارات الرياضيات مقرر ريض (150) Syllabus and Contents of Math 150 (3+1)

The Contents:

Chapter 1: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6

Chapter 2: 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9

Chapter 3: 3.2, 3.3, 3.4, 3.5, 3.6, 3.7.

The Students Should Study the Following Topics:

Chapter 1: Limits and Continuity

The slope, The tangent line, The Concept of  Limit, Computation of  Limits, Continuity and its Consequences, Limits Involving Infinity, Formal  Definition of the Limit.

Chapter 2: Differentiation

The Concept of Derivative, Computation of Derivatives (The Power Rule, Higher Order Derivatives, and Acceleration), the Product and Quotient Rules, The Chain Rule, Derivatives of Exponential and Logarithmic Functions, Implicit Differentiation and Inverse Trigonometric Functions, the Mean Value Theorem.

Chapter 3: Applications of Differentiation

Indeterminate Forms and L’Hopital’s rule, Maximum and Minimum Values,  Increasing and Decreasing Functions, Concavity and the Second Derivative  Test, Curve Sketching, Optimization.

Evaluation:

The Evaluation of the Students will be Continuous during the Course and depends on the following:

 First Mid Term exam 15 Second Mid Term exam 15 Quizzes & Activities 10 Self-learning 10 Final Exam 50 The Total Degrees 100

 Objectives

This course is intended for students who have a thorough knowledge of analytic geometry and elementary functions in addition to college preparatory algebra, geometry, and trigonometry. The purpose of the course is to prepare the student for advanced placement in college calculus. In this course:

·         The student will define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses. Properties of functions will include domains, ranges, combinations, odd, even, periodicity, symmetry, asymptotes, zeros, upper and lower bounds, and intervals where the function is increasing or decreasing.

·         The student will define and apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at

infinity, infinite limits, and nonexistent limits.

·         A Calculus will include the rigorous definitions of a limit.

·         The student will state the definition of continuity and determine where a function is continuous or discontinuous. This will include continuity at a point; continuity over a closed interval; application of the Intermediate Value Theorem; and graphical interpretation of continuity and discontinuity.

·         The student will find the derivative of an algebraic function by using the definition of a derivative. This will include investigating and describing the relationship between differentiability and continuity.

·         The student will apply formulas to find the derivative of algebraic, trigonometric, exponential, and logarithmic functions and their inverses.

·         The student will apply formulas to find the derivative of the sum, product, quotient, inverse, and composite (chain rule) of elementary functions.

·         The student will find the derivative of an implicitly defined function.

·         The student will find the higher order derivatives of algebraic, trigonometric, exponential, and logarithmic functions.

·         The student will use logarithmic differentiation as a technique to differentiate nonlogarithmic functions.

·         The student will state (without proof) the Mean Value Theorem for derivatives and apply it both algebraically and graphically.

·         The student will use L'Hopital's rule to find the limit of functions whose limits yield the indeterminate forms: 0/0 and infinity/infinity

·         A Calculus, these functions will also include functions whose limits yield the indeterminate forms:  0 to the 0th power, 1 to the infinity power, infinity to the infinity power, infinity minus infinity.

·         The student will apply the derivative to solve problems, including tangent and normal lines to a curve, curve sketching, velocity, acceleration, related rates of change, and optimization problems.

 Chapter Section Examples Exercises for Teacher Exercises for student. 1 Limits and Continuity 1.2 The Concept of limit 1,2,3,4,5 1,3, 9,11,15,19,21,25,29 2,6,10,12,16,20,22,31 1.3 Computation of Limits 1,2,3,4,5,6,7,8,9,10 1,3,5,9,13,17,21,23,33,39,63, 65 2,4,6,10,14,18,22,25,34,40,64,66 1.4 Continuity and Consequences 1,2,3,4,5,6,7,9 1,5,7,11,13,17,23,25,27,31,33,41,47 3,6,8,12,14,18,24,26,28,32,34,43,49 1.5 Limits Involving infinity and asymptotes 1, 2, 3, 5, 6, 7 1,5,8,25,27,29,49 3,7,10,26,28,30,50 1.6 Formal definition of the limit 1, 2, 4 1,5,7, 11,13 2,6,8,12,14 2 Differentiation 2.2 The Derivative 1, 2,3,4,5,7 1,7,9,13,14,23,40,57,59,61 2,3,8,15,17,25,39,58,60,62. 2.3 The power rule 1, 2, 3, 4, 6, 7 1,5,7,11,13,15,18,27,29,33,37 2,6,8,12,14,16,19,28,30,34,38 2.4 Product Rule and Quotient Rule 1, 2, 3, 4 1,5,13,19,21,23,49,51 2,5,14,20,22,24,50,52 2.5 Chain Rule 1,2,3 7,11,13,17,23,25,29,33,46,48 8,12,14,18,24,26,31,34,47,49 2.6 Derivatives of Trigonometric Functions 1,2,3,7 3,7,11,17,25,29,37,43,45 4,8,12,18,26,30,38,44,46 2.7 Derivatives of Exponential and logarithmic Functions ,3,4,6 * 1,5,7,11,13,17,21,25,29,31 2,6,8,12,14,18,22,26,30,34 2.8 Implicit Differentiation 1, 2 1,5,9,15,19,23 2,6,13,16,20,24 2.9 Mean Value Theorem 1, 3 1,5,9,35,37 2,6,10,36,38 3 Applications of Differentiation 3.2  Indeterminate Forms (0/0 , ∞/∞,0×∞) 1,2,4,8 1,3,5,9,11,39,41,51 2,4,6,10,12,40,42,52 3.3 Maximum and Minimum values 1, 4, 5, 6, 10 1, 5,31 2, 6,32 3.4 Increasing and Decreasing Functions 1, 2, 3, 4 1,5,11,35,37 2, 6, 12, 36,38 3.5 Concavity and the second derivative test. 1, 2, 3, 4, 7 1,3,13,37,41,47 2, 5, 7, 14, 38, 43,48 3.6 Curve Sketching 1, 2, 6 1,5, 16, 21 2, 6, 11, 15, 22 3.7 Optimization 1, 2 ,6 3,7,15,20 4,8,16,21

Weekly Plan

 Week Section(s) to be Covered 1 ·         Review Precalculus 2 ·         The slope, Tangent line ·         The Concept of Limit 3 ·         Computation of Limits 4 ·         Continuity and its Consequences 5 ·         Limits Involving Infinity. ·         Formal Definition of the Limit 6 ·         The Concept of Derivative . 7 ·         Computation of Derivatives. ·         The product and quotient rules. 8 ·         The chain rule. ·         Derivatives of trigonometric functions. 9 ·         Derivatives of exponential and logarithmic functions. 10 ·         Implicit differentiation. 11 ·         Mean value theorem. 12 ·         Indeterminate Forms and L’Hopital’s rule. ·         Maximum and Minimum Value. 13 ·         Increasing and Decreasing Functions ·         Concavity and the Second Derivative. 14 ·         Curve sketching. 15 ·         Optimization. 16 Final Exam