微積分實驗教材之研究
交大是一所以理工為主的大學，全校除了外文系及管理科學系社會組二班之外， 所有的大一學生皆要修習微積分課程。但由於長期填鴨式教育及速食文化影響， 學生的學習興趣及效果卻不盡理想。1999 年本人訪問美國數所參與微積分改革 之學校後，深深體會，提供學生多元化的微積分學習環境並以電腦科技結合數學 活動實為必然之趨勢。 自九十學年度開始，交大的微積分課程便採取不分系、由學生自由選班上課。並 針對學生的不同需求，提供了三種課程：普通班課程、實驗班課程與榮譽班課程。 其中，實驗班的教材與普通班相同，唯不同於一般只由老師課堂教授的模式，學 生將透過電腦實驗主動學習。 這兩年本人均負責一班的實驗班，部分的教學則以本計劃所設計之problem-based 實驗活動為教材。，讓學生兩人一組在電腦室中，透過網路取得教材，實地操作 學習。 一. 活動教材之設計目標 以學習活動貫穿教材 從活動中引導實做,互相討論. --想法是"做出來"的,而非"獲得的". 強調學習過程而非僅止於學習結果. 二. 教學執行之情形 學生兩人一組在電腦室中，透過網路取得教材，實地操作學習，老師從旁指導。 三. 預期學生在電腦室中的活動觀察(observation) → 認知(identification) → 探索(exploration) →
四. 完成之教材
http://xserve.math.nctu.edu.tw/people/cpai/CalculusLab/index.htm
Lab 1 : Guessing limits Numerically ---Explore the concept of “ limit ” by graphs and numerical data.
Lab 2 : Mathematical Models --- Establish a mathematical model from given data with elementary functions such as polynomials, exponential functions.
Lab 3 : Implicit Functions and Implicit Differentiation --- Understand the concept of “ a function defined implicitly ” , visualize the idea of “ linearization “ and perform the procedure of implicit differentiation.
Lab 4 : Graphical Analysis --- What is a good representative plot of a function and how the derivatives of a function affect its graph.
Lab 5 : Area and Definite Integrals --- Start with the area problem and use the idea to formulate a definite integral.
Lab 6 : Approximation of Integrals --- Left endpoint approximation, right endpoint approximation, Midpoint rule and Simpson’s rule.
Lab 7 : Parametric Curves --- Understand the advantage of parametric descriptions of curves is that they are convenient for "combined motions." Realize that simple functions can do great graphic designs.
附註：此單元學生反應十分良好，學生作品請參考網頁：
http://xserve.math.nctu.edu.tw/people/cpai/demo/gallery/cal91.htm
Lab 8: Polar coordinates --- Be familiar with polar coordinates and explore some interesting curves defined by polar equations.
Lab 9 : Taylor Polynomials --- Explore the fact that a polynomial could be completely determined by its value and the values of its derivatives at
0
=
appropriate coefficients, approximation to the "target" polynomial improves in the sense that the two functions appear to match over a wider domain centered at 0. Further, extend this idea to approximations of a non-polynomial function.
Lab 10 : Cylinders and Quadratic Surfaces --- Explore the graphs of cylinders and quadratic surfaces by their traces. Also, discover the interesting shapes that members of family of surfaces
2 2
cy bxy ax
z = + + can take, by observing how the shape of the surface evolves as we vary the constants.
Lab 11 : Cylindrical and Spherical Coordinates --- Be familiar with cylindrical and spherical coordinates and explore some interesting surfaces parametrized by cylindrical or spherical coordinates.
Lab 12 : Limits of Multivariable Functions --- Understand the concept of “the limit of a two variable function” by level curves and graphs.
Lab 13 : Parametric Representations of Surfaces --- Represent a given surface with suitable parametric equations and identify the grid curves.
Lab 14 : Critical Points and Contour Plots --- Predict the location of the critical points of a two variable function f by its level curves and whether f has a saddle point or a local maximum or a local minimum at each of those points. Find the critical points of f by
two-dimensional Newton’s method.
Lab 15 : Changes of Coordinates --- Investigate how a transformation can do to a region in R2 and realize the “ Jacobian” of a transformation as “ change-in-area factor” for it.
四. 學生的反應
A. 贊成 (1) 可和同學討論，運用課堂上所學解決問題，定義及其運用可更了解。 (2) 可以了解圖形或者可以知道計算的思考路線。 (3) 可以快速的得到結果，而無須以繁瑣的步驟處理。 (4) 可以利用電腦軟體玩一些有趣的東西，提升學習興趣。 (5) 自己動手做，比較能學到東西，加深印象，且不會睡著。 (6) 可用電腦實做出比較難想像的東西。 (7) 可以主動參與學習，自己找尋問題與答案。不會和在一般課堂上一樣只 單方面的教學，無互動。 (8) 我們可以利用電腦的繪圖能力，更佳的了解問題。 (9) 免除處理繁雜計算和作圖的時間，讓人可以更專注於解決問題的方向。 (10) 除了簡單的題目可以手算，大多數的題目都要使用電腦。 (11) 觀念會強，方法會學不少，有價值。 (12) 比一般死氣沉沉的上課方式好玩。 (14) 利用電腦軟體輔助教學可以提高學習效率。 (15) 可以加強空間的概念，由電腦軟體展示圖形，並提供較深刻之理解。 (16) 可以用動畫的方式使學生明白圖形如何隨著某些因素而改變，更加了解 圖形的含意。 17.透過實際操作，可以對函數圖形或是微積分的基本原理有更深的了解。 B.不贊成 (1) 習慣有老師在講解，自己看的話有些都看不出來。 (2) 因為 Lab 佔去部分上課時間，對於課本的內容就會比較不熟，希望有方 法改進。 (3) 利用 Maple 做計算並不會提升自己的數學能力，考試時也不能使用 Maple。 C.其他建議 (1) Lab 內容可以再活潑一點。 (2) 第一次接觸 Lab，覺得滿難的，希望有多一點時間讓大家和教授討論。 (3) 一些較少用或較難的 Maple 指令教學可以放在網頁上，以供忘記時查詢。 (4) 兩個人一組的好處是可以互相討論。但是往往最後會一人做一部分，學 習到的東西就比較少。不分組自己做自己的話，雖然一開始會怕，但是 會強迫自己去學。 (5) 建議不要強迫分組，自己找伙伴，增加學習興趣。 (6) 每次的工作量似乎有點多，希望採小組工作。 (7) 建議以後的學生要買一本 Maple 使用手冊，如此使用 Maple 會較順手。 因為題目不會做都是因為指令不會寫或看不懂，上課說的指令根本不
夠。 (8) 要先上課再上 Lab，否則英文看不懂根本無法了解題意。 (9) 因為我的電腦有問題，無法安裝 Maple，做 Lab 作業時都必須向別人借 電腦，所以除了作業上的指令會用之外，沒有機會看它的設明檔或其他指 令，覺得可惜！ ◎ 印象最深刻的單元
(1) Lab 7： Parametric Curves
a. 圖形很有趣，尤其是最後一個圖。
b. 可看出參數函數圖形上的點隨著參數的改變而移動的情形。 c. 畫圖讓我們想了很久，可是很好玩，很有成就感。
d. 自己動手去設計自己喜愛的圖形，相當有新鮮感。 e. 讓我第一次體會到數學的應用，如畫圖。
(2) Lab 8： Polar Coordinates
a. 可以自由創作出令人意想不到的圖形，很好玩。 b. 起初對極座標的意義不甚了解，但經過 Maple 圖形的輔助，讓我有進一 步的認識。 c. 做了很久，解決了全部的問題，才知道牛頓真厲害，可用極座標加上參 數函數解釋行星運動的現象。 詳細活動內容如下:
Module 1
Guessing limits Numerically
Purpose:
Explore the concept of " limit " by graphs and numerical data.
We write
and say " the limit of f(x), as x approaches a , equals L. "
If we can make the values of f(x) arbitarily close to L (as close to L as we like) by taking
x to be sufficiently close to a, but not equal to a.
In the module, we are going to have fun exploring some interesting limits, such as
, ,
, .
We write
If we can make the values of f( x ) arbitarily close to L (as close to L as we like) by taking x to be sufficiently close to a, but not equal to a .
Part I
We are interested in estimating . 1. Consider the function and plot the graph of
.
> f :=x ->(1-cos(x))/x ^2; plot(f(x),x=-1..1);
> f(0.0);
2. What is the domain of f ?
3. Let's look at the values of f (x) for x < 0 . > for n from 1 to 6
do # This is the beginning of a "do loop".
x:=-1/2 ^n: # Let .
print(evalf(x),evalf(f(x))); # Print the values of x and f (x).
od: # This is the end of our "do loop". 4. Now let's look at some values of f (x) for x > 0.
> for n from 1 to 6 do x:=1/2 ^n: print(evalf(x),evalf(f(x))); od: Remarks:
1. Different rates of convergence can be achieved by
replacing by or .
2. At the end of the do loops in the above code, Maple
will think that n = 6 and x = ± . (You can check this by entering the commands n; and x; after each loop.) This is important to know since if, subsequent to the appropriate do loop, you wanted to reuse n or
x as a variable then you would have to redefine it as
a variable using the command n := 'n' or the command x := 'x'.
> n:='n'; x:='x';
5. On the basis of these data, do you think
exists? If so, what do you think it is ( to 4 decimal places of accuracy)? Justify your answer.
Part II
> restart; # Clear Maple's memory.
1. Define the function f(x) = , and plot the graph of f for x in [-1, 1].
2. Evaluate f(x) for x = 0.1, 0.09, 0.08, 0.07, ..., 0.01.
3. Evaluate f(x) for x = - 0.1, - 0.09, - 0.08, - 0.07, ..., - 0.01.
4. On the basis of these data, do you think
exists ? Justify your answer.
Part III
1. Define the function g(x) = , and plot the graph of g for x in [-2, 2].
3. Evaluate g(x) for x = 2, 2/5, 2/9, 2/13,..., 2/25.
4. What can be said about the behavior of g(x) ? Do
you think exists? Justify your
answer.
Part IV
Explore the functions and h(x) = . Do you
think exists? If so, what do you think it is? Justify your answer.
Module 2
Mathematical Models
Contents
Purpose:
Establish a mathematical model with elementary functions
such as polynomials, exponential functions.
Part I Linear Model
Table Shown below lists the average carbon dioxide level in the atmosphere, measured in parts per million at Mauna Lao Observatory from 1972 to 1990.
Year level (in ppm) 1972 327.3 1974 330.0 1976 332.0 1978 335.3 1980 338.5 1982 341.0 1984 344.3 1986 347.0 1988 351.3 1990 354.0
To enter this data in Maple, we define a list for each column and then "zip" the lists together to make the list of pairs, carbondata.
>Years:=[1972,1974,1976,1978,1980,1982,1984,1986,1988,1990]; co2:=[327.3,330.0,332.0,335.3,338.5,341.0,344.3,347.0,351.3,354.0]; co2data:=zip((x,y)->[x,y],Years,co2);
Edit and use the plot command below to generate a scatter plot of the data.
The style and symbol entries are called "options." You can use plot options to enhance your graphs in a variety of ways. The general format for plot options is
> plot(data, option1, option2, option3, ...);
You can specify the x and y ranges:
x = xmin..xmax y = ymin..ymax
If you use either or both of these options, they must come before other options.
You can set the color of plotted points:
color = red (or green , blue , yellow , violet , etc.)
You can label your axes:
labels = [`Year`, `ppm`]
(Note backward quote, often found above the Tab key.) See ?plot , ?plot,options for more details.
Complete and enter your enhanced plot command below.
> plot(co2data, x=1970..1995, y=320..360, style=point, symbol=circle, labels = [`Year`, `ppm`]);
1. Does the data points appear to lie close to a straight line ? If so, find the equation of the fitting line and explain your fitting procedure here.
The display command in plots package used with several plots will plot them all on the same graph. Complete the following commands to plot your line and the data points together.
> with(plots):
fitline:=plot(???, t=1970..1995, y=320..360, color=blue):
dataplot:=plot(co2data, x=1970..1995, y=320..360, style=point, symbol=circle, labels = [`Year`,
`ppm`]):
display(fitline,dataplot);
2. Does the line look approximately like the data plot ? If not, rework last step.
Maple has a built-in routine for fitting a line to a data set. In the stats package is a fit package that has a command called
leastsquare . If the variables in this
command are specified as [x,y] , then the output for the fitted line is of the form y = b + ax . Put your cursor in the line below, and press Enter to construct the "least squares" fitted line.
> with(stats):
Use copy and paste to define the equation above.
> y1 := t->???;
Next, we include the graph of this least squares line with the other two graphs.
> fitCurve := plot(y1(t), t=1970..1995, color=green): display(dataplot, fitline, fitCurve);
3. Which line fits better ?
4. Predict the level in 1992.
5. According to your model, when will the level exceed 400 parts per million ?
Part II Quadratic model
A ball is dropped from a tower, 450 meters above the ground, and its height h above the ground is recorded at 1-second intervals in the table below.
Time (seconds) Height ( meters) 0 450 1 445 2 431 3 408 4 375 5 332
6 279
7 216
8 143
9 61
1. Generate a scatter plot of the data.
2. Observe that a linear model is inappropriate. Does the data points may lie on a parabola? If so, try to find a parabola that fits the data.
3. Use your model to predict the time at which the ball hits the ground.
Part III Exponential model
World Population in the 20th Century Year Population (millions) 1900 1650 1910 1750 1920 1860 1930 2070 1940 2300 1950 2520 1960 3020 1970 3700 1980 4450 1990 5300 1996 5770
> Y:=[ ??? ]: P:=[ ??? ]:
datP:=zip((a,b)->[a,b], Y, P):
plot(datP, style=point, symbol=circle, color=blue);
There is a semilog command in the plots package called logplot, which works in much the same way as plot, but it does logarithmic (base 10) scaling of the vertical axis.
> with(plots):
logdatP:=logplot(datP, style=point, symbol=circle):
display(logdatP);
2. Does the data points in the semilog plot above look like a straight line ? Can you conclude from this that an exponential model should fit the population data ?
3. Find the equation of the line that fits the semilog plot. (Note that Maple's name for the base-10 logarithm is log10 . Also recall evalf if you want to see numerical values.)
4. Find your population model here and compare it with the population data. Explain why or why not the model you get is a good one.
5. Predict the size of the world population in the year of 2001.
Implicit Functions and Implicit Differentiation
Purpose:
Explore the concept of " a function defined implicitly ", visualize the idea of " linearization " and perform the procedure of implicit differentiation.
Part I
A function can be described either explicitly -- for example,
or
or, in general, y = f(x) . Some functions, however, are defined implicitly by a relation between x and y , such as
x2 + y2 = 25 or x3 + y3 = 3xy .
In some case, it is possible to solve such an equation for y as an explicit function (or several functions) of x. For instance, if we solve x2 + y2 = 25 for y,
> solve(x^2+y^2=25, y);
two functions determined by the equation x2 + y2 = 25 are
are the upper and lower semicircles of the circle x2 + y2 = 25.
The Maple command implicitplot is used for plotting equations. > with(plots):
implicitplot(x^2+y^2=25,x=-5..5,y=-5..5,scaling=co plot(sqrt(25-x^2),x=-5..5,scaling=constrained); plot(-sqrt(25-x^2),x=-5..5,scaling=constrained);
Another example is the folium of Descartes ("folium" means leaf), which is given by the equation x3 + y3 = 3xy. It is difficult to solve this equation for
y explicitly as a function of x by hand. (A computer algebra system has no trouble, but the expressions it obtains are very complicated. If you are really curious, try it!) Here is its graph :
> eq := x^3+y^3=3*x*y;
implicitplot(eq, x=-3..3, y=-3..3, grid=[50,50], scaling=constrained);
Note that this plot contains a loop, which cannot be described globally as the graph of one function y = y(x). However, the plot is the graph of some function near most points. For example, the lower piece of the loop over the interval [-1,1] is the graph of a function y(x). Finding formula for y(x), we need to solve the equation x3 + y3 = 3xy for y in terms of x. This is difficult since this equation involves a cubic. It is possible to find numerical values of y(x) at specific values of x. For example, the values of y at x = 1.5 can be found by using the Maple command fsolve .
> x:=1.5;
fsolve(eq, y, y=1..2);
A. Verify that (1.5,1.5) is on the curve.
A plot over a small range that limits the range of x and y also reveals that the plot satisfies the vertical line test near x = 1.5. Hence, it is a graph of a function.
> x:='x':
implicitplot(eq, x=1..1.75, y=1.25..1.75, scaling=con
Over a very small plot range, the graph looks like a straight line.
B. Do you think that the folium of Descartes has a tangent line at (1.5,1.5)? If so, what is the equation of the tangent line? Justify your answer.
Implicit Differentiation is the procedure used to find the derivative of
an implicitly defined function:
Step 1. Differentiate both sides of the equation with respect to x. ( by viewing y as a function y(x) of x ).
Step 2. Solve the resulting equation for y' ( or ).
The following sequence of commands used for implicit differentiation will be applied to the circle x2 + y2 = 25, but this sequence of commands also applies equally to other implicitly defined expressions.
> x:='x':
eq1:= x^2+y^2=25; subs(y=y(x), %);
> diff(%,x);
solve(%, diff(y(x), x));
The symbol ( diff(y(x),x) ) stands for derivative of y with respect to x.
C. Verify the formula for y' obtained above by differentiating the two
functions and .
D. Find the equation of the tangent line to the circle x2 + y2 = 25 at
( ).
E. Find all the points at which the formula for y' obtained above does not apply. Does the circle have tangent lines at those points?
F. Use the method of implicit differentiation to find the tangent line to the
G. Does the curve have a tangent line at ( ) ? Does the curve have a tangent line at (0,0)? Justify your answers.
Part II
Consider the curve with equation 2y3 + y2 - y5 = x4 - 2x3 + x2. A. Graph this curve and describe what the curve looks like.
B. At what point does this curve have horizontal tangent lines? Justify your answer.
C. Are there any points at which this curve have vertical tangent lines? Justify your answer.
Module 4
Graphical Analysis
Purpose:
Understand what is a good representative plot of a function and how the derivatives of a function affect its graph.
What Does f ' Say about f ?
Play with the animation below and observe how the d function affects the shape of its graph.
回到第一張
<= => >>
On the part of the graph of f which is colored red, the tangent lines have negative slope and so f '( x ) < 0. While on the part of the graph of
f which is colored blue, the tangent lines have positive slope and so f '( x ) > 0. It appears that f decreases when f '( x ) < 0 and increases when
f '( x ) > 0.
If the graph of f lies above all the tangent lines on an interval I, then it is called concave upward on I . If the graph of f lies below all the
tangent lines on an interval I , then it is called concave downward on I . A point P on a curve is called an inflection point if the curve changes from concave upward to concave downward or from concave downward to concave upward at P .
The figure above shows the graphs of two increasing functions, in the graph on the left the curve lies above the tangents, so it is concave upward. In the graph on the right the curve lies below the tangents, so it is concave downward.
回到第一張
<= => >>
Notice that the interval on which the graph of f is colored red, f is concave upward; while the interval on which the graph of f is colored blue, f is concave downward. Do you see how the first and second derivatives help to determine the intervals of concavity and inflection points?
Good representative plots of functions try to exhibit all the changes in shape of the graph and give a strong flavor of the global scale behavior.
A. Plot for x in [0, a] , where a is chosen to be large enough to see the rising and falling of the curve.
> plot(x^3/exp(x),x=0..10);
B. Factor the derivative to find the exact turning point x at which the curve changes direction, and explain why the curve cannot change direction at any other point.
> diff(x^3/exp(x), x); factor(%);
C. Explore functions and as what you have done in A, B.
D. Given a positive number r , factor the derivative of to explain why
the curve first goes up as x advances from 0 and grows until x reaches a point a after which the curve goes down. Find the exact value of the turning point x in terms of r .
E. How does the results above reflect the fact that in the global scale as x
Part II
Consider the function .
Plot the function and its derivative together. > restart;
with(plots):
f:=x->(x^7-58*x^2+8)/(2*x^6+11):
plot([f(x),D(f)(x)], x=-5..5, color=[red,blue], thickne
A. From the graph above, find the intervals of increase and the intervals of decrease of f(x). Verify your answer by factoring the derivative of f(x).
B. Determine the maximum and minimum values of for
x in [-1, 4].
C. Describe how the first derivative tells the concavity of the graph of f(x).
D. Plot f(x) and its second derivative together. Describe how the signs of the second derivative reflect the concavity of the graph of f(x).
E. How does f(x) behave as x approaches ∞ and as x approaches -∞?
or
.
F. Find all the asymptotes of the graph of f(x).
G. Does f(x) have the maximum and minimum values for all x in R ? Justify your answer.
Part III
Plot the graph of the function over [-6, 6] and discuss the important aspects of the function such as the intervals of increase or decrease, local maximum and minimum values, concavity and points of inflection, and asymptotes.
Part IV
Consider f(x) = 2x3 + cx2 + 2x.
A. Plot f(x) for different values of c.
B. Use the command animate in the plots package to create an animation of f(x).
> with(plots):
animate(2*x^3+a*x^2+2*x, x=-10..10, a=-10..10, fra 'view=[-10..10, - 40..100]');
To play an animation you must first select it by clicking on it. Then choose Play from the Animation menu.
C. Describe in words how the graph of f(x) varies as c changes and confirm your answer with the help of calculus.
Summary
A. Why does a good representative plot of a function normally include all points at which its derivative is 0 ?
B. Comment on these statements:
1) If f '(a) = 0, then the plot of f is guaranteed to have a crest or a dip at x
= a.
2) If the plot of f has a crest or dip at (a, f(a)), then it is automatic that f '(a) = 0.
C. Describe how the first derivative tells the concavity of a function?
D. What do you think the sign of f '' tells you about the concavity of the plot of f ?
E. If f has an inflection point at (a, f(a)), and f ''(a) exists. Is it always that f ''(a) = 0 ?
On the other hand, does f ''(a) = 0 guarantee that f has an inflection point at (a, f(a)) ?
Area and Definite Integrals
Purpose:
Start with the area problem and use the idea to formulate a definite integral. 積分問題的起源即是求面積的問題，基本概念十分類似阿基米德的窮盡法 (相關連結一, 相關連結二)。我們想要求出由紅色函數圖形、x 軸、x = 0 與 x = 2 所圍出區域的面積，如下圖所示。 首先將 0 到 2 的區間分割成 n 個子區間，左圖長方形的高度為該子 區間函數的最小值，所以長方形的面積總和必小於所求之面積。相反地， 右圖長方形的高度則為子區間內函數的最大值，所以長方形面積的總和也 會大於所求之面積。隨著 n 越來越大，左圖長方形面積的總和 L 會越來 越大，而右圖長方形面積總和 U 會越來越小，兩者之間的差 E 會趨近
零。也就是說，兩者會同時趨近所求的區域面積。
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In this module we start with the area problem and use it to formulate the idea of a definite integral.
We begin by attempting to find the area of the region that lies under the
Suppose we divide the region into four strips by drawing the vertical lines
, , and .
We can approximate each strip by a rectangle whose base is the same as the strip and whose height is the same as the right edge of the strip.
Here we use Maple's rightbox command to visualize the process of approximating the area under a curve.
In order to use Maple's leftbox (or rightbox ) command, one has to load the
student package first.
> with(student):
rightbox(f(x),x=1..3,4);
Moreover, we can compute the sum of these rectangles with the help of Maple's sum command.
> dx:=(3-1)/4;
Sum(f(1+i*dx)*dx,i=1..4); evalf(%);
> sum(f(1+i*dx)*dx,i=1..4);
In general, we can divide the interval [ ] into subintervals of eqaul
length. The area under the graph of is approximated by the sum
of the areas of rectangles where the base of a rectangle is one of the
subintervals and the height is the value of the function at the right or left endpoint of the subinterval.
The leftbox command to illustrate rectangles that approximates area under
> eftbox(f(x),x=1..3,4);
1. What is the sum of those rectangles illustrated above?
Divide the interval into subintervals of eqaul length, and let and
be the sums of the rectangles with the heights of the right endpoints and left endpoints, respectively.
2. Find and with , , , . What do you find out?
The following commands compute the rightsum for general , and the
limit as goes to infinity. > dx:=(3-1)/n;
right_area:=Sum(f(1+i*dx)*dx,i=1..n); right_limit:=Limit(right_area,n=infinity); value(%);
3. Find the limit of the leftsums as goes to infinity. Does this limit agree with the one of the rightsums?
4. What do you think the area under the graph of is ? Why ?
5. Approximating the area by the sum of the areas of rectangles where
the base of a rectangle is one of the subintervals and the height is
the value of the function at any point , instead of the right or left endpoint of the subinterval, d o you get the same answer as in (4) ?
6. Can you figure out the area under the graph of over the
interval [ ] ? over the interval [ ] for any ?
7. Use the same idea to find the area of the region bounded by the graph of
If is a continuous function defined on [ ], we divide the interval
[ ] into subinterval of equal length . We let
, , ,...., be the endpints of these subintervals, and
let be a point in the subinterval [ ].
Then the definite integral of from to is
The sum is called a Riemann sum .
Remarks :
A. In the definition above can be chosen to be the right endpoint or the
left endpoint of the subinterval [ ].
B. We can view the definite integral to be the " signed area" of over the interval [ ] as follows:
We can use Maple commands int or Int to Integrate functions or expressions.
For example, to integrate over [ ] one should enter > Int(f(x),x=a..b);
value(%);
or
> int(f(x),x=a..b);
The antideivative of f (or indefinite integral) can also be evaluated. > Int(f(x),x);
> F(x):=int(f(x),x);
Note: Maple does not insert the constant of integration.
8. What is the relation between and ?
Approximation of Integrals
Purpose:
Experiment with four different ways: Left endpoint
approximation, right endpoint approximation, Midpoint rule and Simpson's rule, of approximating integrals, and find out which one is most efficient.
There are situations in which it is impossible to find the exact value of a definite integral. For examples :
and .
In these cases we need to approximate values of these definite integrals.
Recall that the definite integral is defined as a limit of Riemann sums. If we divide into n subintervals ,
be any point in the i -th subinterval , then is a
good approximation for when n is sufficiently large.
By choosing ci to be the left endpoint or the right endpoint of
, we have the left endpoint approximation or right endpoint
approximation, respectively.
If we choose ci to be the midpoint of , then we have the
Midpoint Rule approximation , as shown below.
To compromize the difference between the values of left endpoints and right endpoints in each subintervals, we use the sum of areas of the trapezoids lies above the subintervals . This is called Trapezoidal Rule. The idea is shown below :
Simpson's Rule
Another rule for approximation integration results from using the parabolas. As before ,we divide [a, b] into n subintervals ,
( ), of equal length , but
this time we assume that n is an even number. Then on each
consecutive pair of intervals [xi-1, xi] and , we approximate the curve y = f(x) by a parabola passing through the points ,
, and as shown below. Let Sn denote the sum of the areas of these approximating parabolas.
In this module, we are going to explore these from methods and find out which one is most efficient.
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Module 6
Approximation of Integrals
There are situations in which it is impossible to find the exact value of a definite integral. For examples :
and .
In these cases we need to approximate values of these definite integrals.
Recall that the definite integral is defined as a limit of Riemann sums. If we divide [a, b] into n subintervals [xi-1, xi], i = 1, ... , n (x0 = a, xn = b), of
equal length , let ci be any point in the i -th subinterval [xi-1, xi],
then is a good approximation for when n is
sufficiently large.
By choosing ci to be the left endpoint or the right endpoint of [xi-1, xi], we have the left endpoint approximation or right endpoint approximation, respectively.
Divide [a, b] into n subintervals , let Ln and Rn be the left endpoint
approximation and the right endpoint approximation for , respectively.
1. Approximate by L10 and R10 and estimate the error.
Justify your answer.
If we choose ci to be the midpoint of [xi-1, xi], then we have the Midpoint
Rule approximation , as shown below.
Another approximation, called the Trapezoidal Rule. We use the sum of areas of the trapezoids lies above the subintervals . The idea is shown below :
Divide [a, b] into n subintervals , let Mn and Tn be the Midpoint Rule approximation and the Trapezoidal Rule approximation, respectively.
2. Show that .
3. Let on [1, 2], for n = 5, 10, 20, compute Ln, Rn, Mn and Tn by defining Ln, Rn, Mn and Tn as functions of n.
As the command leftsum and rightsum, the commands middlesum,
trapezoid can be found in Maple student package, you may have to
apply the command evalf to get the numerical values. Check your answers by using those commands.
> int(g(x),x=1..2);
Now we can make a table of the errors of the approximation above by the following commands.
> N:=3:
A:=matrix(N+1,5,(Row,Col)->0):
A[1,1]:='n': A[1,2]:='E[L]': A[1,3]:='E[R]': A[1,4]:='E[ A[1,5]:='E[M]': for k from 1 to N do n:=2^(k-1)*5; A[k+1,1]:=n: A[k+1,2]:=0.5-evalf(L(n)): A[k+1,3]:=0.5-evalf(R(n)): A[k+1,4]:=0.5-evalf(T(n)): A[k+1,5]:=0.5-evalf(M(n)): od: eval(A);
4. What do you find out from the table above ?
Error Bounds
Suppose |f "(x)| ≤ K for a ≤ x ≤ b. If
and are the errors involved in using the
and
5. Approximate by the Trapezoidal and Midpoint Rule for n
= 10 and estimate the errors of each approximation ( i.e. and
) by the formula given above.
6. By the formula given above, how large should we take n in order to guarantee that the Midpoint Rule approximation and the Trapezoidal
Rule approximation for are accurate to 10 decimal places. Which approximation is better ?
Simpson's Rule
Another rule for approximation integration results from using the parabolas. As before ,we divide [a, b] into n subintervals [xi-1, xi], i = 1, ... , n (x0 = a, xn = b),
number. Then on each consecutive pair of intervals [xi-1, xi] and [xi, xi+1], we approximate the curve y = f(x) by a parabola passing through the points (xi-1, f(xi-1)), (xi, f(xi)), and (xi+1, f(xi+1)) as shown below. Let Sn denote the sum of the areas of these approximating parabolas.
A typical parabola y = Ax2 + Bx + C passes through three consecutive points (-h, a), (0, b) and (h, c) as shown below.
7. Find the area of the region shown above.
8. Use the result in (7) to get a formula for S6 and give a conjecture of the
formula for Sn.
9. Use Simpson's Rule with n = 6 to approximate . Check your answer with Maple command simpson . What can you say about this approximation?
Error Bound for Simpson's Rule
Suppose that |f(4)(x)| ≤ K for a ≤ x ≤ b. If is
10. By the formular given above, how large should we take n in order to
guarantee that the approximation for using Simpson's Rule is accurate to 10 decimal places?
11. Compare the results from (6) and (10). What is your conclusion?
12. Suppose f(x) is a cubic polynomial. Is the approximation of exact by using Simpson's rule? Justify your answer.
Module 7
Parametric Curves
Purpose:
Understand the advantages of parametric description of curves is that they are convenient for "combined motions." Realize that simple functions can do great graphic designs.
If a particle moves along the curve C shown below, then the x
-coordinates and y -coordinates are functions of time. So we can write x = f(t), y = g(t).
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Notice that the consecutive points marked on the curve appear at equal time intervals but not at equal distances. That is because the particle slows down and speeds up as t increases.
Suppose that x are y are both given as functions of a third variable t (called parameter) by the equations
x = f(t), y = g(t)
(called parametric equations). Each value of t determines a point (x, y), which we can plot in a coordinate plane. As t varies, the point (x, y) = (f(t), g(t)) varies and traces out a curve C, which is called a parametric curve If f(t) and g(t) are defined for all t in [a, b], then (a, f(a)) is called the initial
point of C and (b, f(b)) is called the final point of C. Imagine that a
particle moving along the curve C, we can interpret t as time and (x, y) = (f(t), g(t)) as the position of a particle at time t. We say C is closed if initial point and final point of C are the same.
Take a close look at the following animations, you should be able to tell the difference of a curve, which is a set of points, and a parametric curve, in which the points are traced in a particular way.
C1: x = cos(t), y = sin(t), where t in [0, 2π].
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C3: x = cos(-t), y = sin(-t), where t in [0, 2π].
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Parametric curves are used not only to represent letters and other symbols on the laser printer but also in graphic design. Here is an interesting graphic design using parametric curves by 電物 94 級的歐迪 興同學. Try to make one yourself !
Take a close look at the following animations, you should be able to tell the difference of a curve, which is a set of points, and a parametric curve, in which the points are traced in a particular way.
> with(plots):
animatecurve([cos(t), sin(t), t=0..2*Pi], scaling=con
> animatecurve([cos(2*t), sin(2*t), t=0..2*Pi] ,scaling numpoints=50);
> animatecurve([cos(-t), sin(-t), t=0..2*Pi], scaling=co
1. What are the differences between these three parametric curves
: , for all in [ ] ,
: , for all in [ ] ,
: , for all in [ ] ?
To plot the parametric equations
where is in [ ]
first we define the functions f and g and type the command plot( [f(t), g(t),
t=a..b] ) .
2. Verify that an an ellipse centered at ( ) with horizontal axes radius and vertical axes radius can be parametrized by
and . Plot an arc of an ellipse with the given parametric equations.
> a:=2: b:=5:
The following commands plots a small rectangle with viewing rectangle [0, 2] by [0, 2].
> u[1]:=1+.5*t: v[1]:=1:
line1:=plot([u[1], v[1], t=0..1], 0..2,0..2): u[2]:=1: v[2]:=1-.5*t:
line2:=plot([u[2], v[2], t=0..1], 0..2,0..2): u[3]:=1+.5*t: v[3]:=.5:
line3:=plot([u[3],v[3],t=0..1], 0..2,0..2): u[4]:=1.5: v[4]:=1-.5*t:
line4:=plot([u[4], v[4], t=0..1], 0..2,0..2):
display([line1, line2, line3, line4], scaling=constrai
3. Plot a parallelogram with vertices ( ), ( ), ( ) and
( ) with viewing rectangle [ ] by [ ].
4. Plot the three parts of a capital letter B, using a straight line segment and either two semicircles, or two semi-ellipses.
5. Repeat problem (4), but with the letter moved .5 unit above the x -axis and .5 unit to the right of the y -axis, and with its size doubled.
One of the advantages of parametric description of curves is that they are convenient for "combined motions." This lets us plot curves obtained by adding parametric motions. Here is an example :
The curve traced out by a point P on the circumference of a circle of radius r as the circle rolls with a constant angular speed ω along a straight line is called
Click here to see how to derive the parametric equations for the cycloid and the commands for the animation above.
6. Using the graph above to show that t he cycloid is given by the parametric equations
and
Parametric curves of the form , , with
in , are known as Lissajous curves. Here, is the parameter
and , , and are constants which determine the particular curve in the family. Here are two examples:
> plot([2*cos(3*t), 7*sin(2*t), t=0..2*Pi]);
> plot([cos(5*t), 2*sin(3*t), t=0..2*Pi]);
Trace around these two curves until you understand how they are related to the equations which define them. Then ask Maple to plot one or two other Lissajous curves. See if you can guess what each one will look like before you plot it.
7. For fixed and , describe how the values of and affect the shape of the corresponding Lissajous curve.
8. For fixed and , consider the Lissajous curves with ,
for some integer and is in [ ]. Are the curves closed ? Describe how the shape of the curve changes as varies. 9. For fixed and , consider the Lissajous curves with ,
some integer . Are the curves still closed ? What if is irrational ? Can you explain why ?
When is an even number, the curve looks quite different.
> plot([cos(4*t),sin(5*t), t=0..2*Pi]);
10. Do you see what happened in the last curve? Explain it.
Here are some interesting parametric curves, explore how the shape of the curve varies for different values of and .
> m:=3: n:=2: plot([t^m,t^n,t=-2..2]); > m:=2: n:=5: plot([t+2*sin(m*t),t+2*cos(n*t),t=0..2*Pi],scaling=co > m:=2: n:=3: plot([t+sin(m*t),t+cos(n*t),t=0..2*Pi]);
11. Design an interesting picture with plots of parametric curves.
Module 8
Purpose:
Be familiar with polar coordinates and explore some interesting curves defined by polar equations.
A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. So far we have being using
Cartesian coordinates, which are directed distance from two
perpendicular axes. Here we describe a coordinate system introduced by Newton, called polar coordinate system.
We choose a point in the plane called the pole (or the origin) and is labeled O. Then we draw a half-line starting from O called polar axis. This axis usually drawn horizontally to the right and corresponds to the positive x-axis in Cartesian coordinates.
If P is any other point in the plane, let r be the distance from P to O and let θ be the angle between the polar axis and the line OP as shown below. Then the point P is represented by the ordered pair (r, θ) and r, θ are called the polar coordinates of P.
We use the convention that an angle is positive if measured in the counterclockwise direction from the polar axis and negative in the clockwise direction. If P = O, then r = 0 and we agree that (0, θ) represents the pole for any value of θ.
We extend the meaning of polar coordinates (r, θ) to the case in which
r is negative by agreeing that the points (-r, θ) and (r, θ) lie on the same
line through O and the same distance |r| from O, but on the opposite sides of O. Notice that (-r, θ) and (r, θ + π) represent the same point.
The connection between polar and Cartesian coordinates:
If the point P has Cartesian coordinates (x, y) and polar coordinates (r, θ), then
x = r cos(θ) y = r sin(θ)
r2 = x2 + y2 tan(θ) = y / x
Grids in Cartesian coordinates :
Grids in polar coordinates :
A. Convert the point ( ), ( ) from polar to Cartesian coordinates.
B. Find polar coordinates ( ), where and < , of
the points given by the Cartesian coordinates ( ) and
( ), and find polar coordinates ( ), where and
< , of the points given by the Cartesian coordinates
( ) and ( ).
The graph of a polar equation , consists of all points that have at least one polar representation ( ) whose coordinates satisfy the equation.
We can use the plotcommand the same way as for parametric equations by specifying the coordinates to be polar.
> plot([2, theta, theta=0..2*Pi], coords=polar, scaling=constrained);
> plot([r, Pi/3, r=0..1], coords=polar, scaling=constra
C. What curve is represented by the polar equation ? What curve is
represented by the polar equation ?
We can also use a command in Maple's plots package, called polarplot, to plot polar equations of the form . We start by loading the plots
package.
D. Plot the curve and find a Cartesian equation for this curve.
> with(plots):
f:=theta->2*cos(theta);
polarplot(f(theta),theta=0..2*Pi,scaling=constraine
The animation below will give you a better picture of how the curve goes.
> animatecurve([f(theta)*cos(theta), f(theta)*sin(thet
theta=0..2*Pi], scaling=constrained, numpoints=20
E. Plot the curve for different integer and describe how
the curve varies with .
F. Plot the curve for different value of and observe
how the curve varies with . Find the transitional values of where the basic shape of the curve changes.
The animation below probably will help.
> animate([1+b*sin(t), t, t=0..2*Pi], b=-2..2, coords=p scaling=constrained);
G. Graph the curve by finding a polar equation for the curve.
H. Graph the two ellipses and ,
find the vertices and foci of each of them respectively. What is the relation between these two ellipses.
I. Graph the parabola given in polar form by and find the Cartesian coordinate expression for this parabola.
Remark : Here you will find that the command polarplot will not give you a
good picture. ( Why? )
In order to get a good plot, you should get a proper parametric equation of the curve, then plot the parametic curve.
> polarplot(1/(1-sin(theta)), theta=-Pi/2..Pi/2, numpoi > x:=???;
y:=???;
plot([x(t), y(t), t=0..2*Pi], -5..5, -5..5);
A polar equation of the form
or or or
represents a conic section with eccentricity . The conic section is an ellipse if , a parabola if , or a hyperbola if .
> animate([cos(t)/(1+e*sin(t)), sin(t)/(1+e*sin(t)), t=0.
e=-1.5..1.5, view=[-10..10, -10..10], scaling=constra numpoints=200, frames=50);
Module 9
Taylor Polynomials
Purpose:
Explore the fact that a polynomial could be completely
determined by its value and the values of its derivatives at x = 0. Find out that as terms of higher degree are added with the
appropriate coefficients, approximation to the "target" polynomial improves in the sense that the two functions appear to match over a wider domain centered at 0. Further, extend this idea to
approximations of a non-polynomial function.
A polynomial can be completely determined by its value and the values of its derivatives at x = 0.
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Can we extend this idea to approximations of a nonpolynomial function? Of course, we can't expect to get an exact fit in finite steps. The idea of polynomial approximation is very powerful in later work, and we shall study it in the context of familiar functions like ex and sin(x) in this module.
Part 1. Polynomial Coefficients
The following figure shows the graph of a fourth-degree polynomial, that is,
a function of the form
We are given the following information about and its derivatives at
Our objective is to determine the coefficients , from this information.
1. How is related to , for , , , , ,
respectively. Enter your answer here:
2. One at a time, determine appropriate values for the coefficients , ...,
, and replace the 's in the following definitions. With each new
definition of , the plot will be automatic when you enter the
following block of commands. Compare with the graph of in the web page.
> a[0]:=0; a[1]:=0; a[2]:=0; a[3]:=0; a[4]:=0;
p:=x->a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+a[4]*x^4; plot(p(x),x=-2..10,y=-600..200,thickness=2);
Part 2. Taylor Polynomials
In Part 1 we saw that a polynomial could be completely determined by its value and the values of its derivatives at x = 0. Further, we found that, as we added terms of higher degree (with the appropriate coefficients), our
approximation to the "target" polynomial improved in the sense that the two functions appeared to match over a wider domain centered at 0 . In this part we extend this idea to approximations of a nonpolynomial function. Thus, we don't
expect to get an exact fit in five steps -- or ever.
The idea of polynomial approximation is very powerful in later work, and it makes sense to study it first in the context of familiar functions.
1. How do we know that the exponential function is not a polynomial ? State at least one property of this function that could not be a property of any polynomial.
2. Let , find a polynomial
of degree 4 with the
property that , for , 1 , 2, 3, 4 .
Enter functions and coefficients here, and plot and together. > restart;
with(plots):
x:='x':f:=x->exp(x);
a[0]:=?; a[1]:=?; a[2]:=?; a[3]:=?; a[4]:=?; p:=x->a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+a[4]*x^4; plot1:=plot(f(x),x=-3..3,y=-2..16,thickness=2, color=
plot2:=plot(p(x),x=-3..3,y=-2..16,thickness=2, color display(plot1,plot2);
3. Plot the error function and describe the extent to which
does and does not approximate .
Let's try to find better approximations of with higher-degree polynomials. We look for an nth-degree polynomial
+ ... + such that
, for all , 1, 2,.., n. The resulting polynomial is called
the nth-degree Taylor polynomial of centered at .
4. How is related to the k th-derivatives of , for , 1, 2,.., n ? Enter your answer here:
5. Enter the general formula for and plot and
together. Compare the approximation here and that in (2), which one
looks better ? Try with larger 's, what do you find out ?
6. Find the general formula for the nth-degree Taylor polynomial centered
at for the function . Graph together with the
Taylor polynomials of degree 2, 4, 6, 8 and comment on how well they
approximate .
In general, given a n -defferentiable function ,the polynomial
+ ... +
where for all , 1, 2,.., n ,is called the nth-degree
Taylor polynomial of centered at .
7. Find the general formula for .
8. Find the nth-degree Taylor polynomial centered at for the function
.
9. Suppose that for all .
What is the nth-degree Taylor polynomial of at ?
Module 10
Cylinders and Quadratic Surfaces
Purpose:
Explore the graphs of cylinders and quadratic surfaces by their traces. Discover the interesting shapes that members of family of
surfaces z = a x2 + b x y + c y2 can take, by observing how the shape of the surface evolves as we vary the constants.
In this project we investigate two types of surfaces --- cylinders and quadratic surfaces.
A cylinder is a surface that consists of all lines (called rulings ) that are parallel to a given line and pass through a given plane curve. The animation below shows how the surface is formed by taking the
parabola z = x2 in the xz -plane and moving in the direction of the y -axis.
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A quadratic surface is the graph of a second-degree equation in three variables x, y and z. The most general such equation is
where A, B, C, ... , J are constants. There are six basic shapes : (1) ellipsoid
(3) Hyperbolic Paraboloid
(4) Cone
(6) Hyerboloid of Two sheets
We will also discover the interesting shapes that members of family of surfaces z = a x2 + b y2 + c x y can take, by observing how the shape of the surface evolves as we vary the constants.
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PART I Cylinders
A cylinder is a surface that consists of all lines (called rulings ) that are parallel to a given line and pass through a given plane curve.
You may use the Maple command plot3d to plot an explicit function
or use the command implicitplot3d in the plots package to plot a surface defined by the equation .
1. Graph the parabolic cylinder .
> plot3d(y^2, x=-4..4, y=-4..4, view=0..16, axes=norm
implicitplot3d(z=y^2, x=-4..4, y=-4..4, z=0..16, grid= axes=normal);
2. Graph the parabolic cylinder and compare with the one in 1. 3. Graph and compare the circular cylinders and
.
4. Here is a graph of a cylinder. Observe the graph, make a good guess of its equation and justify your answer.
PART II Quadratic Surfaces
A quadratic surface is the graph of a second-degree equation in three variables x, y and z. The most general such equation is
A x2 + b y2 + C z2 + D x y + E y z + F x z + G x + H y + I z + J = 0
where A, B, C, ... , J are constants. There are six basic shapes :
In order to sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces (or cross-sections) of the surface. The following animation shows the vertical traces in of the surface
1. Graph the ellipsoid and identify the horizontal traces and vertical traces.
2. Graph the elliptic paraboloid and identify the horizontal traces and vertical traces.
3. Graph the hyperbolic paraboloid and identify the horizontal traces and vertical traces.
4. Graph the hyperboloid of one sheet and identify the horizontal traces and vertical traces.
5. Graph the hyperboloid of two sheet and
PART III Families of Surfaces
1. Investigate the family of surfaces . In particular, you should determine the transitional values of for which the surface changes from one type of quadratic surface to another. Justify your answer.
2. Investigate the family of surfaces . In particular,
you should determine the transitional values of , and for which the surface changes from one type of quadric surface to another. Justify your answer.
Module 11
Cylindrical and Spherical Coordinates
Purpose:
Be familiar with cylindrical and spherical coordinates and explore some interesting surfaces parametrized by cylindrical or spherical coordinates.
PART I Cylindrical coordinates
In cylindrical coordinate system, a point in three-dimentional space is
represented by the ordered triple ( ), where and are polar
coordinates of the projection of (as shown below) onto the xy -plane and
is the directed distance from the xy -plane to .
To convert from cylindrical to rectanglular coordinates we use the equations
whereas to convert from rectanglular to cylindrical coordinates we use the equations
1. Find the rectangular coordinates of the point with cylindrical coordinates
( ).
2. Find the cylindrical coordinates of the point with rectangular coordinates
( ).
We can plot a surface with equation in cylindrical coordinates using the Maple command plot3d with the option specifying cylindrical coordinates :
3. What is the surface with equation in cylindrical coordinates ? > plot3d(1,theta=0..2*Pi,z=0..1,coords=cylindrical);
We can also plot a surface given by parametric equations in cylindrical coordinates using the Maple command plot3d with the option specifying cylindrical coordinates :
> plot3d([r,Pi/4,z],r=0..6,z=0..4,axes=normal,scaling=
coords=cylindrical);
4. What is the surface with equation in cylindrical coordinates ?
5. What is the surface with equation in cylindrical coordinates ? > plot3d([r, theta, 1], r=0..6, theta=0..2*Pi, view=0..6,
scaling=constrained, coords=cylindrical);
6. Plot the surface with equation in cylindrical coordinates.
7. Plot the surface with equation in cylindrical coordinates and find the equation of the surface in rectangular coordinates .
8. Plot the surface with equation in cylindrical coordinates.
PART II Spherical coordinates
The spherical coordinates ( ) of a point in space are shown
below, where is the distance from the origin to , is the same
angle as in cylindrical coordinates, and is the angle between the positive z
-axis and the line segment joining the origin and . Note that and
The relationship between rectangular and spherical coordinates is given by the equations:
and
1. Find the rectangular coordinates of the point ( ) given in spherical coordinates.
2. Find the spherical coordinates of the point ( ) given in rectangular coordinates.
We can plot a surface with equation in cylindrical coordinates using the Maple command plot3d with the option specifying cylindrical
cylindrical coordinates using the Maple command plot3d with the option specifying cylindrical coordinates.
> plot3d(1, theta=0..2*Pi, phi=0..Pi, coords=spherica scaling=constrained);
> plot3d([1, theta, phi], theta=0..2*Pi, phi=0..Pi, coord scaling=constrained);
3. Find the equation in rectangular coordinates of the surface given by the
equation in spherical coordinates.
4. Plot the surface with equation in spherical coordinates. What is the surface?
5. Plot the surface with equation in spherical coordinates. What is the surface?
6. Plot the surface with equation in spherical coordinates.
7. Plot the surface with equation in spherical coordinates.
8. Plot the surface with equation in spherical
coordinates. Find the equation of the surface in rectangular coordinates and identify the surface.
1. Draw a picture of the solid that remains when a whole of radius 2 is drilled through the center of a sphere of radius 3.
2. Members of the family of surfaces given in spherical coordinates by the equation
have been suggested as models for tumors and have been called bumpy
spheres and wrinkled spheres .
Investigate this family of surfaces, assuming that and are
positive integers. What roles do the values of and play in the shape of the surfaces?
Module 12
Limits of multivariable Functions
Purpose:
Understand the concept of " the limit of a two variable function " by level curves and graphs of the function.
Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b). Then we say
if for every ε > 0 there is a corresponding number δ > 0 such that |f(x, y) - L| < ε
whenever (x, y) in D and 0 < < δ.
If , then f(x, y) approaches L as (x, y) approaches (a, b) along any path C in D. In other words, if f(x, y)
approaches L1 as (x, y) approaches (a, b) along a path C1 in D and f(x, y)
approaches L2 as (x, y) approaches (a, b) along a path C2 in D, where ,
where L1 ≠ L2, then does not exist.
Let f be a function of two variables whose domain D includes points arbitrarily close to ( a , b ). Then we say
whenever ( x , y ) in D and 0 < < .
If , then f ( x , y ) approaches L as ( x , y ) approaches ( a , b ) along any path C in D . In other words, if f ( x , y )
approaches as ( x , y ) approaches ( ) along a path in D and f ( x , y ) approaches as ( ) approaches ( a , b ) along a path in
1. Let , plot together with the paths you picked as the graph above and determine whether
exists. Explain your answer.
2. Let , plot together with the paths you
picked as the graph above and determine whether
exists. Explain your answer.
The fact that and
does not exist can be detected using contour plots.
3. Execute the following command and execute this commands again with 0.01 replaced by 0.001; does the pattern seem to change ?
>with(plots):
grid=[40,40]);
4. How does those graphs support the conclusion that
exists ?
5. How do the contour plots support the conclusion that
does not exist ?
6. Based on the contour plots, do you think that exists ? Explain your answer.
If ( ) are polar coordinates of the point ( ) with , note that
as . Hence
exists if and only if exists.
7. Let , graph and use polar coordinates to
determine whether exists .
8. Let and be positive integers. Find all the values of and
such that exists.
Module 13
Parametric representations of Surfaces
Purpose:
Represent a given surface with suitable parametric equations and identify the grid curves.
So far we have learned to describe surfaces in three dimensional space as :
level sets for functions of three variables, graphs of equations in three variables.
For example, the surface below
can be described as
the graph of the function f(x, y) = x2 + y2 , the graph of the equation z = x2 + y2, or
a level set of the function f(x, y, z) = x2 + y2 - z
Unfortunately, some surfaces are hard to be represented in any of those ways, for example, the torus shown below.
