What is calculus? Consider the way that Archimedes figured out the formula for the area of a circle:

Transcription

1 What is calculus? Consider the way that Archimedes figured out the formula for the area of a circle: Calculus involves formally describing what happens as a value becomes infinitesimally small (or large). It was simultaneously developed UNM PNM Math Contest This Sunday 11/5, 9 12 Please register! Unit Plan is available on web page Calculus Overview 11/1: Overview 1

2 14C: Investigation 3, #2a (Rate of change) 14D: #1 3 (Derivative function) 1. Distinguish between average and instantaneous rates of change. 2. Recognize instantaneous rate of change as slope of the tangent to a curve at a point. distance (km) Jessie is driving away from school in her new car. The graph to the left represents how far away she is at any given time. At t = 0, how far away from school is she? t = 2?, t = 4? Describe her motion. What is her average speed in the first two minutes? What is her average speed in the next two minutes? The speed limit is 2 km/min. When will she "officially" be speeding? Hint: The equation for the curve is d(t) = ½t Try graphing it on your calculator. Your zoom feature might help. time (min) With a partner, complete the following: What have you concluded so far? The gradient of the tangent to y = f(x) at x = a is the instantaneous rate of change in f(x) with respect to x at that point. 11/1: 4C Rates of change 2

3 1. Understand the meaning of the derivative function. Given a non linear function, f(x), it would be nice to come up with a related function that describes the rate of change (or slope or gradient) of the original function at any point x. The Derivative Function The function that describes the gradient of y = f(x) is called its derivative function and is labeled f'(x) (read as "eff prime of x") The value of f'(a) is the gradient of the tangent to f(x) at the point where x= a We will discuss how to find this function soon. But you can begin to work with certain questions already: UNM PNM Math Contest Sun Nov 5, 9 12 pm Please register at UNM PNM Website 14C: Investigation 3, #1*, 2a (Rate of change) 14D: #1 3 (Derivative function) On #1, use the method of Investigation 3 that we did in class. 11/1: 14D The Derivative Function 3

4 SL2.CalculusDiff.1718.notebook 14C: Investigation 3, #1*, 2a (Rate of change) 14D: #1 3 (Derivative function) Discuss Discuss Pace: 4 14A: #1def,2,3,4cfi (Limits) 14B: #1 3 all (Limits at infinity) 1. Understand how to find limits informally. On your calculator, graph the function What is the value of the function at x = 2? What is the value of the function near x = 2? Hmmm. How can we describe this more formally? Limits exploring a function as the variable approaches a value that might not be well defined. is not the same as evaluating the function f at a. Limits are useful in describing asymptotes Evaluating limits requires special approaches and rules Informal Definition of a Limit: To find limits simplify the expression, sometimes by factoring, to eliminate any discontinuities (a value of x where the function is undefined). If there is no discontinuity at the desired value, just evaluate the expression. Alternatively: It can be shown that certain properties apply to limits: Indeterminant forms Expressions that evaluate to ratios involving 0 and/or are called indeterminant and require special treatment. You must rearrange expressions to see what value the expression will approach. A common trick is to divide the numerator and denominator by some power of the variable. Try some: /3 Courtesy of Alex Kellam (DA '17) 11/3: 14A Limits 4

5 1. Formally represent asymptotes as limits Limits arise when looking at functions with asymptotes (often these are rational functions or functions of the form p(x)/q(x) where p and q are both polynomials). In this course we limit ourselves to functions where both the numerator and denominator are linear (ax + b). Let's review Features of rational functions of the form The shape is a rectangular hyperbola The function has a vertical asymptote at x = d/c The function has an horizontal asymptote at f(x) = a/c The function has a zero at ( b/a, 0) The function has a y intercept at (0, b/d) Division by zero creates a vertical asymptote. Consider the function Vertical asymptote? Pre calculus: as x 5 from the left, y or as x 5, y as x 5 from the right y + or as x 5 +, y + Now: Horizontal asymptote? Pre calculus: as x, y 2 from above or as x, y 2 + as x +, y 2 from below or as x +, y 2 Now We can see the horizontal asymptote algebraically as we showed previously: But how can we tell whether it is approaching from above or below? One way is to use a sign diagram Choose some test points: > Big negative: num, denom = + > 0: + num, denom = > Big positive: + num, + denom = + So how does this help? Recall: VA is at y = 2 the curve comes up from below in Q2 Curve is >0 for x > 5 the curve comes down from above in Q1 There is a second, nifty and more algebraic approach: How does this help us? Notice that, as x +, the fractional part, So the curve, f(x) is a little above 2. it is approaching from above and Similarly, as x, the fractional part, So the curve, f(x) is a little below 2. it is approaching from below and Don't memorize, understand why. The same ideas apply for higher order polynomials in a rational function. But SL will focus on ratios of linear expressions. When this is the case, focus on finding the vertical and horizontal asymptotes and the two axis intercepts. Try one: We've focused on infinite limits in rational functions. But we need to understand other functions as well. And another common one: The number of grams of a chemical in solution (S) decreases during a chemical reaction according to the equation: S(t) = e t where t is the number of minutes after a catalyst has been added. How much of the chemical is in the solution when the catalyst is added? How much is present 24 hours later? No calculator 3SF please t = t = 24 hrs 14A: #1def,2,3,4cfi (Limits) 14B: #1 3 all (Limits at infinity) If you see quadratic forms, try to simplify so that the function is linear over linear. If you can't then you may use your calculator. 11/3: 14B Limits at Infinity 5

6 A more formal definition of limits time permitting Limits demo Another Limits Demo Formal Definition of limits 6

7 14A: #1def,2,3,4cfi (Limits) Present 1f,2,3cd,4cfi 14B: #1 3 all (Limits at infinity) Present 1cde,2 1. Understand the definition of the derivative 2. Find derivatives of certain functions from first principles OK, let's get more formal about this: How do we find the derivative function more rigorously? Geogebra Demo: Derivatives Summary of a Derivative 14E: #1,2bc,3bc,4a,5d,6 (First Principles) Review Set 14A #1 4 Review Set 14B #1 4 is called the limit quotient. Finding the derivative of a function by evaluating the limit quotient is called using first principles. The f' notation is associated with Isaac Newton, and thus is more common in Western European texts (particularly those with a British connection...). Another notation was developed at roughly the same time by Gottfried Leibnitz which uses differentials. This notation is very important in further Calculus. Instead of using h, Leibnitz references the Δx and Δy using the lower case δx and δy. Then the approximate gradient is: The exact derivative function is then...pronounced "dee why, dee ex" Either definition can be used to find the derivative function at any point x. But you can also use first principles to find a derivative at a single point, say, a. Try one 14E: #1,2bc,3bc,4a,5d,6 (First Principles) Review Set 14A #1 4 Review Set 14B #1 4 11/6: 14E First Principles 7

8 14E: #1,2bc,3bc,4a,5d,6 (First Principles) Present 1,3c,4a,5d,6 Review Set 14A #1 4 Present 1c,4 Review Set 14B #1 4 Present 1,2 1. Understand basic rules for differentiating power functions. 15A: #1aeim,2all,3efgh,4ace,5,6dh,7 (Rules) QB #2ab,33a,48a The result of 14E.1 is fundamental and is very useful. Let's prove it: Derivative of a power function We will explore many of the properties of differentiation using a power function to practice. Here are the first set of rules. Can you prove them from first principles? This is also a good time to review notation. The derivative of y with respect to x (meaning that y is the dependent variable and x is the independent variable) is also given by: This notation was developed by Gottfried Leibnitz around He was credited with discovering infinitesimal calculus simultaneously and independently from Isaac Newton, who developed and used the "prime" notation. The following instructions are all different versions of saying the same thing: Find f'(x) Find y' Find Differentiate with respect to x Find the gradient of the tangent to Find the gradient function of f(x)...and others! Try some: What does f'(x) represent? Why do we call it a gradient function? Here's where our fundamentals of algebra come in handy. Don't be lazy! This is where we are reviewing and strengthening our foundations. Find the slope of the line tangent to the curve at x = 2 Let's use a calculator! Option 1: Graph the function Use [2nd][CALC][6: dy/dx] Enter the x value of interest Option 2: Use [MATH][8: nderiv]...or... [ALPHA][WINDOW][3: nderiv] On the updated operating system (2.55), you will see: Enter the function Input the values then press [ENTER] Enter the variable to Enter the value of the differentiate with respect to variable at which to (usually x) evaluate the derivative On a TI 83 or old operating system you will see: nderiv( Enter the parameters, separated by commas: nderiv(<function>,<variable>,<value>) Enter the value of the variable at which to evaluate the derivative Enter the function Enter the variable to differentiate with respect to (usually x) 15A: #1aeim,2all,3efgh,4ace,5,6dh,7 (Rules) QB #2ab,33a,48a 11/8: 15A Basic Rules 8

9 Check out this for next year: derivative_intuitive_chain_rule.html 15A: #1aeim,2all,3efgh,4ace,5,6dh,7 (Rules) Present 1m,3h,4e,5,6h,7 QB #2ab,33a,48a Present all 1. Recognize function compositions 2. Understand and apply the chain rule 15B.1: #1 2 all (Composition) 15B.2: #1,2abcfi,3ace,4 6 all (Chain rule) QB #24a,44ab,46a We begin this section by reviewing composite functions x f f(x) g g(f(x)) = (g f)(x) x Times f(x) = 3x + 7 Square & add 3 (g f)(x) = (3x + 7) The composite of two functions is created by using the output of one function as the input to the other function. Some properties: (f g)(x) is not the same as (g f)(x) in general The range of the first function in a composition is the domain of the second. Try: Given f(x) = x and g(x) = 2x + 4 find (f g)(x) and (g f)(x) (f g)(x) = (2x + 4) (g f)(x) = 2(x 2 + 7) + 4 You'll do a little more practice with composites in 15D.1 Consider the function x 2 whose derivative is 2x. What is the derivative of (2x) 2? Now try differentiating (2x + 3) 2. Do it with and without expanding first. Hmmm... Finally, try differentiating (x 2 + 3x + 4) 2 by guesswork and then by expanding it out. Try finding the derivative of (2x 2 ) 4 two different ways From these short examples, but with no formal proof, we can see the chain rule at work. The Chain Rule If f(x) = g(h(x)) = (g h)(x) then f'(x) = g'(h(x)) h'(x) Not recognizing the need to use the chain rule is probably the single most common source of errors in differentiation! You need to be able to recognize when a function is a composite of other functions. It takes practice. Although the concept is simple, it can get complex. It can help to introduce another variable to keep track of your work. What about Chains can have lots of links! Do 15B.2 thoroughly. This is like learning your times tables. The idea is to master it, not just to know how to do one. PRACTICE! 15B.1: #1 2 all (Composition) 15B.2: #1,2abcfi,3ace,4 6 all (Chain rule) QB #24a,44ab,46a 11/10: 15B Chain Rule 9

10 A lot for a Monday 15B.1: #1 2 all (Composition) Present 1ef,2bd 15B.2: #1,2abcfi,3ace,4 6 all (Chain rule) Present 2 all verbally,3ace,4,5,6 QB #24a,44ab,46a Present Recognize products of functions 2. Understand and apply the product rule 15C: #1,2def,3 5 (Product rule) 15D: #1def,2 4 (Quotient rule) What happens when you take the derivative of a product of two functions? Let's look at this from first principles: Add zero! There are lots of notations for this all boil down to the same thing! Another proof using Leibnitz notation is given in the book. The Product Rule Try a couple: (Don't forget the Chain Rule) 11/13: 15C Product Rule 10

11 1. Recognize quotients of functions 2. Understand and apply the quotient rule in appropriate places! Consider a function Making use of the product rule, derive a formula for f'(x) in terms of u, u', v, & v' The Quotient Rule Try a couple: 15C: #1,2def,3 5 (Product rule) 15D: #1def,2 4 (Quotient rule) Again, this skill needs to be second nature! No pain, no gain......train your brain! 11/1315D Quotient Rule 11

12 15C: #1,2def,3 5 (Product rule) Present #2e,3d,4 15D: #1def,2 4 (Quotient rule) Present #1ef,2d,4 f '(x) = x Proper quotient 1 + 6x 3 Simplify M x 3 ½(25 x 2 ) ½ ( 2x) Substitute 3 M1 Correct working ¾ 1. Find and use derivatives of exponential functions 15E: #1adghkmo,2 6all (Exponentials) QB #9ab We now turn our attention to derivatives of other functions. Like power functions, we will develop shortcuts from the definition. We begin with exponentials. Recall the graphs of exponential functions of the form y = a x For base > 1 the function increases, for 0 < base < 1 the function decreases. Let's find the derivative of this function. Since we don't have a formula yet, we need to start from first principles. Let f(x) = a x. Then: Now we notice something interesting: So we can rewrite our result as: Derivative of an exponential If f(x) = a x then f'(x) = f'(0)a x Whoopdeedo what good is this if we don't know f'(0)? Let that sink in a moment! What is the derivative of 2 x? It's 2 x times the slope of 2 x at x = 0! Point taken. Can I find the slope of 2 x at x = 0? How about by using a calculator! Use MATH/nDeriv(2 x,x,x). It may take a while to calculate. f'(0) Are we going to do that every time? No! Do you recognize that number? Let's find the value of the base that gives us a slope of 1 at x = 0. Experiment with different bases (remember, larger bases create steeper curves) You may have found a guess, but let's look at this algebraically. The question, again, is "for what base will we get f'(0)= 1"? That is what value of a makes: or For this to be true, the numerator has to approach h in the limit. So: Now substitute and notice that as h 0, n So we can rewrite: Raising both sides to the n th power gives our answer! So, as it turns out, the function e x has a slope of one at x = 0. It has the very special property that: It is a function whose derivative is itself! That is, the value of the function at any point x is also the slope of the function at that point! Pretty natural, eh? We will come back to our original question about the derivative of a x but for now let's work a little with the base of e (where we don't have to worry about that pesky f'(0). All the differentiation tools in your belt work for this function even in combinations! Don't forget about the chain rule! The derivative of a function of the form e f(x) requires it! 15E: #1adghkmo,2 6all (Exponentials) QB #9ab Function HW i = 0 Do while (i < 100) Recite ("Repetition is my friend",volume = i) i = i + 1 End Do End HW 11/15 15E Exponentials 12

13 15E: #1adghkmo,2 6all (Exponentials) Present 2gh,3bc,4,5,6 QB #9ab Present them Take time review exponential derivatives well Practice 15F: #1acdefklmn,2ghi,3cfghi,4,5 (Logarithms) 1. Find and use derivatives of logarithmic functions A quick review: log ba is the power to which b is raised to get a b x = a x = log ba The log base e is called the natural logarithm or ln e x = a x = ln a e ln a = a and ln e a = a ln x is the inverse of e x Domain of e x is x R, Range of e x is Domain of ln x is x > 0, Range of ln x is y R y = e x Rules of Logarithms (any base) log bb x = x What is the derivative of y = lnx? We use the chain rule here: Extending this to arguments that are functions of x and using the chain rule, we get the more general case: Derivative of natural logarithms Can you generalize to logs of any base? So, in summary, the most general form is: Derivative of logarithms of base b Of course, this is usually going happen in a larger context. Use laws of logarithms! 15F: #1acdefklmn,2ghi,3cfghi,4,5 (Logarithms) Practice doesn't make perfect. Practice reduces the imperfection. Toba Beta, "Master of Stupidity" 11/17 15F Derivatives of log functions 13

14 15F: #1acdefklmn,2ghi,3cfghi,4,5 (Logarithms) Present #1lmn,2ghi,3hi,4,5 Partner Quick Quiz: Differentiate the functions given in #1 6 below: Find and use derivatives of trigonometric functions 15G: #1 3last col,4 (Trig functions) QB #1 A quick review of trig functions The derivative of sinθ is the rate of change (slope) of sinθ as θ changes. We can see that the slope oscillates from positive to zero to negative to zero and back with the same period as sinθ (2π). This gives us a hint as to what we can expect... Find the derivative of sinx from first principles. A hint: Recall the identity: cos(2θ) = 1 2sin 2 θ so cos(2θ) 1 = 2sin 2 θ and cos(θ) 1 = 2sin 2 (θ/2) Now recall that There is an alternative derivation in the text using the identity: Find the derivative of cos x (Hint: cos x = sin (x + π/2)) Find the derivative of tan x Summary: for x in radians: Don't forget the chain rule (ever!) Some practice: 15G: #1 3last col,4 (Trig functions) QB #1 11/21 15G Trig Func 14

15 15G: #1 3last col,4 (Trig functions) Present 1fl,2il,3fil QB #1 Present Quick Practice: Differentiate the functions below: 1. Understand and apply higher order derivatives. 15H: #1 13 odd (Higher derivatives) QB #45a If we can differentiate a function, can we do it twice? Well, of course! Notation: if y = f(x) then: 1 st derivative = slope of f(x) 2 nd derivative = slope of f'(x) = slope of the slope of f(x) 3 rd derivative = slope of f''(x) = slope of the slope of the slope of f(x) n th derivative = slope of f (n 1) (x) Nothing conceptually new here, except for recognizing the derivatives of the functions involved and properly executing chain rules, product rules, and quotient rules. In some physical applications, the derivatives have important meanings, and thus, special names:...or... So what does the second derivative mean about an abstract function? f''(x) is a function that describes the curvature of f at any value of x. 15H: #1 13 odd (Higher derivatives) QB #45a By doing this HW, you are practicing your derivatives, and your algebra. It is not sufficient to understand that you can do this or how to do it. It is sufficient to do it. 11/27 15H 2nd & Higher Derivatives 15