MSCI 300 FALL 2015 Calculus 1 Charles Rubenstein
MSCI 300 – FALL 2015 Calculus 1 Charles Rubenstein, Ph. D. Professor of Engineering and Information Science Week 14: Session 11: Monday 11/23/15 Mondays 6: 30 pm-8: 50 pm PMC 705 A
Instructor Contact Information Dr. Charles Rubenstein <crubenst@pratt. edu> Professor of Engineering & Information Science Fall 2015 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 © 2015 C. P. Rubenstein 2
MSCI 300 – Fall 2015 - Class Schedule & Due Dates (* Quizzes on Homework due; Reviewed in same session) Copyright © 2015 C. P. Rubenstein 3
* Class Session Archives * http: //www. Charles. Rubenstein. com/300/ 15 fa 11. pdf (Class Power. Point slides) * 15 fa 11 h. pdf (slides in handout format) * *Archive materials normally online by Thursday evenings Copyright © 2015 C. P. Rubenstein 4
FALL 2015 TUTORING AVAILABLE Copyright © 2015 C. P. Rubenstein 5
In Class #11 • • • DUE 23 November: Homework Set #10 Reading: Strang - Chapter 6: Exponentials & Logarithms 2 Do: Using Integrals to find Volumes & Lengths In Class Quiz and Review: Homework Set #10 2 Do: Lecture and Problem Review, Programming For our next class – Week #15 Session #12 • DUE 30 November : Homework Sets #11 – 14 for Extra Credit • Reading: Strang, Chapter 6: Exponentials & Logarithms • 2 Do: Lecture and Problem Review Homework Sets #11 – 14 Week #16 Session #13 7 December In class 3 -hour FINAL EXAM Copyright © 2015 C. P. Rubenstein 6
Questions? Copyright © 2015 C. P. Rubenstein 7
Some Review… Copyright © 2015 C. P. Rubenstein 8
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 © 2015 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 Rule 5. The “Reciprocal Rule” d/dx [1/g(x)] = -dg/dx / [g(x)]2 Rule 6. The “Quotient Rule” d/dx [f(x) / g(x)] = [g(x) df/dx - f(x) dg/dx] / [g(x)]2 Rule 7. The “Chain Rule” Copyright © 2015 C. P. Rubenstein 10
Derivatives of xa There are four possibilities for the exponent ‘a’ in y=xa 1. 2. 3. 4. a can be a positive integer: a = n a can be the reciprocal of a positive integer: a = 1/n a can be a positive integer fraction: a = m/n a can be a negative integer fraction: a = - m/n In all cases: d/dx (x ) = a x a Copyright © 2015 C. P. Rubenstein a-1 11
e – the Magic Constant of Calculus A special value of a is the “magic constant of calculus”, e, for which on your calculator to many decimal places, is e = 2. 7182845. . . The beauty of e is that d/dx (ex) = 1 • ex That is ex is a function whose derivative is itself! Thus the value of the slope equals the function at x: d/dx x (e ) =f x '(e ) Copyright © 2015 C. P. Rubenstein = x e 12
Calculator Program #04 Numerical Integration using the Trapeziodal Method Copyright © 2015 C. P. Rubenstein 13
Numerical Integration using the Trapezoid Method TI calculator program: implements Equation 2, above, to find the area under the curve Y 1 (x) from x = A to x = B, using N strips. PROGRAM: TRAP : Prompt N, A, B : (B-A)/N D : Y 1(A)/2 +Y 1(B)/2 S : For (I, 1, N-1) : S+Y 1(A+I*D) S : End : S*D S : Disp S (Quit) Copyright © 2015 C. P. Rubenstein 14
Trapezoid Method Programming Notes: 1. The “For-loop” executes N-1 times, while the variable I takes on the values 1, 2, 3, … N-1. 2. The “For” statement sets I to 1 and begins the looping. 3. The “End” statement causes the program to loop back to do another iteration, unless I = N-1. In that case, the program proceeds to the first statement after “End. ” This statement multiplies the accumulated value of S by D (which is dx) and displays the result. 4. To use the program, first enter the function to be integrated, f(x) as Y 1 via the “Y=” key. Test Example: f(x) = x 2 “Find the area (integrate) from x = 0 to x = 1 using the trapezoid method” Run the program for N = 100, A = 0, and B =1: Enter Y 1 = x 2 Start TRAP = fourth listed program: Displays: prgm. TRAP At the prompts: After 7 seconds the TI-83+ displays 0. 333350, which is 1/3 (the exact value of the integral) to six decimal place accuracy. Copyright © 2015 C. P. Rubenstein 15
Questions? Copyright © 2015 C. P. Rubenstein 16
Homework #10 QUIZ Logs & Derivatives When you DO NOT show work, I have to guess. When you DO show work, I can try to see what you are doing and give an “O. K. ” #2. Find the derivative of y(x) = ln (6 x 3 + x 2) Hint: Use the chain rule. You have at least 5 minutes to do this example… Copyright © 2015 C. P. Rubenstein 17
Homework #10 Review #2. Find the derivative of y(x) = ln (6 x 3 + x 2) Hint: Use the chain rule. Ans: Chain Rule: recalling that: y'(x) = d/dx ln ( ) = 1 / ( ) y(x) = ln (6 x 3 + x 2) Copyright © 2015 C. P. Rubenstein 18
Homework #10 REVIEW Logs & Derivatives When you DO NOT show work, I have to guess. When you DO show work, I can try to see what you are doing and give an “O. K. ” Copyright © 2015 C. P. Rubenstein 19
Homework #10 Review #1. Find the value of Hint: Since , the answer will be a little greater than one. Ans: Using formula from notes: log 3(3. 1) = log(3. 1) / log (3) = 1. 0298 From scratch: 3. 1 = From which: log (3. 1) = log (3) • rearranging: 1. 0298 Copyright © 2015 C. P. Rubenstein 20
Homework #10 Review #2. Find the derivative of y(x) = ln (6 x 3 + x 2) Hint: Use the chain rule. Ans: Chain Rule: recalling that: y'(x) = d/dx ln ( ) = 1 / ( ) y(x) = ln (6 x 3 + x 2) Copyright © 2015 C. P. Rubenstein 21
Homework #10 Review #3. Check your answer to Problem 2: [ y(x) = ln (6 x 3 + x 2) ] in the usual way: Ans: 3 a. First calculate the value of your derivative function for an arbitrary value of x, say, x = 1. 5 (Let dx = 0. 001) Approximate Slope at x = 1. 5 is 1. 9327 3 b. Then calculate the approximate value of the slope of the function at x = 1. 5. (Let dx = 0. 001): y' (1. 5) = 1. 933 3 c. See if the numbers are in good agreement. YES Copyright © 2015 C. P. Rubenstein 22
Homework #10 Review #4. Find the derivative of y(x) = tan-1(x) Hint: Use the derivative of tan as 1/cos 2 Ans: We know if: x = tan(y) , dx/dy = 1/cos 2(y) , dy/dx = [cos(y)]2 One can also draw a figure to show that if f(x) = tan(y) , then cos(y) = 1/(1+x 2)½ Substituting… dy/dx = 1 / (1+x 2) Copyright © 2015 C. P. Rubenstein 23
Homework #10 Review #5. Find the derivative of y(x) = 12(x) Ans: rewrite the function as an exponential equation: a(x) = [e ln(a)] x = e x[ln(a)] y(x) = 12(x) = [e ln(12)] x = e x[ln(12)] Using the Chain Rule and the constant rule for d/dx (xln(12)): Copyright © 2015 C. P. Rubenstein 24
Homework #10 Review #6. Find the derivative of y(x) = 12(x+2) Ans: In problem #5 we found that and that y(x) = 12(x) = e x[ln(12)] y'(x) = ln(12) 12(x) therefore, if then, y(x) = 12(x+2) y'(x) = ln(12) 12(x+2) Copyright © 2015 C. P. Rubenstein 25
Homework #10 Review #7. Show that is ambiguous without parentheses. That is, show that (Use algebra or just find a numerical counterexample). Ans: Let x = 3, Which is not, 3^(3^2) = 3^(6) = 729 ( ) (3)^(3^3) = 3^9 = 19, 683 ( ) (Note that the equations DO equal each other if x = 1 or x =2 !) Copyright © 2015 C. P. Rubenstein 26
Homework #10 Review #8. Find Hints: Study the notes for the derivation of Use the result in the notes for : Express the first x as eln(x) Ans: Copyright © 2015 C. P. Rubenstein 27
Homework #10 Review #9. Check your answer to problem 8 in the usual way. 9 a. Using your formula for the derivative, find its value when x = 1. 5. (Let dx = 0. 001) Ans: from the last example: Evaluated at the value x = 1. 5: = = 4. 78457 Copyright © 2015 C. P. Rubenstein 28
Homework #10 Review #9 b. Then put the function into your calculator as Y 1 and find the approximate slope at x = 1. 5. (Use 0. 001 for dx). Ans: The Approximate slope or Numerical Derivative at the value x = 1. 5: [f(1. 5001) – f(1. 5)] / 0. 001 = 4. 7949 #9 c. Ans: See if the numbers are in good agreement. YES, indicating the derivative function is correct Copyright © 2015 C. P. Rubenstein 29
Questions? Copyright © 2015 C. P. Rubenstein 30
Review of Calculus 1 Functions and The Derivative Applications of the Derivative Max/Min Problems Linear Approximations to Functions Newton’s Method l’Hopital’s Rule The Integral The Fundamental Theorem of Calculus Integrating Volumes and Lengths Numerical Approximations Copyright © 2015 C. P. Rubenstein 31
1. Function We discussed the concept of a function, f(x), most often a formula which when we feed in a value of x results in the respective value for f(x) Any value of x produces a corresponding value of y A variation in the value of x, which we call dx, produces a variation in the value of y, which we call dy Copyright © 2015 C. P. Rubenstein 32
2. The Derivative The derivative is also a function: df /dx or f '(x) and is derived from f(x) The value of f '(x) at the point x is equal to the slope of a straight line tangent to f(x) at the point x This value is the rate of change of y with respect to x Copyright © 2015 C. P. Rubenstein 33
2. The Derivative The formal definition of the derivative is which we used immediately to find derivatives of elementary functions such as Copyright © 2015 C. P. Rubenstein 34
2. The Derivative When we applied the definition of the derivative to the exponential function f(x) = ax we found where the term in the bracket is a constant and is equal to 1 when the value of a is 2. 71828. . . We call this the fundamental constant e Which therefore means that d/dx (ex) = ex Later we saw that the term in the brackets is ln(a) which is the natural logarithm of a Copyright © 2015 C. P. Rubenstein 35
2. The Derivative We also used the definition of the derivative to find some general rules for combinations of functions - rules such as: Copyright © 2015 C. P. Rubenstein 36
2. The Derivative We showed that the derivatives of inverse functions such as y(x) = a sin(x) can be found by writing x = f(y) and calculating x'(y) For example: y = a sin(x) so x=sin(y) and dx/dy = cos(y) and cos 2 (y) + sin 2 (y) = 1 from which cos(y) = and thus cos(y) = so … Copyright © 2015 C. P. Rubenstein 37
3. Applications of the Derivative 3. 1 Maximum/Minimum Problems: Find the value of x at which the curve y(x) turns around (has a local maximum or minimum) by finding the value of x for which y'(x) = 0 3. 2. Linear Approximation to Functions In the vicinity of a point x 1 , y 1 = f (x 1) the function f (x) can be approximated as y = y 1 + (x-x 1) y'(x 1) (from the point slope formula for a straight line) Copyright © 2015 C. P. Rubenstein 38
3. 2. 1 Newton's Method Newton's method is an algorithm to find roots of a function y(x) , i. e. , values of x which satisfy the equation y(x)=0 To find roots of a function, f(x), Newton used the linear approximation in the vicinity of a root, a. The linear approximation is y = 0 + (x-a) y'(a), since y(a)=0 Newton made the further approximation that y'(a) ≈ y(x) in the vicinity of a. If we guess that an x value, x 1 , is close to a root, Newton's approximation can be solved for a: a = x 1 - y(x 1) / y ' (x 1) This value of a will be a better approximation to the true value of the root. We use this value as a new guess, x 1, and repeat the process. After several repetitions (“iterations”) the value of a will home in accurately on the actual value of the root. Copyright © 2015 C. P. Rubenstein 39
3. 2. 2 l'Hôpital's Rule If f(x 1) and g(x 1) are both zero, then f(x 1) / g(x 1) = 0/0 which is undefined … But using linear approximations, we find that Copyright © 2015 C. P. Rubenstein 40
4. Integral Consider the function F(x), whose derivative is f(x), i, e. , F'(x) = f(x) F(x) is called the antiderivative or indefinite integral of f(x). We denote F(x) by enclosing f(x) between an integral sign and the symbol dx: Example: As the derivative of [sin(x) + C] = cos(x) we have: Copyright © 2015 C. P. Rubenstein 41
5. The Fundamental Theorem of Calculus The fundamental theorem of calculus states that the area enclosed by a curve y(x), the x-axis, the line x = x 1 and the line x = x 2 is given by F(x 2) = F(x 1) We express this symbolically as a “definite” integral Usually we write this as Rearranging the fundamental theorem, we can perfectly extrapolate the value of F(x) at x = x 2 from its value at x = x 1 Copyright © 2015 C. P. Rubenstein 42
5. The Fundamental Theorem of Calculus Example: Velocity is the derivative of position with respect to time. If v(t) is the velocity of an object moving in one dimension then its position at time t 2 , y(t 2), is related to its position at an earlier time t 1 , y(t 1), by Copyright © 2015 C. P. Rubenstein 43
6. Integrating Volumes and Lengths While the (definite) integral is the area under the curve f(x), we can construct the function f(x) such that f(x)dx is a volume element d. V, a length element d. S, or area element d. A on a curved surface. Volume element areas are often disks or shells of thickness dx. The element of length (“arc length”) of a curve g(x) is which comes directly from the Pythagorean theorem. Copyright © 2015 C. P. Rubenstein 44
7. Numerical Approximations 7. 1 We often found an approximate numerical value to the slope of a curve y(x) at a point x 1 by simply calculating the increase in y when x is increased by a small amount dx : This is the same as the definition of the derivative, but without taking the limit as dx goes to zero. Note the distinction: this formula produces a single number which is the slope at x 1, y(x 1). To find the derivative function, which is a formula for the slope at any value x, we leave x as a variable and take the limit as dx goes to zero. Copyright © 2015 C. P. Rubenstein 45
7. Numerical Approximations 7. 2 To find an approximation for the definite integral, we took the area under the curve to be the sum of the areas of vertical strips of finite width dx. We approximated the average height of each strip by the average of its left-hand height and its right hand height the “trapezoid rule. ” Copyright © 2015 C. P. Rubenstein 46
8. Algebra and Trigonometry Review We also reviewed some basic algebra and trigonometry during this semester, including - the ‘completion of the square’ technique … - and several trig identities, such as, sin(a+b) = sin(a)cos(b) + cos(a)sin(b) Copyright © 2015 C. P. Rubenstein 47
In Class #12 Week #15 Session #12 • DUE 30 November : Homework Sets #11 – 14 for Extra Credit • Reading: Strang, Chapter 6: Exponentials & Logarithms • 2 Do: Lecture and Problem Review Homework Sets #11 – 14 Week #16 Session #13 7 December In class 3 -hour FINAL EXAM Copyright © 2015 C. P. Rubenstein 48
Any Questions? Send me an email … crubenst@pratt. edu or c. rubenstein@ieee. org Copyright © 2015 C. P. Rubenstein 49
End Copyright © 2015 C. P. Rubenstein 50
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