MATH90 CALCULUS - NOTES AND AFTERNOTES DR. JOSIP DERADO. Historical background Newton approach - from physics to calculus. Instantaneous velocity. Leibniz approach - from geometry to calculus Calculus as a bridge between physic and geometry. Together with work of Descartes and bridge between algebra and geometry makes a triangle. physics geometry algebra 2. Limits and Limit Processes Understanding the notion of it and convergence through the examples. Geometric its The sequence of regular polygons converges to a circle. Zeno s paradoxes Achilles and tortoise. Walking to the wall. Geometric series formula Discussion of the formula: a) finite sequence case: b) q = 2 or q = or q =. + q + q 2 + q 3 + = q + q + q 2 + + q n = qn+ q Real Numbers The real number in particular irrational numbers could be defined as a it of the series: d n j 0 n j Also it could be viewed geometrically as a it of sequence of nested intervals. j=n Recursive relations x 0 = x n+ = + + x n Calculate the it assuming one exists x n L. Recursive formula leads to L 2 L = 0 and L = + 5 2 as the it.this number is called Golden Ratio Number Engineering Convergence For the previous example calculate the it using Excel or Maple.
2 DR. JOSIP DERADO Continued fractions In the previous example you went from the recursive formula to quadratic equation. Now let us construct recursive formula from the given quadratic equation. For example Let L = 2. Then This yields L 2 = 2 = L 2 + L = 2 + L = L(L + ) = + L + = L = + + L x 0 = Furthermore x n+ = + + x n simplified L = + + + ++ ++ ++ ++ +L L = + 2 + 2+ 2+ 2+ 2+ 2+ 2+L Notice that 2 is represented by the repeating sequence of 2 s, which is similar to the representations of rational number in decimal expansion. The fixed point method x 0 = x n+ = cosx n Use Excel or Maple to calculate the it. Notice the it L is a solution of the equation cos(l) L = 0 Speed of convergence Do the previous example but replace cosine with sine. Notice although we know the it should be 0, the sequence converge very slowly. Compare the speed of convergence with the continued fraction examples and cosine example. Euler Formula 3. Limit Continuation e x = + x! + x2 2! + x3 3! + x4 4! + x5 5! + Evaluate e 0. up to 5 decimal places. Notice an extremely fast convergence.
AFTERNOTES 3 4. Rigorous definition of the it Start with: L is a it of x n if for every nbhd of L eventually all x n fall into the nbhd. Then make words nbhd and eventually precise. ɛ nbhd of L:= (L ɛ, L + ɛ). eventually means that for given ɛ > 0 there is an integer N ɛ such that for every n > N ɛ, x n falls into ɛ nbhd of L. Falling into means x n (L ɛ, L + ɛ) L ɛ < x n < L + ɛ x n L < ɛ. Finally the rigorous definition of the it is: The Rigorous Limit definition. Let x n, n R be a sequence of real numbers. Then L is the it of x n if for every ɛ > 0 there is an integer N ɛ so that x n L < ɛ whenever k > N ɛ 5. Continuous(Function) its Define left it at the point a denoted by x a, and right it at the point a denoted by x a +. Then define general it at point a as equal to the left it and right it. Hence the general it does exist only if both left and right it exist and they are equal. Discuss different possibilities. () left it or right it do not exist (2) left it or right it do exist but they are not equal (3) left it and right it are equal and hence the general it do exist but it is not equal to the function value f(a). (4) all four numbers left it, right it, general it and function value f(a) do exist and they are equal. Point a is referred to as singularity in cases ()-(3), in () and (2) it is an essential singularity and a removable one in(3). In case (4) we say a function is continuous at the point a. sin(x) x and others given just by graphs. x Weird example:a sinus comb sin( x ). Basic its: x = a x a x 0 x = x 0 + x = + Properties of its: x a f(x) + g(x) = x a f(x) + x a g(x) 2 x a c g(x) = c x a g(x) 3 x a f(x) g(x) = x a f(x) x a g(x) f(x) 4 x a g(x) = x a f(x) x a g(x) providing x a g(x) 0 5 Let f(x) be continuous at x a g(x) then x a f(g(x)) = f ( x a g(x) )
4 DR. JOSIP DERADO Examples to show how to use the properties: Show continuity of x, x 2, x 3 and x n for a general integer n. 2 Show continuity of λx n for a every constant λ and integer n. 3 Show continuity of all polynomials 4 Use Property 5 to show that x 2 + is continuous everywhere. 5 Without proof claim that all exponentials, trigonometric and rational functions are continuous on their domains. 5.. Infinite Arithmetic. Infinite arithmetic is a very useful tool for calculating its. However one has to be very careful how and where one applies it. Infinite arithmetic is an arithmetic which includes and deals with it as it is a regular real number according with the following rules: + = = = Let k be a positive real number + k = k = k = = = = k = k = 0 We also use the following convention. x 0 x = 0 = x 0 + x = 0 + = + Notice that there many situations when infinite arithmetic does not give a definite answer. These are called the indeterminate forms: 0 =??? 0 =??? 0 =??? 0 =??? =??? =??? 0 =??? 5.2. Techniques to compute its. Explain why the students favorite technique plugg in works and how you can use it in conjunction with infinite arithmetic. Further explain if at any moment one is faced with with one of those indeterminate forms one has to GO BACK!!! and work to change the expression to avoid the indeterminate forms. Examples one can use: x x5 + 3x 2 + x + x x5 3x 2 + x + x 4 x 4 x 2 3x 4 x x x 2 +
AFTERNOTES 5 5.3. Limits of Rational Functions. x a n x n + a n x n + + a x + a 0 b m x m + b m x m + + b x + b 0 = x x 2x 2 + 3x 2 + x + = 2 3 2x + 3x 2 + x + = 0 2x 2 + x x + = 2x 2 + x x + = a n b m n = m ± n > m 0 n < m 5.4. Asymptotes. Vertical Asymptotes: A function R(x) has a vertical asymptote at the point a if R(x) = ± or R(x) = ± x a + x a For rational function R(x) the possible candidates for vertical asymptotes are at points where the denominator is 0. So x = is a vertical asymptote. 2x 2 + x + x = Horizontal Asymptotes: A function F (x) has a horizontal asymptote y = R and/or y = L if F (x) = R or F (x) = L x x So y = 0 is a horizontal asymptote. 2x 2 + x x = 0 Oblique Asymptotes: O(x) is an oblique asymptote of F (x) if F (x) can be written as F (x) = O(x)+Q(x) and and or Let F (x) = 2x 2 + x. Since y = x 2 is an oblique asymptote. Q(x) = 0 x Q(x) = 0 x ± x x = 0
6 DR. JOSIP DERADO 5.5. Squeezing Theorem. Let f, g, h be functions of real variable such that g(x) f(x) h(x) for all x in some interval containing point a. Further let g(x) = h(x) = L x a x a Then f(x) = L x a Use the theorem and prove sin(x) = x 0 x Using the previous it compute the following its. sin(2x) x 0 3x cos(x) = 0 x 0 x 6. Tangents and Derivatives we start with the following problem: Find the tangent line on the parabola y = x 2 at point (, ). (, )on the line and the parabola = m + b b = m Since the tangent line intersects parabola at one and only one point we have x 2 = mx + ( m) x 2 mx + (m ) = 0 To have only one solution the discriminant D has to be 0. D = m 2 4(m ) = 0 m 2 4m + 4 = 0 m = 2 y = 2x Repeating the calculation for an arbitrary point (x, y) one obtains the general formula for a slope of tangent line m = 2x. This general formula is called derivative. Leibnitz & Newton computation of derivative: Idea: Instead computing the tangent line we compute slopes of secants which passes through the point at which we want to find the tangent line (x, y) and a near by point (x + h, y(x + h)). Then using the it technique, pushing h to 0, we obtain the slope of the tangent line. More precisely, y(x + h) y(x) m tan = h 0 x + h x Compute the derivative for some basic functions: Establish the Newton Power Rule. y = x 2, y = x 3, y = x n Differentiation Rules (x α ) = αx α addition rule (f + g) = f + g homogeneity property (c f) = c f
AFTERNOTES 7 product rule (f g) = f g + f g quotient rule ( f g ) = f g f g chain rule (f(g(x))) = f (g(x)) g (x) g 2 We do few examples to see how we can apply the rules. Derivatives of Constants and Linear functions Derivatives of Polynomials: d ( 7x 5 3x 4 + x 2 + 3x + ) = 35x 4 2x 3 + 2x + 3 dx Derivatives of Rational Functions Derivatives of Exponentials: In Precalculus we define number e as ( ) n e = + n n We did also derived (in Eulerian sense) Euler formula(series) for e. and consequently e = +! + 2! + 3! + + 4! + e x = + x! + x2 2! + x3 3! + +x4 4! + We derive the derivative of e x using the it definition. The derivative can be computed from the series but we leave that for the Hwk. First we show that e h = h 0 h We start by defining z = e h This implies Substituting it in the it we get = h = log( + z) e h z = h 0 h z 0 log( + z) = z 0 z = (log( + z)) z 0 ( log( + z)) z = ( ) = z 0 log( + z)) z log(e) = We are ready to calculate the derivative of e x. ) = d e x+h e x e x e h e x dx (ex ) = = = e x eh = e x e h = e x h 0 h h 0 h h 0 h h 0 }{{ h } = Now we apply chain rule to compute the derivatives of: e x, e x2, xe x2,, ax + e t Implicit derivatives Find the tangent line on curves like Inverse function derivatives: Do the log and arcus tangent example. x 2 + y 2 =, x 2 y y 3 =
8 DR. JOSIP DERADO 7. What we can learn about f from its derivative? We showed the basic relationship between f and f and the original function f. For the second derivative we have f f is f f is f f is fhas a shape of f f is fhas a shape of When f = 0 we have potentially a local extremum. The point c for which f (c) = 0 is called a critical point. All critical points are unfortunately not local extrema. The simple example is point 0 for the function f(x) = x 3. Since the derivative is 3x 2, it is equal to 0 at point 0. It turns out that 0 is an inflection point. The other problem is that many functions do not have derivative everywhere. For example x is not differentiable at x = 0(prove!), Hence we add all the points which are in domain but the function does not have a derivative at this point to the set of critical points. Once when we find all critical point we have to classify them, i.e., we have to decide which one represent local extrema and which one not. For this we can use several tools. Most commonly used are The First Derivative Test and The second derivative Test. Here is how. The First Derivative Test Check the sign of the f around a critical point. If the sign is changed then the point represents a local extremum. More precisely if the derivative changes its sign from + to - then the point is where the function attains local maximum. If the sign is changing from - to + then we have a point where function attains its local minimum. The Second Derivative Test This test is used if it is not too hard to find the second derivative. If c is a critical point and f (c) > 0 then c is where the function attains its local minimum. If c is a critical point and f (c) < 0 then c is where the function attains its local maximum. It is important to emphasize that this test tells you nothing in the case when the second derivative is 0 at c. We did several examples: () P (x) = x 3 4x + (2) R(x) = +x 2 (3) N(x) = e ( x 2 ) Discuss the family of Normal distributions (4) F (x) = e x cos(x) (5) Z(t) = te λt2