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 Coming up...interested? Please register! Calculus Overview Unit Plan is available on web page

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) Geogebra Demo: Derivatives 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.

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 next. But you can begin to work with certain questions already: 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.

4 14C: Investigation 3, #1*, 2a (Rate of change) Discuss 14D: #1-3 (Derivative function) Discuss 1. Understand how to find limits informally. 14A: #1def,2,3,4cfi (Limits) 14B: #1-3 all (Limits at infinity) 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

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). Consider Review rational functions! Redo this lesson using hyperbolas only. The rest is not SL material. To explore asymptotes, factor as much as you can and cancel terms. You might see quadratics, even though it's really a ratio of linear expressions (with a twist). Note a problem at x = 7. Division by zero creates a vertical asymptote. To explore it we need to look at the sign of f very close to but on either side of x = 7. When x is just < 7, num is >0, denom is <0) so f is big and negative: When x is just > 7, num is >0, denom is >0) so f is big and positive: Horizontal asymptotes Let's also look at what happens as x gets very large. You may recall that: > As x gets large, the highest power of x in a polynomial dominates the value of the function. In our case, the numerator and denominator are both quadratic Thus, as x gets large, they both approach infinity at the same rate. The lower order terms become irrelevant because: So this tells us that the limit as x gets large is 2. But how can we tell whether it is approaching from above or below? One way is to use a sign diagram. This is simply a record of where the function is positive, negative, or changes sign. In our case, the function can only change sign at two places: x = 7 (an asymptote) or x = 3 (a zero) For + : f is very big at 7, so it has to come down to get to 2. Choose some test points: > Big negative: get squared: positive > A bit above 3: + num, denom = > Big positive: get squared: positive For : f comes up through 0 at 3 so it's approaching 2 from below. (Note: to ensure that this approach is correct, we must confirm that the function never crosses the asymptote. So we look at whether Is this ever true? Nope: So f never crosses the asymptote and we are done Another approach to this involves rewriting the expression Note that as x +, f(x) 2 from above since Likewise, as x -, f(x) 2 from below since Don't memorize, understand why. SL will focus on ratios of linear expressions. When this is the case, you can expect to find one vertical and one horizontal asymptote. Try one: 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.

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

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 14E: #1,2bc,3bc,4a,5d,6 (First Principles) Review Set 14A #1-4 Review Set 14B #1-4 OK, let's get more formal about this: How do we find the derivative function more rigorously? Geogebra Demo: Derivatives Summary of a Deriva.ve 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

8 14E: #1,2bc,3bc,4a,5d,6 (First Principles) Present 1,3c,4a,5d,6 Review Set 14A #1-4 Present 1c,4 UNM-PNM Math Contest Review Set 14B #1-4 Present 1,2 Nov 4...interested? Please register! 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: 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 differentiate with respect to (usually x) 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 Enter the function which to evaluate the derivative Enter the variable to differentiate with respect to (usually x) Enter the value of the variable at which to evaluate the derivative 15A: #1aeim,2all,3efgh,4ace,5,6dh,7 (Rules) QB #2ab,33a,48a

9 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 g f x f(x) g(f(x)) = (g f)(x) Square x Times f(x) = 3x + 7 & (g f)(x) = (3x + 7) add 3 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

10 UNM-PNM Math Contest Fri, Nov 4, 2-5 pm Register at UNM-PNM Website before 10/31 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 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!

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 A1A1A x 3 A1 Simplify M x 3 A1 ½(25 x 2 ) ½ ( 2x) A1A1A1 Substitute 3 M1 Correct working A1 ¾ A1 A1 A1 A1 A1 A1 this is missing a factor in the numerator A1 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 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

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 bx is the power to which b is raised to get x 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"

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) Also 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 applications: 15G: #1-3last col,4 (Trig functions) QB #1

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) In some physical applications, the derivatives have important meanings, and thus, special names:...or... Nothing conceptually new here, except for recognizing the derivatives of the functions involved and properly executing chain rules, product rules, and quotient rules. 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.