**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