Chapter 2

February 29 2016

Chapter 2 The Derivative of a Function 2.1 What is Calculus? The Problem of Tangents - almost all the ideas and applications of Calculus revolve around two generic problems that are very easy to understand - both refer to the graph of a function y = f(x) - differential calculus - the problem of tangents - find the slope of the tangent line to the graph at point p - integral calculus - the problem of areas - calculate the area under the graph between the points x = a and x = b - the rest of the book focuses on these two problems, the ideas and techniques that have been developed for solving them, and on the applications that arise from them - the Fundamental Theorem of Calculus says in effect that the solution of the tangent problem can be used to solve the area problem - this is certainly the most important theorem in the whole of mathematics - it allows the two halves of Calculus to be weld together into a problem solving art of astonishing power and versatility - the modern concept of a tangent line was originated by Fermat around 1630. this concept is not only a reasonable statement about the geometric nature of tangents, it is also the key to a practical process for the construction of tangents - let P and Q be points on the curve. the tangent line at P can be thought of as the limiting position of the variable secant line PQ as Q slides along the curve towards P - this way of thinking about tangents is not a minor technical point in the geometry of curves, on the contrary, it is one of the three or four most fruitful ideas that any mathematician ever had 2.2 How to calculate the slope of the tangent - points PQ, slope m - msec = y1 - y0 / x1 - x0 - let x1 approach x0 so that Q approaches P by sliding along the curve - intuitively clear that the slope m of the tangent is the limiting value approached by the slope of the secant - then m = lim q>p msec = lim x1>x0 y1 - y0 / x1 - x0 - cannot solve by setting x1 = x0, would give 0/0 - must think of x1 as coming arbitrarily close to x0 but remaining distinct from it - way to solve is to use the equation of the curve - y0 = x02 y1 = x12 , msec = x1 + x0 , m = lim x1>x0 msec = 2x0 - delta notation: - shorthand for the amount of change from x1 to x0 - dx = x1 - x0 - x1 = x0 + d - msec = x12- x02 / x1 - x0 = (x0 + d)2 - x02 / dx - expand top to dx (2x0 + dx) - msec = 2x0 + dx - m = lim dx>0 msec = 2x0 - both methods resulted in the same answer - first worked by factoring and second worked by expansion - in general expanding is easier, so use the method of increments - msec = f(x0 + dx) - f(x0) / dx - then m = lim dx>0 msec - value of limit is usually denoted f'(x0), depends both on f(x) and on x0 - if f(x) = x2, then f'(x0) = 2x0 - this assumes the tangent exists at point P, that we are getting the same limiting value for both directions of approach 2.3 The Definition of the Derivative - if we now leave the geometric motivation and also drop x0 - f'(x) = lim dx>0 f(x + dx) - f(x) / dx - the limit may exist for some values of x and not exist for some others - if the limit exists at x = a, we say it is differentiable at a - a differentiable function is one that is differentiable over the entire domain - the geometric interpretation is only one interpretation, we must be prepared to consider f'(x) purely as a function - the process of actually forming the derivative f'(x) is called the differentiation of the function f(x) - this is the fundamental operation of Calculus, upon which everything else depends - in principle, we merely follow the computational instructions specified above - the instructions can be arranged into a systematic procedure: Three-Step Rule 1) write down the difference f(x + dx) - f(x) for the function, and if possible simplify to point where dx is a factor 2) divide by dx to form the difference quotient f(x + dx) - f(x) / dx, and manipulate to prepare for evaluating the limit of dx>0. Often, this is just cancelling the dx. 3) Evaluate the limit of the difference quotient as dx>0. Usually done by simple inspection - generally, derivatives are capable of telling us a great deal about the behaviour of functions and the properties of their graphs Leibniz Notation - write difference quotient f(x + dx) - f(x) / dx as dy/dx - where dy is not just any change in y, it is the specific change that results when the independent variable is changed from x to x + dx - then Leibniz wrote f'(x) as dy/dx - then the definition of the derivative becomes dy/dx = lim dx>0 dy/dx - or d f(x)/dx and d/dx f(x) - d/dx f(x) = f'(x) - the symbol d/dx can be read as the derivative with respect to x of ... - it is a single indivisible symbol - dy/dx reminds us of the whole process of forming the difference quotient dy/dx and calculating its limit as dx>0 - some fundamental formulas will be easier to remember and use with this notation - but dy/dx does not have an easy way to specify the specific point we are talking about e.g. (dy/dx) @ x = 3 2.4 Velocity and Rates of Change Velocity - concept of derivative is closely related to the problem of computing the velocity of a moving object - this made Calculus an essential tool of thought for Newton in his efforts to uncover the principles of dynamics - these ideas provide a fairly easy introduction to the general concept of rate of change - consider special case of the general velocity problem, where the object is moving along a straight line, so that the position of the point is determined by a single coordinate s - the motion is fully known if we know here the moving point is at each moment, that is, if we know the position s as a function of time t, s = f(t) - for a free falling object, it is known from many experiments that this rock falls s = 16t2 feet in t seconds - two questions can be asked about this motion 1) what is meant by velocity at a given instant 2) how to computer this instantaneous velocity - average velocity is distance / time, v ~= ds / dt, the approximation gets better as dt is smaller - then instantaneous velocity v = lim dt>0 ds/dt - so velocity is a direct and intuitive concept - however, it is also possible to see this derivative as the definition of velocity - for general motion s = f(t) - v = ds/dt = f'(t) - this instantaneous velocity can be positive or negative - speed is the absolute value of velocity - consider a projectile fired straight up with initial velocity of 128 ft / s - experimental evidence suggests that the height is s = f(t) = 128t - 16t2 - if decompose as s = 16t (8 - t) - then can see that s = 0 when t = 0 and t = 8 - so the object returns to earth 8 seconds after start up - using the derivative, the velocity at time t is - v = ds/dt = 128 - 32t - at top of the flight, the projectile is momentarily at rest: v = 0 - this happens at t = 4, s = 256, this is the maximum height - as t increases from 0 to 8, it is clear v decreases from 128 ft/s to - 128 ft/s - so v decreases by 32 ft/s every second - we call this the acceleration, it is 32 ft/s2 Rate of Change - velocity is an example of the general concept of the Rate of Change, which is basic for all sciences - for any function y = f(x), the derivative dy/dx is called the rate of change of y with respect to x - intuitively, this is the change in y that would be produced by an increase of one unit in x if the rate of change remained constant (slope) - so, velocity is simply the Rate of Change in position with respect to time - when time is the independent variable, we often just ignore it and call it simply the Rate of Change - examples of rates of change: - acceleration - the rate of change of velocity a = dv/dt - pouring water into a cone at 5 ft3/min, the rate of change in volume per time is dV/dt = 5 - economics, marginal cost is the rate of change in cost by quantity produced dC/dx - area of circle is A = pir2, the rate of change of Area with respect to the radius is dA/dr = 2pir 2.5 Limits and Continuous Functions Limits - clarifying the concept of limits - consider f(x) that is defined for values of x near a point a, but not necessarily at a itself - suppose there is a number L with the property that f(x) gets closer and closer to L as x gets closer and closer to a - then we say L is the limit of f(x) at a, that - lim x>a f(x) = L - or f(x) > L as x > a - if there is no number L with this property, then we say f(x) has no limit as x approaches a, or lim x>a f(x) does not exist - key is to understand it doesn't matter what happens to f(x) when x equals a, all that matters is the behaviour of f(x) for x's that are near a - we can quantify the limit as a relationship between the distance to x and the distance to L (epsilon-delta definition) - for each positive number epsilon there exists a delta - |f(x) - L| < epsilon such that 0 < |x - a| < delta Continuous Functions - a function is continuous if it displays similar behaviour, that is, a small change in x produces a small change in the corresponding value f(x) - f(x) is continuous at a if f(x) can be made as close as we please to f(a): - lim x>a f(x) = f(a) - requires three things: 1) a is in the domain of the function so f(a) exists 2) f(x) must have a limit as x approaches a (from both sides) 3) lim x>a f(x) must = f(a) - a continuous function is a function that is continuous over every point in its domain (can include gaps in domain) - includes all polynomials and rational functions (1/x) - we will be especially interested in functions that are continuous at intervals - a function which is differentiable at a point is continuous at that point - if it is not differentiable, it is not continuous - converse is not true, a continuous point is not necessarily differentiable (a curve meets a straight line) - use x instead of a: - lim dx > 0 f(x + dx) = f(x) or lim dx > 0 f(x + dx) - f(x) = 0 - lim dx > 0 dy = lim dx > 0 dy/dx dx = [ lim dx > 0 dy dx ] [ lim dx > 0 dx ] = dy/dx 0 = 0