on the shoulders of giants a course in single variable calculus - smith & mcleland

[...]... each allowable value of the independent variable To each value of the independent variable in the domain, we get one and only one value of the dependent variable ! ! In the next chapter, we shall formalise these ideas into the concept of a function, one of the most important ideas in mathematics EXERCISES 1.6 1 The pressure on the hull of a submarine at a depth is Explain how we can regard either of. .. Kingsford -Smith several times before landing Plot a graph of the distance of the airplane from Melbourne against time from the moment of takeoff to the moment of landing 6 A rectangle of height is inscribed in a circle of radius Find an expression for the area of the rectangle Plot a graph of this area as a function of and decide from your graph at what value of the area of the rectangle is a maximum... to the problem of nding the range of a function As we have mentioned earlier, nding the range of a function can be difcult In the case of rational functions with quadratic terms, we can sometimes use our knowledge of quadratic equations to nd the range A rational function is one that can be expressed as the quotient of two polynomials 26 FUNCTIONS EXAMPLE 2.17 Find the range and natural domain of the. ..   The (natural) domain of the function is and it can be shown that the range is also We can plot the graph either by plotting points by hand and joining them or, more conveniently, by using a graphics package such as Mathematica The graph of the function is shown in Figure 2.3 h h i A diagram such as Figure 2.3 may give a good idea of the appearance of the graph, but it does not tell us certain critical... natural domain and to be a solution to a practical problem, a rule must have the property that every physically reasonable value of the independent variable is in the natural domain It is sometimes useful to consider a rule in which the set of allowable values of the input number is smaller than the natural domain The new rule is called a restriction of the original rule and such restrictions may have properties... Displacement of a moving body As we have mentioned earlier, the small circle centred at the point indicates that the point at its centrein this case is excluded from the graph The physical interpretation of such a graph seconds, the body has no position, because there is no point on the axis is that at the time corresponding to the point seconds This behaviour is another example of lack of continuity and... commands 2 The GRAPHS OF FUNCTIONS 19 2 Plotting a graph over a part of its domain can lead to deceptive conclusions about the nature of the graph in its entire domain By selecting a suitable interval, obtain the following plots for and In each case, the dotted plot is the graph of What would you have concluded had you plotted only one of these graphs?  C - +  - + Y 3 Here is a table of values...  H - C   To sketch the graph, it is easier to let Mathematica do all the work of evaluating function values and plotting the curve The result is shown in Figure 2.5 and are not part of the graph They are called vertical asymptotes The dotted lines and we had to give Mathematica special instructions in order to include them on the diagram  E C B 24 FUNCTIONS 30 20 10 -4 -3 -2 -1 1 2 3 -1 0 -2 0... 2.5: A Mathematica plot of the graph of 2   g f 2 d E 2 C 2  Functions dened by algebraic formulas arise in some simple applications, but in most applications the functions we obtain as solutions have to be dened in other ways We give an example of such a function, whose rule of evaluation may seem strange, articially contrived and of no practical use In fact the rule is very useful in applications... close together, the straight line segments joining them are very short, and the overall impression we get from looking at the graph is that a curve has been drawn 2.3.1 Plotting graphs with Mathematica The instruction for plotting graphs with Mathematica is Plot The essential things that Mathematica needs to know are the function to be plotted, the independent variable and the range of values for the 18 . each value of the independent variable in the domain, we get one and only one value of the dependent variable. In the next chapter, we shall formalise these ideas into the concept of a function ,. independent variable and the other as the dependent variable. Suggest a reasonable domain when is the independent variable. 2. Hypothetical data values for the Tower of Terror are given in the table. independent variable will be called the domain of the rule. There has to be a procedure which enables us to calculate a value of the dependent variable for each allowable value of the independent variable. To