Chapter 4

October 24 2016

Chapter 4 Applications of Derivatives 4.1 Increasing and Decreasing Functions. Maxima and Minima - we begin to justify the effort we have spent on learning how to calculate derivatives - our first applications are based on the interpretation of the derivative as the slope of the tangent line to a curve - the purpose of this work is to enable us to use the derivative as a tool for quickly discovering the most important features of a function and sketching its graph - this art of curve sketching is essential in the physical sciences - it is also one of the most useful skills that calculus can provide for those who need to use mathematics in their study of economics or biology or psychology - the sign of the derivative tells us whether a function is increasing or decreasing - a function f(x) is said to be increasing on a certain interval of the x-axis if x1 < x2 implies f(x1) < f(x2) - a function f(x) is increasing on any interval in which f'(x) > 0, and it is decreasing on any interval in which f'(x) < 0 - this is geometrically evident if we keep in mind the fact that a straight line points upwards and to the right if its slope is positive - it is clear that a smooth curve can make the transition from rising to falling only by passing over a peak where the slope is zero - at such points we have a maximum value of the function - we locate these values by finding the critical points of a function, which are the solutions to f'(x) = 0 - that is, we force the tangent to be horizontal by equating the derivative to zero, and we then solve the equation f'(x) = 0 to discover where this happens - the critical points are the values of the function at these points, such as f(x1) - the critical point is not necessarily either a maximum or a minimum, it could be a momentary flattening - these are relative (or local) maximum or minimum values compared only with nearby points on the curve - the zeros of a function, where f(x) = 0, where the curve crosses the x-axis, are always valuable aids in curve sketching when they can be found, but finding them can be quite difficult - as a matter of fact, we sometimes sketch the graph of a function to help us guess the approximate location of its zeros - these examples, as well as our past experience, suggest a few informal rules that are useful in sketching the graph of a function f(x) - if possible and convenient, we should determine: 1. the critical points of f(x), where f'(x) = 0 2. the critical values of f(x), the value of f(x) 3. the sign of f'(x) between critical points 4. the zeros of f(x) 5. the behaviour of f(x) as x > inf and as x > -inf 6. the behaviour of f(x) near points at which the function is not defined - maxima and minima can occur in three ways not covered by the preceding discussion: at endpoints, cusps and corners - in seeking the maxima and minima of functions, equate the derivatives to zero by all means, but do so carefully, keeping these three possibilities in mind as well - our statements about increasing and decreasing functions and maxima nad minima are supported only by geometric plausibility arguments - the statements themselves are true, but these arguments are a far cry from the proofs that would satisfy a mathematician 4.2 Concavity and Points of Inflection - one of the most distinctive features of a graph is the direction in which it curves or bends (up or down) - the sign of the second derivative gives us this information - a positive second derivative f''(x) > 0 tells us that the slope f'(x) is an increasing function of x - this means that the tangent turns counterclockwise as we move along the curve from left to right - it is said to be concave up - such a curve lies above its tangent except at the point of tangency - most curves are concave up on the same intervals and concave down on other intervals - a point at which the direction of concavity changes is called a point of inflection - if f''(x) is continuous and has opposite signs on each side of P, then it must have a zero at P itself - the search for points of inflection is mainly a matter of solving for f''(x) = 0 and checking the direction of concavity on both sides of each root - knowing that f''(x0) = 0 is not enough to guarantee that x = x0 furnishes a point of inflection - we must also know that the graph is concave up on one side of x0 and concave down on the other - the simplest function that shows this difficulty is y = f(x) = x4 - here f'(x) = 4x3 and f''(x) = 12x2 so f''(x) = 0 at x = 0 - but since f''(x) is clearly positive on both sides of the point x = 0, it is a minimum, and not a point of inflection - in searching for points of inflection, we must consider not only points at which y''=0, but also points (if there are any) at which y'' does not exist (such as divide by zeroes) - in the so-called second derivative test, a sign of the second derivative is used to decide whether a critical point furnishes a maximum or minimum value - this test is sometimes useful, but its importance is often exaggerated - we will see that in most applied problems it is clear form the context whether we have a maximum or minimum value, and no testing is necessary 4.3 Applied Maximum and Minimum Problems - among the most striking applications of calculus are those that depend on finding the maximum or minimum values of functions - practical everyday life is filled with such problems, and it is natural that mathematicians and others should find them interesting and important - a business person seeks to maximise profits and minimise costs, an engineer designing a new automobile wishes to maximise its efficiency - an airline pilot tries to minimise flight times and fuel consumption - in science, we often find that nature acts in a way that maximises or minimises a certain quantity - for example, a ray of light traverses a system of lenses along a path that minimises its total time of travel, and a flexible hanging chain assumes a shape that minimises its potential energy due to gravity - whenever we use such words as largest, smallest, most, least, best, and so on, it is a reasonable guess that some kind of maximum or minimum problem is lurking nearby - if this problem can be expressed in terms of variables and functions - which is not always possible by any means - then the methods of calculus stand ready to help us understand it and solve it - many of our examples and problems deal with geometric ideas, because maximum and minimum values often appear with particular vividness in geometric settings - find two positive numbers whose sum is 16 and whose product is as large as possible - x + y = 16 - we are asked to find the particular values of x and y that maximise their product - P = xy - our initial difficulty is that P depends on two variables, whereas our calculus of derivatives works only for functions of a single independent variable - x + y = 16 enables us to get over this difficulty, so we can express P as a function of x alone - P = 16x - x2 - this function is concave down, as verified by d2P/d2x = -2 < 0 - so the highest point is where the tangent is horizontal - dP/dx = 0, x = 8 - as these examples show, the mathematical techniques required in most maximum-minimum problems are relatively simple - the hardest part of such a problem is usually "setting it up" in a convenient form - this is the analytical, thinking part of the problem, as opposed to the computational part - we emphasise this because it is clear that calculus is unlikely to be of much value as a tool in the sciences unless one learns how to understand what a problem is about and how to translate its words into appropriate mathematical language 4.4 More Maximum-Minimum Problems. Reflection and Refraction - we continue to develop the basic ideas by means of additional examples - e.g. what dimensions will minimise the total surface area of a can - e.g. what angle does light reflect at if it travels in the shortest path 4.5 Related Rates - if a tank is being filled with water, then the water level rises - to describe how rapidly the water level rises, we speak of the rate of change of the depth dh/dt - further, the volume V of water in the tank is also changing dV/dt - similarly, any geometric or physical quantity Q that changes with time is a function of time, say Q = Q(t) and its derivative dQ/dt is the rate of change of the quantity - the problems that we now consider are based on the fact that if two changing quantities are related to one another, then their rates of change are also related - gas is being pumped into a large spherical balloon at the constant rate of 8 ft3/min, how fast is the radius r of the balloon increasing when r = 2 ft and r = 4 ft - volume is given by V = 4/3pir3, dV/dt = 8 - we want to find dr/dt - it is important to understand that V and r are both dependent variables with the time t as the underlying independent variable - with this in mind, it is natural to introduce the rate of change of V and r by differentiating with respect to t - we summarise the lessons of these examples - in solving a problem involving related rates of change, it is usually a good idea to begin by drawing a careful sketch of the situation being considered - next, add to the sketch all numerical quantities that remain fixed throughout the problem - then, denote by letters any quantities - the dependent variables - that change with time, and seek a geometric or physical relationship among these variables - finally, differentiate with respect to time t to obtain a relation among the various rates of change, and use this to determine the unknown rate 4.6 Newton's Method for Solving Equations - consider the cubic equation x3 - 3x - 5 = 0 - it is possible to solve this equation by exact methods, that is, by formulas yielding a solution such as the solutions to the quadratic equation ax2 + bx + c = 0 - however, if we need a numeric solution that is accurate only to a few decimal places, then it is more convenient to find the solution by approximation - furthermore, while formulas that yield exact solutions for equations of degree 2, 3, and 4 do exist, it is known to be impossible to solve the general equations of degree 5 or more - therefore, in order to solve a fifth-degree equation like x5 - 3x2 + 9x - 11 = 0, we would be forced to use an approximation method, since no other method is available - we let f(x) = x3 - 3x - 5, we are trying to find the root where f(x) = 0 - f(2) = -3 and f(3) = 13, it is intuitive that there is a root f(x) = 0 between these two values - we can take either of these numbers as the first approximation for this root - in general, suppose we have a first approximation x = x1 to a root r of an equation f(x) = 0 - this root is a point where the curve y = f(x) crosses the x-axis - the idea of Newton's method is to use the tangent line to the curve of the point where x = x1 as a stepping-stone to a better approximation x = x2, and so on - the slope of the first tangent line is f'(x) - if we consider this line to be determined by the points (x2,0) and (x1, f(x1)), then the slope is also - 0 - f(x1) / x2 - x1, wihch = f'(x1) - this yields x2 = x1 - f(x1)/f'(x1) - in this way our first approximation x1 leads to a second approximation x2 and so on - f(x) = x3 - 3x - 5, f'(x) = 3x2 - 3, x1 = 2 - f(x1) = -3, f'(x1) = 9 - x2 = x1 - f(x1)/f'(x1) = 2 - -3/9 = 21/3 - Newton's method is not restricted to the solution of polynomial equations, but can also be applied to any equation containing functions whose derivatives we can calculate - in some cases, the sequence of approximations produced by Newton's method may fail to converge to the desired root