The Heart of Mathematics: An Invitation to Effective Thinking
4th Edition
ISBN: 9781118156599
Author: Edward B. Burger, Michael Starbird
Publisher: Wiley, John & Sons, Incorporated
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Textbook Question
Chapter 7.3, Problem 9MS
Any right triangle. Take any right triangle. It can be broken into four similar triangles, each one having edges that are 1/2 the length of their corresponding edges in the big triangle. Use this fact to show that, using the dimension definition, any filled-in right triangle has dimension 2.
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Students have asked these similar questions
7. (a) (i) Express y=-x²-7x-15 in the form
y = −(x+p)²+q.
(ii) Hence, sketch the graph of
y=-x²-7x-15.
(b) (i) Express y = x² - 3x + 4 in the form
y = (x − p)²+q.
(ii) Hence, sketch the graph of y = x² - 3x + 4.
28
CHAPTER 1
-
(c) Suppose V is a solution to the PDE V₁ – V× = 0 and W is a solution to the PDE
W₁+2Wx = 0.
(i) Prove that both V and W are solutions to the following 2nd order PDE
Utt Utx2Uxx
= 0.
(ii) Find the general solutions to the 2nd order PDE (1) from part c(i).
(1)
Solve the following inhomogeneous wave equation with initial data.
Utt-Uxx = 2, x = R
U(x, 0) = 0
Ut(x, 0):
= COS X
Chapter 7 Solutions
The Heart of Mathematics: An Invitation to Effective Thinking
Ch. 7.1 - The incredible shrinking duck. On the Quacked...Ch. 7.1 - Multiplicity (S). In the Sierpinski Triangle,...Ch. 7.1 - Different sizes. In the fern, find three reduced...Ch. 7.1 - Blooming broccoli. In the bunch of broccoli, find...Ch. 7.1 - Prob. 5MSCh. 7.1 - Prob. 6MSCh. 7.1 - Maybe moon. What features of the fractal forgeries...Ch. 7.1 - Exposing forgeries. What features of the fractal...Ch. 7.1 - Natures way. Find some examples of self-similarity...Ch. 7.1 - Do it yourself (H). Draw a figure that contains...
Ch. 7.1 - With a group of folks. In a small group, discuss...Ch. 7.1 - Prob. 12MSCh. 7.2 - A search for self. What does self-similarity mean?Ch. 7.2 - Desperately seeking similarity. Which of these...Ch. 7.2 - Too many triangles? At stage 0, the Sierpinski...Ch. 7.2 - Counting Koch. Look at the early stages of the...Ch. 7.2 - Argyle art. Take four line segments and place them...Ch. 7.2 - Natures way. Find several examples of objects in...Ch. 7.2 - Whos the fairest? Can you position three mirrors...Ch. 7.2 - Billiards and mirrors. On an idealized, square...Ch. 7.2 - MTV. Youve become a rock star and consequently...Ch. 7.2 - Photo op. Suppose you arrange two mirrors facing...Ch. 7.2 - How many mes? Arrange mirrors and camera as in...Ch. 7.2 - Quacker, Quacker, Quacker. Suppose you had the job...Ch. 7.2 - Sierpinski hexed (S). Take an equilateral triangle...Ch. 7.2 - The Kinks (ExH). Kochs kinky curve is created by...Ch. 7.2 - Four times (H). Draw a picture on a square piece...Ch. 7.2 - Burger heaven (S). Sketch a picture of a...Ch. 7.2 - Ice cream cones. Draw a picture of an ice cream...Ch. 7.2 - Sierpinski boundary. Take the boundary of a...Ch. 7.2 - Catching Zs (H). Take a Z. Put in nine smaller Zs,...Ch. 7.2 - Replacement pinwheel. Take a 1,2,2 right triangle....Ch. 7.2 - Koch Stool (ExH). Start with a line segment, mark...Ch. 7.2 - Koch collage stool. Given the Koch Stool described...Ch. 7.2 - Sierpinski shooting. Suppose that you were playing...Ch. 7.2 - Sierpinski target practice (H). What sequence of...Ch. 7.2 - Cantor Set. Start with the interval [0, 1]. Build...Ch. 7.2 - Cantor luck (H). Start with the point 0. Flip a...Ch. 7.2 - Cantor Square (S). Take a square. Make four...Ch. 7.2 - Cantor Square shrunk. Take the Cantor Square....Ch. 7.2 - Cantor Squared. Draw the four corners of a square,...Ch. 7.2 - Hexed again. Suppose you start with the three...Ch. 7.2 - Prob. 31MSCh. 7.2 - Antoines necklace. Antoines necklace is a delicate...Ch. 7.2 - Menger jacks. For the game of jacks, lets imagine...Ch. 7.2 - A tighter weave. Look back at the Tight Weave...Ch. 7.2 - A looser weave. Lets modify the carpet-designing...Ch. 7.2 - From where? Look at this fractal. What collage...Ch. 7.2 - Treed. Describe collage-making instructions that...Ch. 7.2 - Flaky (H). Describe collage making instructions...Ch. 7.2 - How big a hole? In the Tighter weave (34) and...Ch. 7.2 - 4D fractal. Describe a fractal in 4-dimensional...Ch. 7.2 - Personal perspectives. Write a short essay...Ch. 7.2 - With a group of folks. In a small group, discuss...Ch. 7.2 - Graphing Koch. Plot the following points in the...Ch. 7.2 - Koch adds life. Look at the points given in the...Ch. 7.2 - White out. Each stage of the Sierpinski triangle...Ch. 7.2 - Measuring the length of your Koch. The Koch stool...Ch. 7.2 - Counting Koch segments (H). Suppose you start with...Ch. 7.3 - Parallel grams. At right we see a parallelogram....Ch. 7.3 - Moving on up. Heres a triangle along with some...Ch. 7.3 - Bigger rug. Heres a picture of the Sierpinski...Ch. 7.3 - Dimension connection. What equation relates the...Ch. 7.3 - Divining dimension. If a fractal object requires...Ch. 7.3 - Stay inbounds. Give two consecutive integers that...Ch. 7.3 - Regular things (H). Find the fractal dimension of...Ch. 7.3 - More regular things. Find the fractal dimension of...Ch. 7.3 - Any right triangle. Take any right triangle. It...Ch. 7.3 - Sierpinski carpet (S). Compute the fractal...Ch. 7.3 - Koch Stool. Compute the fractal dimension of the...Ch. 7.3 - Cantor Set (H). The Cantor Set was constructed by...Ch. 7.3 - Cantor reduced. Suppose you take a unit interval,...Ch. 7.3 - Long Koch (ExH). The first stage in the...Ch. 7.3 - Pius (H). This fractal plus sign is self-similar...Ch. 7.3 - Tiniler triangles (S). Suppose you make something...Ch. 7.3 - Menger Sponge (ExH). Compute the fractal dimension...Ch. 7.3 - Thinning. Take a square. Make two copies, each...Ch. 7.3 - Not much. What is the fractal dimension of the...Ch. 7.3 - Koched (H). Create a Koch-like curve with fractal...Ch. 7.3 - Find a fractal. Describe a fractal having...Ch. 7.3 - Prob. 22MSCh. 7.3 - Personal perspectives. Write a short essay...Ch. 7.3 - With a group of folks. In a small group, discuss...Ch. 7.3 - Prob. 25MSCh. 7.3 - Power beyond the mathematics. Provide several...Ch. 7.3 - Varying dimensions. Using any method, including...Ch. 7.3 - Dimensional thinking. The text defines the...Ch. 7.3 - Power play (H). Using any method, including...Ch. 7.3 - Marching madness. The marching band wants to...Ch. 7.3 - More marching madness. For the second drill in the...Ch. 7.4 - Not Raul. Who was Gaston Julia?Ch. 7.4 - Use your imagination. What does the number i...Ch. 7.4 - Complex plots. The complex number 3+2i is plotted...Ch. 7.4 - Prob. 4MSCh. 7.4 - Outie or innie? If you use the point a+bi to build...Ch. 7.4 - Arithmetic. Compute the following sums and...Ch. 7.4 - More arithmetic (ExH). Compute the following sums...Ch. 7.4 - Quick draw. Two complex numbers are marked and...Ch. 7.4 - III iterate I (12:S). Feeling queasy? Then compute...Ch. 7.4 - III iterate I (12:S). Feeling queasy? Then compute...Ch. 7.4 - III iterate I (12:S). Feeling queasy? Then compute...Ch. 7.4 - III iterate I (12:S). Feeling queasy? Then compute...Ch. 7.4 - III iterate II (18:S). Compute the first three...Ch. 7.4 - III iterate II (18:S). Compute the first three...Ch. 7.4 - III iterate II (18:S). Compute the first three...Ch. 7.4 - III iterate II (18:S). Compute the first three...Ch. 7.4 - 1III iterate III. Compute the first three iterates...Ch. 7.4 - III iterate III. Compute the first three iterates...Ch. 7.4 - III iterate III. Compute the first three iterates...Ch. 7.4 - III iterate III. Compute the first three iterates...Ch. 7.4 - Orange Julias. Pictured here are some Julia Sets...Ch. 7.4 - Julia Webbed. Visit the Heart of Mathematics Web...Ch. 7.4 - Great escape? For each of the complex...Ch. 7.4 - Great escape? For each of the complex numbers...Ch. 7.4 - Great escape? For each of the complex numbers...Ch. 7.4 - Mandelbrot or not? (H). The images on the...Ch. 7.4 - Mandelbrot or not? (H). The images on the...Ch. 7.4 - Zero in (S). For each picture of the Julia Sets in...Ch. 7.4 - Zero in (S). For each picture of the Julia Sets in...Ch. 7.4 - Zero in (S). For each picture of the Julia Sets in...Ch. 7.4 - Zero in (S). For each picture of the Julia Sets in...Ch. 7.4 - Mandelbrot origins. Using your insights from...Ch. 7.4 - Prob. 36MSCh. 7.4 - Bounded Julia (ExH). Why is no Julia Set...Ch. 7.4 - Prisoner. Find a complex number a+bi that is fixed...Ch. 7.4 - Mandelbrot connections. Can every two points on...Ch. 7.4 - Personal perspectives. Write a short essay...Ch. 7.4 - With a group of folks. In a small group, discuss...Ch. 7.4 - Power beyond the mathematics. Provide several...Ch. 7.4 - Quadratic complexity (H). The quadratic formula...Ch. 7.4 - Special products. Compute the following products...Ch. 7.4 - Somethings fishy. The treasurer of the Marine...Ch. 7.4 - On the line. In the xy-plane, sketch the graph of...Ch. 7.4 - A line with a complex. Plot the points in the...Ch. 7.5 - Any interest? Suppose you put $1000 in a bank...Ch. 7.5 - Urban expansion. In the year 2004 the village of...Ch. 7.5 - Pre-sushi. Suppose a pond can sustain a maximum of...Ch. 7.5 - A booby trap. A small landmass off the Galapagos...Ch. 7.5 - Too many. The island of Birdburg has an area of...Ch. 7.5 - Call your shots. For each picture below, trace the...Ch. 7.5 - Getting cornered. Starting from the spot indicated...Ch. 7.5 - Double your money (H). Suppose you I deposit $1000...Ch. 7.5 - Too many(S). Earths surface is approximately...Ch. 7.5 - Rice bowl (H). One day long ago, the Emperor of...Ch. 7.5 - Natures way. Find some examples, other than those...Ch. 7.5 - The Game of Life (12:S). For each initial...Ch. 7.5 - The Game of Life (12:S). For each initial...Ch. 7.5 - The Game of Life (12:S). For each initial...Ch. 7.5 - The Game of Life (12:S). For each initial...Ch. 7.5 - Life cycles. For each initial population in the...Ch. 7.5 - Life cycles. For each initial population in the...Ch. 7.5 - Life cycles. For each initial population in the...Ch. 7.5 - Life cycles. For each initial population in the...Ch. 7.5 - Explosion. Devise a new initial population in the...Ch. 7.5 - Extinction. Construct a new initial population in...Ch. 7.5 - Periodic population (H). Construct a new initial...Ch. 7.5 - Programmed population. On a computer or...Ch. 7.5 - Programmed population: the next generation. Using...Ch. 7.5 - How many now? Suppose that a population is modeled...Ch. 7.5 - Fibonacci. Fibonacci numbers are constructed using...Ch. 7.5 - Fibonacci again. For each of the two columns you...Ch. 7.5 - Prob. 29MSCh. 7.5 - Cobweb plots (ExH). Here we visualize the...Ch. 7.5 - More spiders (S). Here is the graph of...Ch. 7.5 - Arachnids. Here is the graph of y=4x(1x). Start on...Ch. 7.5 - Making dough. Many delicious French desserts, such...Ch. 7.5 - Making dough. Many delicious French desserts, such...Ch. 7.5 - Making dough. Many delicious French desserts, such...Ch. 7.5 - More cobwebs (H). Consider the inverted V graph...Ch. 7.5 - Yet more cobwebs. Given the inverted V graph...Ch. 7.5 - Cantors cuts. Start with the unit interval [0, 1]....Ch. 7.5 - How much ¡s gone? In the construction of the...Ch. 7.5 - How much remains? Consider numbers that are not in...Ch. 7.5 - Personal perspectives. Write a short essay...Ch. 7.5 - With a group of folks. In a small group, discuss...Ch. 7.5 - Prob. 43MSCh. 7.5 - Power beyond the mathematics. Provide several...Ch. 7.5 - Life happens. A particular starting configuration...Ch. 7.5 - Perplexing population predictions. Verhulst model...Ch. 7.5 - More prediction peril. Consider the previous...Ch. 7.5 - Cobweb parabola. Mindscapes 3032 describe cobweb...Ch. 7.5 - Two sad and one happy parabolas. Building on your...Ch. 7.6 - Does this thing come with a warranty? If you use...Ch. 7.6 - Root repeater. Find a calculator with a square...Ch. 7.6 - Transforming experience. The equation y=4x(1x) is...Ch. 7.6 - Up and over. In this graph, we see a red diagonal...Ch. 7.6 - Over and up. Here is a simpler version of the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - The cobweb tent (ExH). Take a square in which the...Ch. 7.6 - Too high. Consider a tent function that got too...Ch. 7.6 - More gone. In the too high tent function in the...Ch. 7.6 - Too short. Consider a short tent function. Will...Ch. 7.6 - Where to? Using the transformation y=3.5x(1x),...Ch. 7.6 - Calculator slips. Using the logistic...Ch. 7.6 - Take stock. Pick a stock. Some Web sites will...Ch. 7.6 - Repulsive. If you take the number 1 and square it,...Ch. 7.6 - Attractive. Again consider the process of repeated...Ch. 7.6 - Sierpinski attractor. Remember how the Sierpinski...Ch. 7.6 - Two step. Consider the transformation that takes...Ch. 7.6 - Periodic attraction (S). Again consider the...Ch. 7.6 - Periodic attraction. Again consider the...Ch. 7.6 - Four-peat (H). Consider the equation y=3.5x(1x)....Ch. 7.6 - Nearly fourly. Consider the equation y=3.5x(1x)....Ch. 7.6 - Tent attraction? Consider the point 0.4, which is...Ch. 7.6 - Becoming periodic. The point 0.4 is of period 2 in...Ch. 7.6 - The Earth moved (ExH). Consider a transformation...Ch. 7.6 - Poles apart. Consider the same transformation as...Ch. 7.6 - Logistic cobwebs. Lets explore the relatively...Ch. 7.6 - Logistic cobwebs. Lets explore the relatively...Ch. 7.6 - Logistic cobwebs. Lets explore the relatively...Ch. 7.6 - With a group of folks. In a small group, discuss...Ch. 7.6 - Power beyond the mathematics. Provide several...Ch. 7.6 - V-quation. Look at the graph for Mindscape 4. The...Ch. 7.6 - Meeting up. Look at the graph for Mindscape 38....Ch. 7.6 - We meet again. Look at the graph for Mindscape 39....Ch. 7.6 - Calculating percentages (H). Return to Table 1 on...Ch. 7.6 - Calculating percentages II. Repeat Mindscape 48...
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