Calculus 1 with Dr. Janet Semester 1, 2021/22 1.1 What is Calculus? Syllabus • Arithmetic, algebra, geometry, etc. are useful for describing quantities that are not changing or moving. 1. Functions, Limits & Continuity • But we live in a world full of change! 2. Differentiation • Calculus gives us tools to describe change. 3. Applications of Differentiation Examples: 4. Integration 5. Applications of Integration 3 Calculus is the best way to describe most of the 'laws of nature' as well as many relationships in finance, engineering and other fields. Chapter 1 Functions, Limits and Continuity 1.1 1.2 1.3 1.4 1.5 1.6 1.7 What is Calculus? Straight Lines. Equations of Lines Functions and Graphs New Functions from Old Functions. Inverse Functions Parametric Curves Definition of a Limit. One-sided Limits Laws of Limits. Evaluating Limits. The Squeeze Theorem 1.8 Limits Involving Infinity 1.9 Continuity 1.10 The Intermediate Value Theorem 2 "Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast coffee prices will rise, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas." James Stewart 4 Example 1 1.2 Straight Lines. Equations of Coordinates and Graphs Lines Draw line segments with slope a) 3, b) 5, c) 1/2, d) -1. O is the origin Ox is the x-axis Oy is the y-axis (x, y) are the coordinates of a point y = x2 The graph of an equation is the set of all points (x, y) whose coordinates satisfy the equation. 5 The SLOPE of a Straight Line Example 2 Consider any two points (x1, y1) and (x2, y2) on a straight line segment. On the interval [x1, x2], is the change in x is the change in y. Find the slope of the graph a) on the interval (2, 5) The slope (or gradient) of the line is b) on (0, 2) • E.g. if m = 5 then Dy = 5 Dx, so for every unit increase in x, y increases by 5 units. • m is a constant, characteristic of the line segment. • m tells us the rate of change of y with respect to x. http://www.mathwarehouse.com/algebra/linear_equation/interactive-slope.php +slope gif c) at x = 6 6 8 The Equation of a Straight Line Example 3 • Suppose a straight line with slope m crosses the y -axis at y = c. We call c the y-intercept. • For any two points on the line, • Setting (x1, y1) = (0, c) and letting (x2, y2) be a Sketch the following graphs: (a) y = x + 2 (c) y = 1 – x (b) y = 2x – 6 (d) 2y = x + 2 general point (x2, y2) = (x, y), we get and so This is the most common way of writing the equation of a straight line. It is called the slope-intercept form. The slope-intercept form is very convenient for graphsketching. slope y y = 3x y=x+1 Point-Slope Form For a line with slope m passing through point (x1, y1): y=x y=x 2 1 1 1 -1 OTHER FORMS The equation of a straight line can also be rearranged or written in other ways, for example: intercept y 3 11 9 x y=x–1 Two Point Form For a line passing through points (x1, y1) and (x2, y2): x -1 (These results follow directly from y=-x 10 ) 12 Example 4 1.3 Functions and Graphs Find the equation of the straight line passing through points (2, 0) and (0, 3). 1.3.1 Functions A function arises when one quantity depends on another. E.g. the height H of a child varies with age t. the cost C of mailing a parcel depends on its mass m. the area A of a circle depends on the radius r. Given the value of x, there is a rule which determines the value of f. We say f is a function of x. It is like a machine: 13 Practical Application x is called an independent variable. f(x) is a dependent variable. (It depends on x.) 15 Functions are often expressed by formulae. Example 5 On a certain day, the temperature of air at ground level was 20 ºC and the temperature at a height of 1 km was 10 ºC. Assume temperature varies linearly with height. a) Sketch a graph of the temperature T (in ºC) as a function of height h in kilometres. b) Find the equation of the line. c) What is the slope? What are its units? What does it mean? d) Find the temperature at a height of 2.5 km. Example 6 Given , find: a) b) c) 14 16 Definition A function f is a rule that assigns to each element in some set D(f) exactly one element f(x) in a set R(f). The element f(x) is called the value of f at x. Many functions can be represented by their graph. The graph of a function f is the graph y = f(x). It can also be visualized as an arrow diagram: But not every graph or equation represents a function! The domain D is the set of values x can take, the range R is the set of values f(x) can take. If not explicitly given, D(f) is the set of numbers for which f(x) makes sense. To be a function, each x must correspond to a single value of y = f(x). 17 Example 7 19 Vertical Line Test. A curve in the xy-plane is the graph of a function of x if and only if any vertical line intersects the curve not more than once. State the domain and range of the given functions. a) f(x) = x2 + 3 Yes! No! b) Example 8 Sketch the graphs (a) y = x2, (b) y2 = x. State whether or not each curve represents a function of x. c) h(x) = 2 + 3 sin(πx) 18 20 Example 10 Representing Functions A box with an open top is made from a rectangular piece of card, 15 cm 20 cm, by cutting out squares of side length x at each corner, then folding up the sides, as shown in the figures. Find a formula for the volume of the box as a function of x. A function can generally be represented in one or more of the following four ways: (1) a verbal description (2) a table of values (3) a graph (4) a formula You need to be able to move between these forms. 21 23 Example 9 Functions and Mathematical Modelling a) Sketch an approximate graph of your height H as a function of your age t. In many practical situations, data does not fit a formula exactly, but we can use an approximate formula to ‘model’ the data. When we plot this data, we find it lies approximately on a straight line CO2 level (ppm) b) Find a formula for the area A of a circle as a function of the circumference l. For example, the table shows the CO2 level measured at a certain place 1980 – 2002. 22 year 24 So we could assume a linear model for this data. - We could find the equation of the straight line through the end points. - Then use our equation to predict the 2021 CO2 level, etc.. C = 1.545t - 2721 This is an example of mathematical modelling. real problem formulate maths model A polynomial of degree 1 has form f(x) = mx + c so is a linear function. A polynomial of degree 2, f(x) = ax2 + bx + c, is called a quadratic function. A polynomial of degree 3 is called a cubic function. Example Sketches of four polynomials are shown below. What degree do you think each has? solve maths solution interpret real prediction test Here, data was modelled with a linear function. Sometimes other functional forms will be appropriate. Models are never absolutely accurate but a good model yields predictions close to reality. 25 1.3.2 Some Common Functions We will revise some common classes of functions. You should be able to sketch these types of functions quickly and know their basic properties. 27 POWER FUNCTIONS have the form a is a constant. where You should know the graphs of common functions such as: y = x3 POLYNOMIALS A polynomial is a function of the form y = x2 where n is a non-negative integer and the numbers an are constants. The numbers an are called coefficients. The value of the highest power, n, is the degree of the polynomial 26 28 TRIGONOMETRIC FUNCTIONS • You should know the sine (sin), cosine (cos) and tangent (tan) functions EXPONENTIAL FUNCTIONS have the form x is the exponent (or power or index) a is the base • Also The most common exponential function (often called the exponential function) is f(x) = ex. e is an irrational number called the exponential constant, e = 2.7182818…. (Its importance will become clearer later!) • In calculus, USE RADIANS unless told otherwise. • Complete the table: Graphs sin q q 0 . cos q tan q y = ex You should know the graphs of a) y = ex (exponential growth) p/6 b) y = e-x (exponential decay) p/4 p/3 p/2 y = e-x 29 31 Trigonometric functions are periodic. • sin x, cos x have period 2p, e.g. sin x = sin(x + 2p) • sin wx has period T = 2p/w, LOGARITHMIC FUNCTIONS If x = ay then y = loga x. This is a logarithmic function. a is again called the base. Graphs: If no base is given, log x should be understood to mean log10 x (log to the base 10). -p y = sin x 1 p 0 But in calculus we almost always natural logs, notated ln, which are logs to the base e. That is ln x = loge x. 2p -1 y = cos x y = tan x -p 0 p Graphs You should know the graph y = ln x 2p 30 32 1.3.3 Piecewise Functions & Symmetry Extra note: CIRCLES PIECEWISE FUNCTIONS A circle of radius r centred at (a,b) has equation A piecewise function is defined by different formulae in different parts of its domain. Two common examples are: Note that a circle cannot be described by writing a single function of x or y. (Why not?) 1) The Modulus Function |x| is called the modulus or absolute value of x. However we can write functions for the upper half of the circle and the lower half We have 2) A Step Function You should be able to sketch circles from their equations. You may need to first rearrange an equation into the standard circle form using the technique of 'completing the square'. 33 On graphs, indicates that the end point is included, indicates that the end point is not included. Example 11 Example 12 Sketch the graph of the equation and describe it in words. The table below gives the cost C of mailing a parcel as a function of its mass m. Write a formula for C(m) and sketch the graph of the function. 34 Mass of Parcel Cost (USD) Up to 100 g 1.25 100 to 250 g 2.30 250 to 500 g 4.10 500 to 1000 g 6.90 35 36 Example 13 Example 15 a) Sketch the graph of the function Give examples of even and odd functions. Draw their graphs. b) Write a formula for the function g. c) State the value of i) f(3) ii) g(5) Symmetry An even function satisfies An odd function satisfies 39 37 fe(-x) = fe(x) fo(-x) = – fo(x) Note: The graph of an even function is symmetric with respect to reflection in the y-axis. The graph of an odd function is symmetric with respect to rotation by 180° about the origin. Example 14 Show that f(x) = x3 – 1/x is an odd function. It is easily proved that: 38 • Any sum of two or more even functions is even • Any sum of two or more odd functions is odd • For products, even × even = even odd × odd = even odd × even = odd Example 16 1.4 New Functions from Old Sketch a) Functions 1.4.1 New Graphs from Old Graphs Suppose we know the graph of a certain function. We can quickly obtain the graphs of some related functions by some simple transformations. b) Investigation Exercise Plot the following graphs. What patterns do you notice? 1. a) y = x2, b) y = x2 + 3, c) y = (x – 3)2. 2. a) b) c) http://www.meta-calculator.com/online/ 41 TRANSLATIONS For a function f(x) and positive constant c, 43 Example 17 to obtain the graph of Figure A is the graph of f(x) = x2. What is the equation of graph B? y = f(x) + c, shift the graph of y = f(x) UP by c units y = f(x) – c, shift the graph of y = f(x) DOWN c units y = f(x + c), shift the graph of y = f(x) LEFT c units y = f(x – c), shift the graph of y = f(x) RIGHT c units A 42 B 44 Investigation Exercise Example 20 The graph of f(x) is shown. Match the Plot the following graphs. What patterns do you notice? other graphs with their equations: 1. a) y = sin x, b) y = 3 sin x, c) y = sin 2x. 2. a) b) c) STRETCHES To obtain the graph of y = cf(x), stretch y = f(x) vertically by a factor c y = 2f(x) y = f(2x) y = f(x) y = f(cx), compress y = f(x) horizontally by a factor c 47 45 Example 21 Sketch: (a) y = 1 – sin x , REFLECTIONS (b) y = |sin x| To obtain y = – f(x), reflect y = f(x) in the x-axis To obtain y = f(–x), reflect y = f(x) in the y-axis Example 19 Sketch: a) y = – x2 , b) Note that y = |f(x)| means 46 So where f(x) is positive, the graph is unchanged. Where f(x) is negative the graph is reflected in the x-axis (to become positive). 48 1.4.2 Combinations and Compositions of Functions Compositions of Functions Let f and g be functions with domains A and B respectively. These functions can be combined or composed to make new functions. Suppose and By substitution, This procedure is called composition. The new function is called the composition or composite of f and g, denoted f ० g. Combinations of Functions Algebraic operations on f and g are defined as follows: (f+g)(x) = f(x)+ g(x) with domain A B (f ० g)(x) = f(g(x)) (f – g)(x) = f(x) – g(x) with domain A B (fg)(x) = f(x)g(x) with domain A B (f /g)(x) = f(x)/g(x) with domain A B {x: g(x) 0}. Addition and subtraction of functions can also be done graphically. f ० g is defined whenever both f and g are defined. I.e. Its domain is the set of all x in the domain of g such that g(x) is in the domain of f. Note: In general f ० g g ० f 49 51 Example 22 Example 23 Let Let a) State the domains of f and g. Find a) f ० g , b) g ० f , c) (f ० g ० f )(0) . b) Find f + g and its domain. c) Find f / g and its domain. 50 52 One-to-One Functions 1.4.3 Inverse Functions • We know that if y is a function of x then for every x there is exactly one value of y = f(x) (see slide 19-20). Remember a function can be thought of as a machine: • If it is also true that for every y there is exactly one value of x, then f(x) is called a one-to-one function. Examples y=x y = x2 y is NOT a function of x y is a function of x y is a function of x but is NOT one-to-one and is one-to-one Q: Can we have another machine which does the reverse process? f(x) ? x ? A: Yes if the original function is one-to-one. The ‘reverse’ function is called the inverse function. 53 Definition A function f is called one-to-one if it never takes the same value twice. That is, f(x1) ≠ f(x2) whenever x1≠ x2. Horizontal Line Test A function is one-to-one if and only if no horizontal line intersects its graph more than once. 55 Definition Let f be a one-to-one function with domain A and range B. Then its inverse function, f –1, is defined by for any y in B, and has domain B and range A. x f(x) Example 24 Are the following functions one-to-one? a) y = sin x b) y = x3 + 1 f -1 x Notes 1. f –1 is a special symbol for the inverse. The -1 is NOT an exponent. f –1(x) [f(x)] –1 = 1/ f(x). 2. 54 56 Finding an Inverse Function Graphs of Inverse Functions To find the inverse of a given function f(x): • If f maps a onto b, then f –1 maps b onto a. • So if the graph of f includes (a, b) then the graph of f –1 includes (b, a). 1. Write y = f(x). 2. Solve the equation to find x in terms of y. 3. To express f –1 as a function of x, interchange x and y. This gives y = f –1(x). • Point (b, a) is obtained from (a, b) by reflecting in the line y = x. Example 25 a) Find the inverse of the function f(x) = x2 + 3, x ≥ 0. • So the graph f –1 is obtained by reflecting the graph f in the line y = x. Your answers to Ex25 should illustrate this! 59 57 Example 25, cont. Non-one-to-one functions and Inverses b) Find the inverse of the function g(x) = e x Many important functions are not one-to-one! But if we restrict the domain (as in Example 25a) we can obtain a one-to-one then find the inverse of this function. For example … Inverse Trigonometric Functions Inverse Sine Function The function is one-to-one. The inverse of this restricted sine function is denoted by sin-1 or arcsin: c) Sketch graphs of the functions f and g and their inverses. NOTE: do not confuse 58 with 60 Inverse Cosine Function is one-to-one on [0, p], so we define Inverse Tangent Function For tangent we take the interval (-p/2, p/2), and define Example 26 a) Sketch the curve x = t2 – 2t , y = t + 1. We can construct a table of values and thus plot the curve: t x y -2 -1 0 1 2 3 4 8 3 0 -1 0 3 8 -1 0 1 2 3 4 5 b) Eliminate the parameter to find a Cartesian equation for the curve in the form x = f(y). The graphs are the reflections of the original graphs in the line y = x. 61 1.5 Parametric Curves Introduction Suppose a particle moves along the curve C. C cannot be described by an equation of the form y = f(x). (Why not?) But the x- and y- coordinates of the particle are both functions of time: x= f(t) and y= g(t). t is called a parameter. C is called a parametric curve. C has parametric equations x= f(t) and y= g(t). We can also write c(t) = (f(t), g(t)). Generally, a parameter may be any quantity on which two other quantities depend. Time and angle are common parameters. 62 63 Notes • The parameter can sometimes be eliminated (as in Example 26). But this is not always possible. • The direct equation and parametric equations describe the same curve. • But the parametric equations also tell us when the particle was where, i.e. how the curve is traced. • The parameter domain can be restricted. E.g. x = t2 – 2t, y = t + 1, 0 ≤ t ≤ 4. • Parametric forms are especially useful for complicated curves which are not functions (or not one-to-one). 64 Parametric curves are easily drawn by computers and are widely used in computer-aided design (CAD). Some Common Parametrizations 1) A circle of radius R centred at the origin has Cartesian equation x2 + y2 = R2. Letting t be the angle a point makes with Ox, parametric equations to traverse the circle once anti-clockwise are: 2) The straight line segment that joins (x1, y1) and (x2, y2) can be described by the parametric equations For example, for the line segment from (1, 2) to (4, 9), we can write [Graphs drawn at https://www.desmos.com/ ] 65 Example 27 1.6 Definition of a Limit Sketch the curve with parametric equations x = sin t 67 Introduction y = sin2 t Suppose a scientist wants to know the value of a certain physical quantity at zero air pressure. In his laboratory he can produce low air pressures but he cannot achieve a perfect vacuum. What might he do? We are often interested in the value of a function f(x) when x is very close to a value x0 but not necessarily equal to x0. This requires the concept of the limit of a function. 66 68 Limits: A Working Definition One-Sided and Two-sided Limits We ask: As x gets closer and closer to x0 (but x x0), does f(x) get closer and closer to some finite number L? For the function above, we get the same answer whether we approach from above or below. This is not always the case. So we need the concept of one-sided limits. If ‘yes’, we say the limit of f(x) as x approaches x0 equals L. Written A limit from the right (x approaching x0 from above): or Equivalently: we can make the value of f(x) as close as we like to L by taking x sufficiently close to x0. Note: A limit from the left (x approaching x0 from below): depends only on the values of f(x) near x0. The two-sided limit exists if and only if both one-sided limits exist and are the same, i.e. if and only if The value of f(x0) is not relevant! f(x0) may have a different value or be undefined. 71 69 Three Examples Example 31 We will consider the following functions: (II) Sketch a graph of What is the value of , and ? These functions are not defined at x = 0. But we can look at their behaviour close to x = 0. (I) Consider f(x). Using a calculator or computer we can draw a table of values or plot the graph. It seems that 70 72 (III) The graph of is shown below. What can we say about , and Limits: Formal Definition [Optional] ? The definition given above is rather informal. More formally, the concept of a limit may be defined as follows. Definition Let f be a function that is defined on an open interval containing x0, except possibly at x0. We say if for every small quantity e > 0 there exists a d > 0 such that | f(x) – L |< e for all x satisfying 0 < | x – x0|< d. • As x 0+, 1/x gets bigger and bigger … ... and sin(1/x) continues to oscillate in the range [-1,1]. I.e. the function does not tend towards any fixed value. • This means I.e. graphically, if f(x) lies inside the horizontal strip of the width 2e around L then x lies inside the vertical strip of the width 2d around x0 (irrespective or whether or not point (x0, L) belongs to the graph of f). does not exist. • Similarly does not exist. • So also does not exist. 75 73 Example 32 Similar definitions can be written for one-sided limits. Use the given graph of the function f to state the value of the following limits. If a limit does not exist, explain why. These definitions can be used to find limits. Example 33 (Optional) Use the definition above to prove that 74 76 Example 34 1.7 Evaluating Limits. Laws of Using the theorem and laws above, find Limits. In section 1.5 we used tables and graphs to ‘guess’ limits. Then we met a formal proof but this is hard work to use! Now we will develop tools for finding limits precisely and relatively easily. 1.7.3 Limits of Elementary Functions 1.7.1 An Initial Theorem Most of the functions we meet are elementary functions: polynomials, power functions, rational functions (ratios of two polynomials), exponentials, logarithms, trigonometric and inverse trigonometric functions, and all the functions which can be obtained from these by addition, subtraction, multiplication, division and composition. E.g. From the definition of a limit, the following simple but important result can be proved: For any constants x0 and c, and 79 77 1.7.2 Laws of Limits Direct Substitution Property If f is an elementary function and x0 is in the domain of f , then So if f is elementary and x0 is in its domain, the limit can be found simply by substituting x0 into the formula for f. If x0 is not in the domain then this property cannot be used! In some cases the limit can still be found by algebraic manipulation. Other techniques will be studied in Chapter 3. Example 35 (For proofs, see textbooks.) Find the following limits: From these basic laws, further results can be derived. E.g. can be proved by repeated application of (iii) with f(x)=g(x). 78 80 1.7.4 The Squeeze Theorem (or sandwich theorem) If f(x) ≤ g(x) ≤ h(x) for all x in an open interval containing x0, except possibly at x0, and if then . If g is trapped between f and h, and if f and h have the same limit L at x0, (i.e. f and h meet at x0), then g must also have the same limit L at x0. 81 Example 36 Given 83 Example 37 , find and Use the squeeze theorem to show that 82 84 An asymptote is a straight line which a graph approaches arbitrarily close to at long distances from the origin. It may be approached in many different ways: 1.8 Limits involving Infinity A cup of hot tea is placed in a room which is air conditioned at 25 ºC. After a long time, what will the temperature of the tea be? 1.8.1 Limits at Infinity Example Consider E.g. For What happens to the value of f(x) as x becomes arbitrarily large (approaches infinity)? so as x , f(x) approaches the straight line y = 2. We say y = 2 is a horizontal asymptote of the graph of f. - Both numerator and denominator become large - But the quotient does not become large … Dividing throughout by x2, f(x) , we have (x 0). As x , 1/x2 0 so f(x) 2. So we say . 85 Limits at Infinity (Informal Definition) 1.8.2 Infinite Limits Let f be a function defined on some interval (a, ∞). Then means the value of f(x) gets closer and closer to L1 as x gets bigger and bigger. Example Consider the function Let g be a function defined on some interval (−∞, a). Then means the value of g(x) gets closer and closer to L2 as x gets more and more negative. • As x → 0+ , the value of h(x) gets bigger and bigger, without bound. • So h(x) do not approach any fixed value L. • So the limit does not exist. [x may be read as “x approaches infinity”, “x becomes infinite” or “x increases without bound”.] Graphically, such a limit corresponds to a horizontal asymptote, y = L1 or y = L2. Sketch the graph. What is ? • However it is convenient to say that [as x → 0+, h(x) “approaches infinity” or “tends to infinity”] • Similarly, it is convenient to say that • The graph of h(x)=1/x has a vertical asymptote x = 0. 88 Example 39 Infinite Limits (Informal Definition) Find the following limits: The notation means f(x) becomes larger and larger as x gets closer and closer to x0; And means f(x) becomes more and more negative as x gets closer and closer to x0. Note: Whenever a limit has the value ∞ or −∞, this means the limit does not exist. (∞ is a useful concept but is not a real number!) Where a function has an infinite limit, the graph has a vertical asymptote. I.e. if and/or then the graph y = f(x) has a vertical asymptote at x = x0. Example 38 91 Sketch the graphs of f(x) = 1/x2 and g(x) = ln x. Asymptotes – summary i) State the values of the following limits: • If limx→ f(x) = L1 and/or limx→- f(x) = L2 then y = L1 and/or y = L2 is a horizontal asymptote to the graph • If limx→a+ f(x) = ± and/or limx→a- f(x) = ± then x = a is a vertical asymptote to the graph. • Horizontal asymptotes can be identified by looking at the behaviour of the function as x → ± . • Values where a function is undefined may indicate vertical asymptotes. Example 40 Identify the horizontal and vertical asymptotes of ii) What asymptote(s) does each graph have? 90 92 Example 40 1.9 Continuity a) Consider again the graph shown. At what values of x is f discontinuous? What type of discontinuities are these? Definition A function f is continuous at x0 if . I.e. To be continuous, f(x) must satisfy three conditions: 1) f is defined on an open interval containing x0 2) exists 3) b) Consider . Is f continuous at x = 1? Graphically, f is continuous at x0 if its graph extends some distance to the right and left of the point (x0, f(x0)) and has no break at that point. 93 95 Further DEFINITIONS A function f is continuous from the right at x0 if Conversely, f is discontinuous at x0 if there is a break, or the left and right limits are not equal or do not exist. Discontinuities are classified into three types: A function f is continuous from the left at x0 if (a) Removable Discontinuities could be ‘removed’ by redefining the function at a single number. E.g. in Example 40, at x = 1 is f continuous from the left. (b) Infinite Discontinuities A function f is continuous on the open interval (a, b) if it is continuous at every interior point of the interval. (c) Jump Discontinuities A function f is continuous on a closed interval [a, b] if it is continuous on the open interval (a, b), continuous from the right at x = a and continuous from the left at x = b. 94 Graphically, a function is continuous on (a, b) if you can draw that part of the graph without lifting your pen off the paper! 96 A common use of this theorem is in locating solutions or roots* of equations: if f(x) is continuous on [a, b] and if f(a) and f(b) have opposite signs so f(a)f(b) < 0, then there must exist a number c in (a, b) such that f(x) = 0. Further Theorems If functions f and g are continuous at x0, then so are f + g, f − g, fg, and f /g (provided g(x0) 0). If g is continuous at x0 and f is continuous at g(x0), then the composite function f ◦ g is continuous at x0. Example 41 Show that the equation x4 + x2 − x − 3 = 0 has a root in the interval (1, 2). Every elementary function is continuous on its domain. The inverse of any continuous function is also continuous. (For proofs, see textbooks.) The last theorem can be established graphically: the graph of f −1 is the reflection of f in the line y = x, so if f has no break then f −1 will also have no break. 97 Preparation for Chapter 2: Questions to think about 1.9 The Intermediate Value Theorem *A root of a function f(x) is a solution to the equation f(x) = 0. Theorem If f is continuous on a finite closed interval [a, b] and if M is a real number lying between f(a) and f(b), then there exists a number c in (a, b) such that f(c) = M. I.e., in the interval [a, b], a continuous function takes on every value between f(a) and f(b) at least once. The idea is obvious graphically: if a graph starts at height f(a) and finishes at height f(b) and is continuous, it must cross the a line of height M at least once. (For formal proof, see textbooks.) 98 • What is meant by the ‘speed’ or ‘velocity’ of a moving object? • How is it calculated? • If we have a graph of distance as a function of time, how does speed relate to the graph?