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 14: Session 11: Monday 04/18/16 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) 25 January (2) 1 February (3) 8 February (4) 15 February (5) 22 February (6) 29 February (7) 7 March (8) 14 March (9) 21 March (10) 28 March (11) 4 April (12) 11 April (13) 18 April (14) 25 April (15) 2 May (16) 9 May (17) NO CLASS – Martin Luther King Day Introduction; Review of Syllabus, Algebra & Trigonometry Functions and Slopes (Quiz 1) Approximating Slopes, Limits, The Derivative (Quiz 2) Rules of Differentiation (Q 3) Max/Min Prob, 2 nd + Derivatives (Q 4) Trig, Trig Derivatives, Limit of sin(x)/x (Q 5) Derivatives of Exponentials, Constant "e"; Take Home Exam (Q 6) NO CLASSES – Pratt Spring Break – 14 -20 March 2016 NO CLASS – Instructor out sick l'Hôpital's Rule; Kinematics: Position, Velocity, Acceleration; Midterm Due (Q 7) CMFM Seminar Midterm Review (Q 8) Linear Approximation, Newton's Method; Review Logs (Q 9) Inverse Function Derivatives, Implicit Differentiation (Q 10) Areas, Intro to Integrals, Fundamental Theorem of Calculus Using Integrals to find Volumes and Lengths, Review 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: Mondays 4: 30 -6: 00 pm PMC 403 Email: jguadagni@pratt. edu Fridays 3: 00 -5: 00 pm WTC North Hall Call: 718 -636 -3459 Copyright © 2016 C. P. Rubenstein 4
* Class Session Archives * http: //www. Charles. Rubenstein. com/300/ 16 sp 11. pdf (Class Power. Point slides) * 16 sp 11 h. pdf (slides in handout format) * *Archive materials normally online by Thursday evenings Copyright © 2016 C. P. Rubenstein 5
In Class #11 • DUE: Homework Set #10 • Reading: Strang - Chapter 6: Exponentials & Logarithms • 2 Do: The Fundamental Theorem of Calculus, Using Integrals to find Volumes & Lengths • In Class Quiz and Review: Homework Set #10 For class Session #12: • DUE: Homework Set #11 – 14 for extra credit • In Class Review: Optional Homework Set #11 -14 For class Session #13: Any remaining materials For class Session #14: Monday 9 May 2016 FINAL EXAM – In Class; 3 -hours – Open notes* 20@5 -point parts in 15 questions, plus a 10 point bonus (* You are better off having up to 4 pages of notes and formulas…) Copyright © 2016 C. P. Rubenstein 6
Questions? Copyright © 2016 C. P. Rubenstein 7
Some Review… Copyright © 2016 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 © 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 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 © 2016 C. P. Rubenstein 10
Derivatives of xa There are four possibilities for the exponent ‘a’ in y = x a 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 a a-1 In all cases: d/dx (x ) = a x Copyright © 2016 C. P. Rubenstein 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: x x x d/dx (e ) = f '(e ) = e Copyright © 2016 C. P. Rubenstein 12
Calculator Program #04 Numerical Integration using the Trapeziodal Method Copyright © 2016 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 © 2016 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 © 2016 C. P. Rubenstein 15
Questions? Copyright © 2016 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 © 2016 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 © 2016 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 © 2016 C. P. Rubenstein 19
Homework #10 Review #1. Find the value of Hint: Since Ans: , the answer will be a little greater than one. 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 © 2016 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 © 2016 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. 933 333 3 c. See if the numbers are in good agreement. YES Copyright © 2016 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 © 2016 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 © 2016 C. P. Rubenstein 24
Homework #10 Review #6. Find the derivative of y(x) = 12(x+2) Ans: In problem #5 we found that y(x) = 12(x) = e x[ln(12)] and that y'(x) = ln(12) 12(x) therefore, if then, y(x) = 12(x+2) y'(x) = ln(12) 12(x+2) Copyright © 2016 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, 3^(3^2) = 3^(9) = 19, 683 ( ) Which is not, (3)^(3^3) = 3^27 = 7. 62 x 10^12 ! ( ) (Note that the equations DO equal each other if x = 1 or x =2 !) Copyright © 2016 C. P. Rubenstein 26
Homework #10 Review #8. Find Hints: Study the notes for the derivation of : Express the first x as eln(x) Use the result in the notes for Ans: Copyright © 2016 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 © 2016 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. See if the numbers are in good agreement. Ans: YES, indicating the derivative function is correct Copyright © 2016 C. P. Rubenstein 29
Questions? Copyright © 2016 C. P. Rubenstein 30
More about uses of Integrals, The Fundamental Theorem of Calculus Numerical Approximation to the Value of a Definite Integral Numerical Integration using the Trapezoid Method Using the Integral to find Volume Calculation of the Volume of a Cone Extrapolation using the Derivative Copyright © 2016 C. P. Rubenstein 31
Numerical Approximation of the Value of a Definite Integral We often found approximate values for the derivative by using an “approximate slope formula, ” the derivative with a finite, rather than infinitesimal, value of dx. We simply took a conveniently small number for dx, e. g. , dx=0. 001, and calculated f ' (x) ≈ [f(x+dx) - f(x)] / dx as an approximation to value of the derivative at x. Similarly, we can approximate the value of a definite integral by summing the areas of the strips, d. A = f(x) dx using a finite rather than infinitesimal value of dx. Copyright © 2016 C. P. Rubenstein 32
Numerical Approximation of a Definite Integral Since the strips are finite, their tops are not horizontal; with f(x) varing slightly across the top. Should we take the height to be y(x), y(x+dx)? One fairly obvious choice is to take the height as the average of y(x) and y(x+dx), i. e. , h = (y[x] + y[x+dx])/2 which is the same as assuming that the strip is a trapezoid, i. e. , that y(x) is a linear (straight line) function between x and x+dx. Summing the areas of N strips spanning the interval from x = a to x = b the trapezoid method gives us Equation 1: where x 1 = a, x 2 = (a + dx, ) x 3 = (a + 2 dx), . . . and x. N = (a+Ndx ) = b. Since a + Ndx = b, rearranging the equation, we see that dx = (b-a)/N Copyright © 2016 C. P. Rubenstein 33
Numerical Approximation of a Definite Integral Usually we choose N to be at least ten or greater, so calculating and adding up the terms on the right side of Equation 1 is best done with a program in your calculator. A digital computer will be able to perform the calculation faster if we rearrange Equation 1: as follows recognizing that all terms except the first and last are repeated ‘half’ terms, thus f(x) / 2 + f(x) / 2 = f(x)… to get Equation 2: Copyright © 2016 C. P. Rubenstein 34
Calculator Program #04 Numerical Integration using the Trapeziodal Method Copyright © 2016 C. P. Rubenstein 35
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 © 2016 C. P. Rubenstein 36
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 © 2016 C. P. Rubenstein 37
Questions? Copyright © 2016 C. P. Rubenstein 38
Using the Integral to find Volume The notation reminds us that this quantity the sum of the areas of vertical strips of height f(x) and width dx. When we are asked to calculate an area, we often start by writing But, in many problems, f(x) is chosen so that f(x) dx also represents some quantity other than area. Let’s look at an example where we calculate a volume. Copyright © 2016 C. P. Rubenstein 39
Calculation of the Volume of a Cone Example: Calculate the volume of a cone with height h and radius R The figure shows the cone, tipped to lie along the x-axis. (The z-axis is out from the page). Note that the cone cuts the x, y plane, making a straight line whose slope is R/h, and that the equation of this line is just y(x) = (R/h)x We can think of the cone being formed when this line is rotated about the x-axis. The volume of the cone can be regarded as the volume of a (horizontal) stack of coins. One of these coins is shown in the figure. The volume of this coin, d. V, is given by the area of its face: y 2 times its thickness, dx. But y = Rx/h, so d. V = y 2 dx = ( Rx/h)2 dx = ( R 2 /h 2) x 2 dx Copyright © 2016 C. P. Rubenstein 3) 40
Calculation of the Volume of a Cone The individual volume of each coin is d. V = ( R 2 /h 2) x 2 dx The total volume of the cone is the sum of the volumes of these stacked coins. As , R, and h, are NOT functions of x, they will be considered as constants and removed from within the integral. In the limit that the thickness dx goes to zero (and the number coins becomes infinite), the sum is the integral shown here as Equation 4: 4) Since the antiderivative of x 2 = x 3/3 the value of the integral at the right is h 3/3 - 03/3 = h 3/3, so 5) Equation 5 is the familiar formula, “one third the height times area of the base” which gives the volume of a cone with a base of any arbitrary plane shape. Copyright © 2016 C. P. Rubenstein 41
Extrapolation using the Derivative If we know a function’s derivative and the value of the function at one point, x 1, we can find the value at any other point, x 2. We can start with the recipe for finding area under a curve g(x) between x 1 and x 2, 6) where G(x) is the antiderivative of g(x). We can write f (x) in place of G(x) of we write f ′ (x) in place of g(x), i. e. , 7) Copyright © 2016 C. P. Rubenstein 42
Extrapolation using the Derivative 7) Rearranging Equation 7 (above) gives us the desired result, Equation 8: This formula allows us to find the value of the function at x = x 2 given the value at x = x 1 Copyright © 2016 C. P. Rubenstein 43
Extrapolation using the Derivative Example Suppose that at t= 0 hours, the position of a car is 50 miles down the road. Let the car’s velocity be given by the function v(t) = 60 + 10 sin(30 t) (Maybe the driver is trying to maintain a steady 60 mph (miles per hour) but he is speeding up to 70 mph and back down to 50 mph every two minutes). Find his position at t = 15 minutes. Solution: Velocity is the derivative of position with respect to time, i. e. , v(t) = dx/dt. Rewriting Equation 8 in these terms we have Equation 9: 9) where x′(t) is the derivative function dx/dt which is the velocity v(t) Copyright © 2016 C. P. Rubenstein 44
Extrapolation using the Derivative Example, continued 9) where x′(t) is the derivative function dx/dt which is the velocity v(t) To recap what we know: v(t) = x′(t) = 60 + 10 sin(30 t) also t 1 =0, t 2 = 0. 25 hr, and x(t 1) = 50 With these substitutions, Equation 10 becomes: We use the antiderivative recipe to evaluate the integral, giving us: or in 15 minutes the car will be 65. 22 miles down the road. Copyright © 2016 C. P. Rubenstein 45
Extrapolation using the Derivative Note that we have solved problems like this using just the derivative / antiderivative relation between velocity and position. But using the formalism of the integral points out that the change in position over a given time interval is the time integral of velocity of that interval. Using Equation 9, one could track the position of a car along a road by observing only the speedometer. By taking frequent readings of the speedometer, spaced by a small interval, you can construct the running x(t) curve by using numerical integration. In an inertial navigation system, the speedometer is not needed. Instead, the position of a mass held by spring is used to indicate the instantaneous acceleration. Copyright © 2016 C. P. Rubenstein 46
Extrapolation using the Derivative Readings of the acceleration are taken at closely-space intervals in time and numerically integrated to produce a running estimate of the velocity. The successive velocity estimates are, in turn, numerically integrated to provide a running estimate of the position. Note: A real 3 -d inertial guidance system contains three accelerometers, one each for each direction. It also contains gyroscopes to keep track of the orientation of the vehicle with respect to the x, y, and z directions. Copyright © 2016 C. P. Rubenstein 47
Questions? Copyright © 2016 C. P. Rubenstein 48
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 © 2016 C. P. Rubenstein 49
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 © 2016 C. P. Rubenstein 50
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 © 2016 C. P. Rubenstein 51
2. The Derivative The formal definition of the derivative is which we used immediately to find derivatives of elementary functions such as Copyright © 2016 C. P. Rubenstein 52
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 © 2016 C. P. Rubenstein 53
2. The Derivative We also used the definition of the derivative to find some general rules for combinations of functions - rules such as: Copyright © 2016 C. P. Rubenstein 54
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 © 2016 C. P. Rubenstein 55
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 © 2016 C. P. Rubenstein 56
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 © 2016 C. P. Rubenstein 57
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 © 2016 C. P. Rubenstein 58
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 © 2016 C. P. Rubenstein 59
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 © 2016 C. P. Rubenstein 60
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 © 2016 C. P. Rubenstein 61
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 © 2016 C. P. Rubenstein 62
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 © 2016 C. P. Rubenstein 63
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 © 2016 C. P. Rubenstein 64
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 © 2016 C. P. Rubenstein 65
In Class #12 • DUE: Homework Set #11 – 14 for extra credit • In Class Review: Optional Homework Set #11 -14 For class Session #13: Materials not yet covered, if any… For class Session #14: Monday 9 May 2016 FINAL EXAM – In Class; 3 -hours – Open notes* 20@5 -point parts in 15 questions, plus a 10 point bonus (* You are better off having up to 4 pages of notes and formulas…) Copyright © 2016 C. P. Rubenstein 66
Any Questions? Send me an email … crubenst@pratt. edu or c. rubenstein@ieee. org Copyright © 2016 C. P. Rubenstein 67
End Copyright © 2016 C. P. Rubenstein 68
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