I. Numbers, Points, Lines and Curves1. What is a number?Another reason to believe in 2Why are real numbers called real? Exercises2. The real number line and intervals2.1. Intervals2.2. Set notationExercises3. Sets of Points in the Plane3.1. Cartesian Coordinates3.2. Sets3.3. LinesExercises4. Functions4.1. Example: Find the domain and range of f(x) = 1/x24.2. Functions in ``real life''5. The graph of a function5.1. Vertical Line Property 5.2. Example6. Inverse functions and Implicit functions6.1. Example6.2. Another example: domain of an implicitly defined function6.3. Example: the equation alone does not determine the function6.4. Why use implicit functions?6.5. Inverse functions6.6. Examples6.7. Inverse trigonometric functionsExercisesII. Derivatives (1)7. The tangent to a curve8. An example -- tangent to a parabola9. Instantaneous velocity10. Rates of changeExercisesIII. Limits and Continuous Functions11. Informal definition of limits11.1. Example11.2. Example: substituting numbers to guess a limit11.3. Example: Substituting numbers can suggest the wrong answerExercise12. The formal, authoritative, definition of limit12.1. Show that limx33x+2=11 12.2. Show that limx1x2 = 112.3. Show that limx41/x = 1/4Exercises13. Variations on the limit theme13.1. Left and right limits13.2. Limits at infinity. 13.3. Example -- Limit of 1/x 13.4. Example -- Limit of 1/x (again) 14. Properties of the Limit15. Examples of limit computations15.1. Find limx2x215.2. Try the examples 11.2 and 11.3 using the limit properties15.3. Example -- Find limx2x 15.4. Example -- Find limx2x 15.5. Example -- The derivative of x at x=2. 15.6. Limit as x of rational functions15.7. Another example with a rational function 16. When limits fail to exist16.1. The sign function near x=0 16.2. The example of the backward sine16.3. Trying to divide by zero using a limit16.4. Using limit properties to show a limit does not exist16.5. Limits at which don't exist17. What's in a name?18. Limits and Inequalities18.1. A backward cosine sandwich19. Continuity19.1. Polynomials are continuous19.2. Rational functions are continuous19.3. Some discontinuous functions19.4. How to make functions discontinuous19.5. Sandwich in a bow tie20. Substitution in Limits20.1. Compute limx3x3-3x2+2Exercises21. Two Limits in TrigonometryExercisesIV. Derivatives (2)22. Derivatives Defined22.1. Other notations23. Direct computation of derivatives23.1. Example -- The derivative of f(x)= x2 is f'(x) = 2x 23.2. The derivative of g(x) = x is g'(x) =1 23.3. The derivative of any constant function is zero 23.4. Derivative of xn for n=1, 2, 3, … 23.5. Differentiable implies Continuous23.6. Some non-differentiable functionsExercises24. The Differentiation Rules24.1. Sum, product and quotient rules24.2. Proof of the Sum Rule24.3. Proof of the Product Rule24.4. Proof of the Quotient Rule 24.5. A shorter, but not quite perfect derivation of the Quotient Rule 24.6. Differentiating a constant multiple of a function 24.7. Picture of the Product Rule25. Differentiating powers of functions25.1. Product rule with more than one factor25.2. The Power rule 25.3. The Power Rule for Negative Integer Exponents 25.4. The Power Rule for Rational Exponents 25.5. Derivative of xn for integer n 25.6. Example -- differentiate a polynomial 25.7. Example -- differentiate a rational function25.8. Derivative of the square root Exercises26. Higher Derivatives26.1. The derivative is a function26.2. Operator notationExercises27. Differentiating Trigonometric functionsExercises28. The Chain Rule28.1. Composition of functions28.2. A real world example28.3. Statement of the Chain Rule28.4. First example28.5. Example where you really need the Chain Rule28.6. The Power Rule and the Chain Rule28.7. The volume of a growing yeast cell28.8. A more complicated example28.9. The Chain Rule and composing more than two functionsExercisesIV. Derivatives (2)“Leibniz never thought of the derivative as a limit. This does not appearuntil the work of d’Alembert.”http://www.gap-system.org/∼history/Biographies/Leibniz.htmlIn chapter II we saw two mathematical problems which led to expressions of the form00.Now that we know how to handle limits, we can state the definition of the derivative ofa function. After computing a few derivatives using the definition we will spend most ofthis section developing the differential calculus, which is a collection of rules that allowyou to compute derivatives without always having to use basic definition.22. Derivatives DefinedDefinition. 22.1. Let f be a function which is defined on some interval (c, d) and let abe some number in this interval.The derivative of the function f at a is the value of the limit(15) f0(a) = limx→af(x) − f(a)x − a.f is said to be differentiable at a if this limit exists.f is called differentiable on the interval (c, d) if it is differentiable at every point a in(c, d).22.1. Other notationsOne can substitute x = a + h in the limit (15) and let h → 0 instead of x → a. Thisgives the formula(16) f0(a) = limh→0f(a + h) − f(a)h,Often you will find this equation written with x instead of a and ∆x instead of h, whichmakes it look like this:f0(x) = lim∆x→0f(x + ∆x) − f(x)∆x.The interpretation is the same as in equation (8) from §10. The numerator f(x + ∆x) −f(x) represents the amount by which the function value of f changes if one increases itsargument x by a (small) amount ∆x. If you write y = f (x) then we can call the increasein f∆y = f(x + ∆x) − f(x),so that the derivative f0(x) isf0(x) = lim∆x→0∆y∆x.Gottfried Wilhelm von Leibniz, one of the inventors of calculus, came up with theidea that one should write this limit asdydx= lim∆x→0∆y∆x,the idea being that after letting ∆x go to zero it didn’t vanish, but instead became aninfinitely small quantity which Leibniz called “dx.” The result of increasing x by thisinfinitely small quantity dx is that y = f(x) increased by another infinitely small quantity4546dy. The ratio of these two infinitely small quantities is what we call the derivative ofy = f(x).There are no “infinitely small real numbers,” and this makes Leibniz’ notation difficultto justify. In the 20th century mathematicians have managed to create a consistent theoryof “infinitesimals” which allows you to compute with “dx and dy” as Leibniz and hiscontemporaries would have done. This theory is called “non standard analysis.” Wewon’t mention it any