MSCI 300 SPRING 2016 Calculus 1 Charles Rubenstein

MSCI 300 – SPRING 2016 Calculus 1 Charles Rubenstein, Ph. D. Professor of Engineering and Information Science Week 6: Session 5: Monday 02/22/16 Mondays 6: 30 pm-8: 50 pm PMC 705 A

Instructor Contact Information Dr. Charles Rubenstein <crubenst@pratt. edu> Professor of Engineering & Information Science Office hours (by appointment *) • Mondays: 5: 00 pm-6: 00 pm Pratt Manhattan Campus Office: PMC 604 -C • Tuesdays: 12: 00 pm - 2: 00 pm Pratt Brooklyn Campus Office: ARC G-45 (or E-08 Lab) (*Please email me at least a day in advance if you plan on coming to office hours…) Send me an email … crubenst@pratt. edu Subject line: 300 Calc Copyright © 2016 C. P. Rubenstein 2

MSCI 300 – Spring 2016 - Class Schedule & Due Dates Monday (Week) NOTES 18 January (1) NO CLASS – Martin Luther King Day 25 January (2) Introduction; Review of Syllabus, Algebra & Trigonometry 1 February (3) Functions and Slopes (Quiz 1) 8 February (4) Approximating Slopes, Limits, The Derivative (Quiz 2) 15 February (5) Rules of Differentiation (Q 3) 22 February (6) Max/Min Prob, 2 nd + Derivatives (Q 4) 29 February (7) Trig, Trig Derivatives, Limit of sin(x)/x (Q 5) 7 March (8) Derivatives of Exponentials, Constant "e"; Take Home Exam (Q 6) 14 March (9) NO CLASSES – Pratt Spring Break – 14 -20 March 2016 21 March (10) Linear Approximation, Newton's Method; Midterm Due (Q 7) 28 March (11) l'Hôpital's Rule, Kinematics: Position, Velocity, Acceleration ; Midterm Review (Q 8) 4 April (12) CMFM Seminar Review Logs (Q 9) 11 April (13) NO CLASS – Instructor out of town 18 April (14) Inverse Function Derivatives, Implicit Differentiation (Q 10) 25 April (15) Areas, Intro to Integrals, Fundamental Theorem of Calculus 2 May (16) Using Integrals to find Volumes and Lengths, Review 9 May (17) In-class Final Examination (* Quizzes on Homework due; Reviewed in same session) Copyright © 2016 C. P. Rubenstein 3

Spring 2016 Math/Science TUTORING WHO: Professor Joe Guadagni BY APPOINTMENT ONLY: WHEN and WHERE: MONDAY 4: 30 -6: 00 pm PMC 403 Email: jguadagni@pratt. edu FRIDAY 3: 00 -5: 00 pm WTC North Hall NOTE: REVISED Friday Hours Call: 718 -636 -3459 Copyright © 2016 C. P. Rubenstein 4

* Class Session Archives * http: //www. Charles. Rubenstein. com/300/ 16 sp 05. pdf (Class Power. Point slides) * 16 sp 05 h. pdf (slides in handout format) * *Archive materials normally online by Thursday evenings Copyright © 2016 C. P. Rubenstein 5

In Class #05 • DUE: Homework Set #04 • Reading: Strang - Chapter 3: Applications of the Derivative • Lecture and Problem Review: Max/Min Problems, Second and Higher Derivatives For class Session #06: • • • Homework Set #05 Reading: Strang - Chapter 3: Applications of the Derivative In Class Quiz and Review Homework Sets #03 and #04 Lecture and Problem Review: Trigonometry Review Remaining Class Note Sets all online Copyright © 2016 C. P. Rubenstein 6

Questions? Copyright © 2016 C. P. Rubenstein 7

Class #05: Rules for Differentiation (continued) Rules (and Proofs) for Differentiation 1. Derivative of a Constant (and Proof – last week) 2. Derivative of a constant times a function (and Proof – last week) 3. The Sum Rule: Proof and Application Graphing a Function and its Derivative First Application of Differential Calculus: Max/Min problems 4. The Product Rule: Proof and Application 5. The Reciprocal Rule: Proof and Application 6. The Quotient Rule: Proof The Chain Rule Slopes to Derivatives to rules The Derivative of x Copyright © 2016 C. P. Rubenstein 8

REVIEW: Rules for Differentiation Rule 1. The Derivative of a Constant is zero d/dx [C] = 0 (The graph of a constant is a horizontal line whose slope = 0) Rule 2. Constants can be removed from equation d/dx [C f(x) ] = C df/dx Where C is any constant Rule 3. The “Sum Rule” d/dx [ f(x) + g(x) ] = df/dx + dg/dx Rule 3 a. Combining Rules 2 and 3 above we have d/dx [C 1 f(x) + C 2 g(x)] = C 1 df/dx + C 2 dg/dx Copyright © 2016 C. P. Rubenstein 9

More Rules for Differentiation Rule 4. The “Product Rule” d/dx [f(x) g(x)] = g(x)df/dx + f(x)dg/dx Today, we will prove: Rule 5. The “Reciprocal Rule” d/dx [1/g(x)] = -dg/dx / [g(x)]2 Rule 6. The “Quotient Rule” Combining rules 4 and 5 produces: d/dx [f(x) / g(x)] = [g(x) df/dx - f(x) dg/dx] / [g(x)]2 Rule 7. The “Chain Rule” Copyright © 2016 C. P. Rubenstein 10

The Reciprocal Rule Let us now show that the derivative of one over a function equals The negative reciprocal squared times the derivative of the function: Using the derivative definition is: Cross multiplying: d/dx [1/f(x+dx) – 1/f(x)] = d/dx [-f(x) + f(x+dx)] / [f(x+dx) f(x)] = [-f(x) + f(x+dx)] / dx [f(x+dx) f(x)] Thus: d/dx [1/f(x)] = f(x+dx) - f(x) / f(x+dx) f(x) dx or : Copyright © 2016 C. P. Rubenstein 11

The Reciprocal Rule… continued Only dx is involved with resolving the zero-over-zero limit, so we can pull the two terms f(x+dx)f(x) outside the parentheses and then take the product of the limits of the two factors: The limit of f(x+dx)f(x), as dx goes to zero is just f(x)2 so we have which proves the reciprocal rule . Copyright © 2016 C. P. Rubenstein 12

Example of the Reciprocal Rule Example: Let y(x) = 1/x 3 (i. e. , y = 1/f(x) where f(x) = x 3) The derivative of f(x) is 3 x 2 so the reciprocal rule gives us If we write the equation as d/dx (x -3) we see that this follows the familiar pattern d/dx (xn) = n • x n-1 with n= -3: d/dx (x-3) = -3 • x -3 -1 d/dx (x-3) = -3 x -4. Copyright © 2016 C. P. Rubenstein 13

The Quotient Rule The quotient rule, d/dx [ f(x)/g(x) ] = f ' /g - g ' f / g 2 can be seen by combining the product rule and the reciprocal rule: NOTE: It is probably easier to apply the product rule to f(x) • 1/g(x) than to remember the quotient rule formula. Copyright © 2016 C. P. Rubenstein 14

The Chain Rule The chain rule lets us find the derivative of a function such as f(x) = [ sin(x 3) ] which is the sine function with an argument of x 3 instead of just x If we let g(x) denote the “cube function” and h(x) denote the sine function, then f(x) = sin( [x 3] ) = h(g[x]) Therefore, for any arbitrary functions h and g: Copyright © 2016 C. P. Rubenstein 15

The Chain Rule… continued The chain rule for arbitrary functions h and g is: If we regard the right-hand side as the product of two fractions, we cancel the dg terms, leaving just dh/dx If dx is small but not infinitesimally small, this formula is only a good approximation In the limit that dx goes to zero, this formula is the derivative of h(g[x]) and in the same way [g(x+dx) -g(x)]/dx is only an approximation until we take the limiting case where dx becomes infinitesimally small. Copyright © 2016 C. P. Rubenstein 16

Example of the Chain Rule Example: Find the derivative of f(x) = sin(x 3) = h(g[x]) Use the chain rule and that the derivative of the sine is the cosine. Applying the chain rule we have Let us test this result by numerically approximating the slope of sin(x 3) at x = 2: Approx. slope = (sin[(2 + 0. 0001)3] - sin[23] ) / 0. 0001 = -1. 75 The value of the our derivative function at x = 2 is cos(23) • 3 • 22 = -1. 746 which is good agreement so we have confidence that we applied the chain rule correctly. (Check this approximation using the TI calculator program ‘SLOPE’ we set up in Session 4. Be sure you have the calculator set to radians and not degrees!) Copyright © 2016 C. P. Rubenstein 17

Slopes to Derivatives to Rules We have moved through three levels of abstraction while studying the slopes of curves. We began by finding the numerical value of the slope at a particular point on a particular curve, f(x). Next, we learned to find a formula, the derivative function, whose value at any point x is equal to the slope of the curve at the point x, f(x). Finally, we derived rules such as the sum, product, and chain rules, which let us find the derivatives of functions constructed from arbitrary functions Copyright © 2016 C. P. Rubenstein 18

The Derivative of the Square Root of x We have learned so far to find the derivative of xn and (with the aid of the reciprocal rule) x-n. But what do we do about fractional powers? Let us look at the derivative of y(x) = x½ which is known as the square root function, y(x) = x By squaring both sides we can rewrite the equation y = x½ equals y 2 = x We can now regard x as a function of y: x(y) = y 2 . Copyright © 2016 C. P. Rubenstein 19

Derivative of the Square Root of x … continued We know how to find the derivative of the square function, so we can write What we are after is dy/dx, which is the ratio of a change in y to a change in x (when the changes are infinitesimally small). But this ratio, dy/dx, is just the reciprocal of dx/dy. That is, Another way to write this is: y'(x½) = ½x(½ - 1) y’ = ½x(-½) Note that this again fits the familiar pattern d/dx (xa) = a xa-1 which is also written: Copyright © 2016 C. P. Rubenstein 20

Questions? Copyright © 2016 C. P. Rubenstein 21

For Class Session #06: For class Session #06: • • • Homework Set #05 Reading: Strang - Chapter 3: Applications of the Derivative 2 Do: In Class Quiz and Review Homework Set #05 Lecture and Problem Review Trigonometry Review, Derivatives of Exponentials, “e” Remaining Class Note Sets online For class Session #07: • DUE: Homework Set #06 • Reading: Strang - Chapter 5 : Integrals • 2 Do: In Class Quiz and Review Homework Set #06 • Lecture and Problem Review Linear Approximations, Newton’s Method, Midterm Review Copyright © 2016 C. P. Rubenstein 22