July
2005
MATH 2200
Syllabus and Detailed Syllabus
Modeled on MWF schedule
I.
Prelude to
calculus (2 weeks)
2.1 Tangent Lines and Slope Predictors
2.2 The Limit Concept Day 1.
2.2 The Limit Concept Day 2.
2.3 More About Limits Day 1.
2.3 More About Limits Day 2.
2.4 Concept of Continuity Day 1.
2.4 Concept of Continuity Day 2.
3.1 The Derivative and Rates of Change Day 1.
3.1 The Derivative and Rates of Change Day 2.
3.1 The Derivative and Rates of Change Day 3.
3.2 Basic Differentiation Rules Day 1.
3.2 Basic Differentiation Rules Day 2.
3.3 The Chain Rule Day 1.
3.3 The Chain Rule Day 2.
3.4 Derivatives of Algebraic Functions
III.
Applications
of the derivative; derivatives of transcendental functions.
(3 weeks)
3.5 Maxima and Minima of Functions on Closed Intervals.
3.6 Applied Optimization Problems Day 1.
3.6 Applied Optimization Problems Day 2.
3.6 Applied
Optimization Problems Day 3.
3.7 Derivatives of Trigonometric Functions Day 1.
3.7 Derivatives of Trigonometric Functions Day 2.
3.8 Exponential and Logarithmic Functions.
3.9 Implicit Differentiation and Related Rates Day 1.
3.9 Implicit
Differentiation and Related Rates Day 2.
3.9 Implicit Differentiation and Related Rates Day 3.
IV. Mean Value Theorem and
applications (2 weeks)
4.2 Increments, Differentials, and Linear Approximation
4.3 Increasing and Decreasing Functions and the Mean value Theorem Day 1.
4.3 Increasing and Decreasing Functions and the Mean value Theorem Day 2.
4.4 The First Derivative Test and Applications Day 1.
4.4 The First Derivative Test and Applications Day 2.
4.4 The First Derivative Test and Applications Day 3.
IV.
Curve
sketching (1 ½ weeks)
4.5 Simple Curve Sketching
4.6 Higher Derivatives and Concavity Day 1.
4.6 Higher Derivatives and Concavity Day 2.
VI. Anti-derivatives (1 ½
weeks)
5.2
Antiderivatives and Initial Value Problems Day 1.
5.2 Antiderivatives and Initial Value Problems Day 2.
5.2 Antiderivatives and Initial Value Problems Day 3.
8.3 Separable
Equations and Applications
2.1
Tangent Lines and Slope Predictors
Purpose: Develop the notion of tangent line.
Outline for Lecture:
Review point-slope formula, and slopes of parallel and perpendicular lines. Introduce slope of tangent line as limit of slopes of secant lines. Develop formula for the slope of the secant of a general quadratic.
Skip the animal pen problem and numerical investigation of slopes.
Examples: 1, 2, 3.
Core Problems: p. 62: 7, 8, 9, 11, 5, 1, 19, 21, 33, 34, 28.
2.2 The
Limit Concept Day 1.
Purpose: Understand the definition of limit and those applications of limit laws which dont require manipulating the function.
Outline for Lecture:
1. Introduce the idea of the limit on p. 64 as the definition of limit.
2. Observe that limits dont always exist.
Example 4.
3. Introduce limit laws (not including substitution) using example 6 and problem 4.
Examples: 1, 4, 6, 7.
Core Problems: p. 73: 1, 2, 3, 5, 6. p. 99, 5, 6.
2.2 The
Limit Concept Day 2.
Purpose: Applying the limit laws to functions which require some algebraic manipulation.
Outline for Lecture:
1. Introduce substitution law.
2. Introduce factoring and multiplying numerator and denominator by conjugate tricks.
3. Define Slope Predictor Function and apply these tricks to compute slope predictor limits for polynomial functions, rational functions, and functions involving square root.
Examples: 10, 12, 13. (The authors choose a poor way to do the algebra in Ex. 12; warn them and do it more economically yourself.)
Core Problems: p. 73: 9, 13, 17, 19, 25, 29, 31, 33, 37, 41, 43, 45.
2.3 More About Limits Day 1.
Purpose: Learning advanced limits, including trigonometric and one-sided limits.
Outline for Lecture:
1. Brief Trig Review
2. State the basic trigonometric limit in Theorem 1.
Examples 1,2.
3. Define one-sided limits and observe that the limit laws of 2.2 apply to them.
Examples 6,8.
4. State Theorem 2, giving the relationship between one and two-sided limits.
Example 4 of Section 2.2.
Core Problems: p. 85: 1, 3, 5, 7, 10, 15, 17, 19, p. 86: 59, 60.
2.3 More About Limits Day 2.
Purpose: Learning advanced limits, including infinite limits, and formal definition of tangent lines.
Outline for Lecture:
1. Introduce criteria for existence of tangent lines.
Example 9.
2. Introduce infinite limits, including Equation 14.
Examples 10,12.
3. Give the proof of Theorem 1 (optional).
Core Problems: p. 86: 29, 37, 39, 41, 45, p. 86: 49, 51, 56.
2.4
Concept of Continuity Day 1.
Purpose: Understand the definition of continuity and the continuity of combinations of elementary functions.
Outline for Lecture:
1. Introduce the definition of continuity at a point.
Examples 1,2,3.
2. Introduce the definition of continuity for a function.
(Skip removable discontinuities.)
3. State continuity of sums, products, and quotients of continuous functions,
including polynomials and rational functions.
Examples 5,6.
4. State Theorem 1 (continuity of trig functions).
5. State Theorem 2 (continuity of compositions of continuous functions).
Example 7.
Core Problems: p. 97: 1, 3, 5, 7, 9, 15, 19, 25, 33, 35.
2.4
Concept of Continuity Day 2.
Purpose: To be able to check whether a function is continuous on a closed interval, and to know the Intermediate Value Theorem.
Outline for Lecture:
1. Define continuity from the left and right.
2. Define continuity on a closed interval (this will be important for max-min problems).
3. State Intermediate Value Theorem.
Example 9.
3. Show how to use this to show that solutions of f(x) = 0 exist, always checking the
hypothesis of the IVT.
Examples 10,11.
Note that by IVT, a function can change sign only at discontinuities or zeros of f.
Core Problems: p. 99: 21, 27, 17, 29, 11. p. 97: 53, 55, 58, 59 (be sure to define a function f(x), and verify that the IVT applies to this function). p. 100: 61, 63.
3.1 The
Derivative and Rates of Change Day 1.
Purpose: To define the derivative with geometric and rate interpretations.
Outline for Lecture:
State the definition of derivative in the usual h form.
Use the definition to compute the derivative of polynomial functions, rational functions, and functions involving square roots.
Introduce geometric interpretation of derivative as the slope of the tangent line.
Example 1, Rule 6, Example 2 and 2 (continued).
Introduce differential notation (briefly).
Core Problems: p. 112: 5, 3, 9, 11, 15, 17, 19, Find equation of tangent line at (1, f(1)) for 11, 15, 17, 19. p. 112: 30-35, Example 8
3.1 The
Derivative and Rates of Change Day 2.
Purpose: To define the instantaneous rate of change of a function as the limit of its average rates of change, and to observe that this is the derivative with respect to time.
Outline for Lecture:
Define average rate of change from t to t + Dt, and instantaneous rate of change at time t.
Identify instantaneous rate of change with the derivative at time t.
Point out that a function is increasing when f(t) > 0 and decreasing when f(t) < 0.
Example 3.
Definition of position function x = x(t).
Definition of average velocity from t to t + Dt and velocity at time t.
Observe that the velocity is the derivative of position.
Note that a particle is stopped when velocity is zero.
Example 5, Problem 26 on pg. 112.
Core Problems: p. 112: (note the hint before problem 44), 41, 42, 52, 53. p. 112: 21, 23, 25, 27, 29, 39.
3.1 The
Derivative and Rates of Change Day 3.
Purpose: To discuss previous homework problems and introduce general rates of change.
Outline for Lecture:
Discuss homework problems.
Definition of average rate of change from x to x + Dx, and instantaneous rate of change at time t.
Examples 7.
Core problems: p.112: 37, 44, 45 (note the hint before problem 44), 47, 50, 51.
3.2 Basic
Differentiation Rules Day 1.
Purpose: State all of the rules without proof, and do some examples.
Outline for Lecture:
State constant, power, linear combination, polynomial, product, and quotient rules.
Examples 2,3,4,6,8,9.
Remember to simplify the function being differentiated, when possible.
Core problems: p. 123: 1,3,13,19,35,27,31,15,(43-47 requires Example 4).
3.2 Basic
Differentiation Rules Day 2.
Purpose: Provide proofs and do more examples.
Outline for Lecture:
Prove all laws except linear combination (which is a homework exercise).
Examples 4,5.
Be sure to include the product rule and Leibniz rule (Equation 16) this requires differentiability => continuity, which will be covered in Section 3.4.
Core problems: p. 123: 55, 56, 57, 59, 51, 53, 62, 63, 65, 5, 17 and prove the linear combination law for derivatives.
3.3 The
Chain Rule Day 1.
Purpose: State and motivate chain rule, with examples.
Outline for Lecture:
Motivate chain rule by Leibniz rule in dy/dx notation, following book.
State chain rule in functional notation (without proof).
Introduce generalized power rule.
Examples 1,3,4,5.
Core problems: p. 132: 1, 3, 7, 9, 11, 13, 17, 21, 25, 29.
3.3 The
Chain Rule Day 2.
Purpose: Introduce rate-of-change applications with examples.
Outline for Lecture:
Discuss homework.
Introduce rate-of-change applications.
Examples 6,7.
Core problems: p. 132: 49, 51 ,53, 55, 57, 59, 61.
3.4
Derivatives of Algebraic Functions
Purpose: Learn to differentiate functions with rational exponents.
Outline for Lecture:
Derive Equation 4 using the chain rule.
State generalized power rule.
Examples 1,2,3.
Discuss the proof that differentiability implies continuity.
(Vertical tangents will be covered in curve sketching.)
Core problems: p. 138: 1, 9, 19, 33, 39, 63, 65.
3.5
Maxima and Minima of Functions on Closed Intervals.
Purpose: Learn the theory of finding maxes and mins on a closed interval.
Outline for Lecture:
Define absolute maxima and minima only.
(Local maxima and minima will be covered in curve sketching.)
State Theorem 1. (existence of a max and min of a continuous fn. on closed interval)
Recall that we learned to recognize continuity on a closed interval in Section 2.3.
State and prove Theorem 2 for absolute maxima and minima.
Be sure to include the Beware statement on page 144.
Introduce the notion of critical point.
State and prove Theorem 3.
Example 6 (stress factoring out the smallest exponent).
Core problems: p. 148: 13, 17, 11, 19, 33, 35, 37, 39, 47, 48, 49, 50, 51, 52.
3.6
Applied Optimization Problems Day 1.
Purpose: Learn the 5-step plan of attack for max-min problems with examples involving volume or area constrained by boundary (polynomials and rational functions).
Outline for Lecture:
State the 5-step plan of attack for max-min problems on page 151.
Emphasize the end of step 2- closing the domain if you have to.
Examples 1,4.
Core Problems: p. 159: 11, 5, 20, 9, 25.
3.6
Applied Optimization Problems Day 2.
Purpose: Understand harder max-min problems involving algebraic functions and more difficult rational functions which are harder to set up.
Outline for Lecture:
Review homework problems.
Examples 5,3.
Core Problems: p. 159: 1 or 3, 13, 21, (2 out of 3 of 27, 31, 29)
3.6
Applied Optimization Problems Day 3.
Purpose: Understand even harder max-min problems where the setup is a substantial portion of the difficulty of the problem.
Outline for Lecture:
Review homework problems.
Example 6.
Core Problems: p. 159: 45, 47, 33, 48.
3.7
Derivatives of Trigonometric Functions Day 1.
Purpose: Learn the differentiation formula for cosine and sine, with proof.
Outline for Lecture:
Recall the fundamental trig limit from 2.3 (and the limit of 1-cos(x)/x), without proof.
State and prove Theorem 1.
Examples 2,4.
Core Problems: p. 171: 1, 3, 5, 9, 11, 13, 67, 72.
3.7
Derivatives of Trigonometric Functions Day 2.
Purpose: Learn to differentiate the other trigonometric functions.
Outline for Lecture:
State Theorem 2, with proof of tangent and secant formulae.
State the chain rule versions of these (Formula 11, Equations 12-17).
Examples 9, 10, 12.
Core Problems: p. 171: 15, 41, 43, 51, 59, 73, 75, 77.
3.8
Exponential and Logarithmic Functions.
Purpose: Learn to differentiate exp(x) and log(x).
Outline for Lecture:
Recall definition of exponents and laws of exponents.
Define e as the limit on page 179, under formula 8.
Recall the sketch of the graph of f(x) = ex.
State Equations 8 and 9 (dont follow books exposition).
Example 2.
Define log as inverse of e, cover Equation 10, and the laws
of logs.
Explain Theorem 1 geometrically.
State Equations 18 and 20.
Core Problems: p. 187: 1,3,5, 9, 17, 19, 23, 33, 37, 59.
3.9
Implicit Differentiation and Related Rates Day 1.
Purpose: Learn basic implicit differentiation skills.
Outline for Lecture:
Theory of implicit differentiation.
Examples 2, 3.
Core Problems: p. 195: 3, 7, 11, 13, 19, 23, 25, 31.
3.9
Implicit Differentiation and Related Rates Day 2.
Purpose: Learn to do simpler related rates problems.
Outline for Lecture:
Give 5-step plan of attack for related rates problems on p. 193.
Example 6, 7.
Core Problems: p. 195: 45, 55, 39, 43, 51.
3.9
Implicit Differentiation and Related Rates Day 3.
Purpose: Get more practice with related-rates problems.
Outline for Lecture:
Review homework.
Example 8 (optional).
Core Problems: p. 195: 53, 38, 57, 61, 56, 68, 47.
4.2
Increments, Differentials, and Linear Approximation
Purpose: Learn to use the tangent line as an approximation to the function.
Outline for Lecture:
Introduce linear approximation of Equation 6, 7. (Do not use delta notation).
Example 1, 2.
Core Problems: p. 225: 17, 21, 23, 25, 29, 33.
4.3
Increasing and Decreasing Functions and the Mean Value Theorem Day 1.
(Please see the attached, Supplement for Instructors of MATH 2200)
Purpose: The Mean Value Theorem with application to antiderivatives.
Outline of Lecture:
Statement and geometric interpretation of the Mean Value Theorem (without proof).
Example 4.
State and prove Corollary 1 and 2.
Example 5.
Core Problems: p. 235: 7, 8, 9, 10, 45, 46, 47.
4.3 Increasing and Decreasing Functions and the
Mean Value Theorem Day 2.
Purpose: Understand increasing and decreasing functions and sign analysis.
Outline of Lecture:
State and prove Corollary 3.
Examples 6, 7, 8 (stressing algebra and sign analysis).
Core Problems: p. 235: 1, 3, 5, 13, 19, 21, 41, 43.
4.4 The
First Derivative Test and Applications Day 1.
Purpose: Learn to classify critical points on an interval.
Outline of Lecture:
Define global and local extrema.
State and prove Theorem 1.
Example 1, 2
Be sure to draw figures on Page 240, giving examples of various types of critical points.
Core Problems: p. 245: 1, 3, 5, 11, 15, 19, 21, 23.
4.4 The
First Derivative Test and Applications Day 2.
Purpose: Learn max-min theory on an open interval.
Outline of Lecture:
Introduce open interval max-min theory.
Examples 3, 4 be sure to cover Example 4.
Core Problems: p. 245: 27, 33, 35, 29, 43.
4.4 The
First Derivative Test and Applications Day 3.
Purpose: Understand what to do when functions have a single critical point.
Outline of Lecture:
Review homework problems.
Example 5.
State and prove Theorem 2.
Core Problems: p. 245: 45, 55, 40.
p. 249: 83.
4.5
Simple Curve Sketching
Purpose: Learn to graph polynomials with first derivative information only.
Outline of Lecture:
Describe behavior of polynomials at infinity.
Introduce method for finding out where functions are increasing and decreasing by sign analysis of first derivative.
Recall definition of critical points.
Examples 2, 3.
Make sure to stress the algebra of factoring out the smallest power of x.
Core Problems: p. 255: 3, 4, 7, 11, 19, 23, 43, 45, 51, 53.
4.6
Higher Derivatives and Concavity Day 1
Purpose: Learn to graph functions with concavity information.
Outline of Lecture:
Define higher derivatives.
Example 1, and sin(x).
Discuss meaning of sign of second derivative.
Define concavity.
State (but do not prove) Theorems 2 and 3.
(Notice that we are skipping the second derivative test.)
Core Problems: p. 268: 1, 9, 15, 23, 29, 35, 77-82.
p. 271: 75
4.6
Higher Derivatives and Concavity Day 2.
Purpose: Sketch the graph of algebraic functions.
Outline of Lecture:
Cover material on vertical tangent lines from p. 136.
Example 8 on p. 137.
Do Example 7 in detail.
Core Problems: p. 268: 65, 69, 75 without using the second derivative test to classify the critical points. In addition, sketch a graph of the function in problem 35.
4.7 Curve
Sketching and Asymptotes
Purpose: Learn to graph rational functions.
Outline of Lecture:
Introduce and define vertical asymptotes.
Example 1.
Define limits at infinity.
Introduce horizontal asymptotes.
Example 2.
Do Example 8 in detail, introducing the strategy for curve sketching on page 276 along the way.
(Notice that we are skipping slant asymptotes.)
Core Problems: p. 281: 1, 3, 21, 25, 35, 41, 49.
5.2
Antiderivatives and Initial Value Problems Day 1.
Purpose: Introduce the idea of integrals and do simple integrals.
Outline of Lecture:
Define antiderivatives.
Example 1.
State Theorems 1 and 2.
Prove Equations 15 and 16.
Example 3.
Prove Equation 17.
Example 6.
Core Problems: p. 314: 1, 5, 9, 13, 17, 23, 27, 19, 21, 33.
5.2
Antiderivatives and Initial Value Problems Day 2.
Purpose: To complete simple initial value problems.
Outline of Lecture:
Cover Equations 19 and 20.
Example 7.
Define initial value problems (using the statement in Equation 22).
Example 8.
Introduce position, velocity, and acceleration functions.
Example 9.
Core Problems: p. 314: 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55.
5.2 Antiderivatives
and Initial Value Problems Day 3.
Purpose: Analyze motion with constant acceleration.
Outline of Lecture:
Introduce theory for constant acceleration problems.
Example 10.
Cover vertical motion.
Example 11.
Core Problems: p. 314: 57, 61, 65, 69, 71, 75, 77, 78, 79.
8.3 Separable Equations and Applications
Purpose: Learn to solve separable differential equations.
Outline of Lecture:
Introduce separable equations (be sure to cover Equations 1-5).
Example 1
Cover
Example 4.
Solve general linear differential equation (including Equation 13).
Example 5.
Core Problems: p. 576: 3, 7, 11, 15, 17, 31, 32, 41.