 7.7.5.1: For answer the following: a. As x changes from 1 to 4, y changes fr...
 7.7.6.1: A cube of edge length x has a surface area S(x) 6x2 and a volume V(...
 7.7.7.1: Match each function with its standard linearscale graph and its lo...
 7.1: Suppose you have a solid rectangular box, with dimensions x by 2x b...
 7.7.1.1: The twodimensional analog to the ratio of (surface area)/volume fo...
 7.7.2.1: Evaluate the following functions when x 2 and : h(x) 5 23x2 f(x) 5 ...
 7.7.3.1: Sketch by hand and compare the graphs of the following: y3 5 2x2 y2...
 7.7.4.1: (Graphing program recommended.) Graph y x4 and y 4x on the same gri...
 7.7.5.2: For and , find: a. (2x), (3x), and b. g(2x), g(3x), and
 7.7.6.2: a. Sketch (without technology) a graph for each of the following fu...
 7.7.7.2: The Brown University Standard Corpus of PresentDay American Englis...
 7.2: Inside a bottle of Bayer aspirin is a small plastic cylinder (about...
 7.7.1.2: Consider the formulas (volume of a sphere) and V (volume of a cylin...
 7.7.2.2: Identify which of the following are power functions. For each power...
 7.7.3.2: (Graphing program option.) For each part sketch by hand the three g...
 7.7.4.2: Examine the following two different versions of the graphs of both ...
 7.7.4.3: The following figures show two different versions of the graphs of ...
 7.7.5.3: For and , find: a. h(2x), , and b. j(2x), , and jA 1 3 hA (j 23x) x...
 7.7.6.3: a. Sketch (without technology) a graph for each of the following fu...
 7.7.7.3: Does the accompanying graph suggest that an exponential or a power ...
 7.3: The website for La Riviera Gourmet in Lexington, Massachusetts, sta...
 7.7.1.3: A tiny sphere has a radius of 0.000 000 000 1 10 10 meter, which is...
 7.7.2.3: Each of the following represents direct proportionality with powers...
 7.7.3.3: Match the function with its graph. Explain why you have chosen each...
 7.7.4.4: a. Which eventually dominates, y (1.001)x or y x1000? b. In general...
 7.7.5.4: Find the value for k, the constant of proportionality, if: a. y 5 a...
 7.7.6.4: (Graphing program optional.) In each part, sketch the three graphs ...
 7.7.7.4: Given the following three power functions in the form , a. Use the ...
 7.4: Which of the following functions represent power functions? Which p...
 7.7.1.4: The radius of Earth is about 6400 kilometers. Assume that Earth is ...
 7.7.2.4: Find the constant of proportionality, k, for the given conditions. ...
 7.7.3.4: Match the function with the appropriate graph. Explain why you have...
 7.7.4.5: (Graphing program optional.) Use a table or graphing utility to det...
 7.7.5.5: Create an equation that meets the given specifications and then sol...
 7.7.6.5: (Graphing program optional.) Sketch on the same grid, the graphs of...
 7.7.7.5: Using rules of logarithms, convert each equation to its power funct...
 7.5: A bathtub holds about 15 gallons of water. a. Construct a function ...
 7.7.1.5: If the radius of a sphere is x meters, what happens to the surface ...
 7.7.2.5: Write the general formula to describe each variation and then solve...
 7.7.3.5: Begin with the function . a. Create a new function g(x) by vertical...
 7.7.4.6: Match the following data tables with the appropriate function. (x) ...
 7.7.5.6: Assume y is inversely proportional to the cube of x. a. If x double...
 7.7.6.6: Match each of the following functions with its graph (below and top...
 7.7.7.6: . Given linear equations in the form Y 5 B 1 mX, where Y 5 log y, X...
 7.6: In some areas wind turbines offer a viable clean source of energy. ...
 7.7.1.6: Assume a box has a square base and the length of a side of the base...
 7.7.2.6: The data in the table satisfy the equation , where n is a positive ...
 7.7.3.6: (Graphing program optional.) Plot the functions and on the same gri...
 7.7.4.7: (Graphing program recommended.) If x is positive, for what values o...
 7.7.5.7: a. B is inversely proportional to . What is the effect on B of doub...
 7.7.6.7: Match each function with its graph and specify its domain and range...
 7.7.7.7: a. Create a table of values for the function for x 5 1, 2, 3, 4, 5,...
 7.7: Some northern cities celebrate New Years eve with First Night event...
 7.7.1.7: A box has volume . a. Find the volume of a cereal box with dimensio...
 7.7.2.7: Assume Y is directly proportional to . a. Express this relationship...
 7.7.3.7: (Graphing program recommended.) Plot the functions and on the same ...
 7.7.4.8: For x 0, match each function with its graph. a. c. b. d. y 5 4(3x y...
 7.7.5.8: The time, t, required to empty a tank is inversely proportional to ...
 7.7.6.8: Match each function (where x 0) with the appropriate graph. Explain...
 7.7.7.8: Match each power function with its log y vs. log x plot. 2x 4 y 5 1...
 7.8: 8. (continued) inversely proportional to the length L of the string...
 7.7.1.8: A box with a lid has a square base with each base side x inches and...
 7.7.2.8: The distance, d, a ball travels down an inclined plane is directly ...
 7.7.3.8: Consider the accompanying graph of , where n is a positive integer....
 7.7.4.9: Match each function with its graph at the top of the next column. a...
 7.7.5.9: The intensity of light from a point source is inversely proportiona...
 7.7.6.9: Begin with the function (x) x3 . Then: a. Create a new function g(x...
 7.7.7.9: (Graphing program required that can convert from a linear to a loga...
 7.9: Construct a formula for each situation. a. The time, t, taken for a...
 7.7.1.9: Consider a cylinder with volume What happens to its volume when you...
 7.7.2.9: a. Assume L is directly proportional to x5 . What is the effect of ...
 7.7.3.9: Consider the accompanying graph of , where n is a positive integer....
 7.7.4.10: Given the functions (x) x4 and g(x) (4)x : a. Find (3x) and 3(x). S...
 7.7.5.10: In Exercise 9, at what distance would the light quadruple its inten...
 7.7.6.10: (Graphing program optional.) Sketch the functions , , and . For par...
 7.7.7.10: Find the equation of the line in each of the accompanying graphs. R...
 7.10: The following four graphs are power functions with positive exponen...
 7.7.1.10: Consider two solid figures, a sphere and cylinder, where each has r...
 7.7.2.10: In Love That Dirty Water (Chicago Reader, April 5, 1996), Scott Ber...
 7.7.3.10: Consider the accompanying graphs of four functions of the form . Wh...
 7.7.4.11: Consider a power function (x) xn where n is a positive integer, and...
 7.7.5.11: A light fixture is mounted flush on a 10foothigh ceiling over a 3...
 7.7.6.11: (Graphing program optional.) Sketch the functions and and then inse...
 7.7.7.11: Use the accompanying graphs to answer the following questions. a. A...
 7.11: Describe any symmetries and asymptotes for each function graphed in...
 7.7.1.11: The volume, V, of a cylindrical can is and the total surface area, ...
 7.7.2.11: When a variable is directly proportional to the product of two or m...
 7.7.3.11: Sketch by hand the graph of each function: , ,, a. Identify the k v...
 7.7.4.12: Given the points (1, 2) and (6, 72): a. Find a linear function of t...
 7.7.5.12: The frequency, f (the number of oscillations per unit of time), of ...
 7.7.6.12: Consider the accompanying graph of , where n is a positive integer....
 7.7.7.12: Graphing program required that can convert from a linear to a logar...
 7.12: Of the following three functions, as x , which would eventually dom...
 7.7.1.12: (Hint: See Exploration 7.1.) To celebrate Groundhog Day, a gourmet ...
 7.7.2.12: Write a general formula to describe each variation. Use the informa...
 7.7.3.12: a. Complete the partial graph in the accompanying figure in three d...
 7.7.4.13: Find power, exponential, and linear equations that go through the t...
 7.7.5.13: Boyles Law says that if the temperature is held constant, then the ...
 7.7.6.13: Consider the accompanying graph of , where n is a positive integer....
 7.7.7.13: (Graphing program required.) For each table, create a linear, expon...
 7.13: (Graphing program optional.) a. In Nashua, New Hampshire, you can r...
 7.7.1.13: A cylinder (with no top or bottom) has a radius of x feet and a hei...
 7.7.2.13: The cost, C, in dollars, of insulating a wall is directly proportio...
 7.7.3.13: (Graphing program optional.) Given , construct a function that is a...
 7.7.4.14: Think of the graphs of (x) ax3 and g(x) a(3)x , where a 0, and then...
 7.7.5.14: The pressure of the atmosphere around us is relatively constant at ...
 7.7.6.14: Consider the following graphs of four functions of the form , where...
 7.7.7.14: (Graphing program required.) Examine the following table. Treating ...
 7.14: (Requires a calculator that can compute square roots.) Film studios...
 7.7.1.14: A circle has a radius of x units. Which graph could represent the r...
 7.7.2.14: Suppose you are traveling in your car at speed S and you suddenly b...
 7.7.3.14: (Graphing program optional.) Are and reflections of each other acro...
 7.7.4.15: If you know that one of the graphs in the accompanying figure is a ...
 7.7.5.15: a. Construct an equation to represent a relationship where x is dir...
 7.7.6.15: (Graphing program optional.) Sketch the graph of the function . The...
 7.7.7.15: The accompanying graph shows oxygen consumption versus body mass in...
 7.15: (Requires a calculator that can compute square roots.) Surgeons use...
 7.7.2.15: (Graphing program required.) Assume a person weighing P pounds is s...
 7.7.5.16: a. Construct an equation to represent a relationship where w is dir...
 7.7.6.16: a. Complete the partial graph on the next page three different ways...
 7.7.7.16: a. The lefthand graph below shows the relationship between populat...
 7.16: Which of the following graphs could depict power functions with a n...
 7.7.2.16: (Graphing program required.) In Exercise 15 we looked at how much a...
 7.7.5.17: Waves on the open ocean travel with a velocity that is directly pro...
 7.7.6.17: (Graphing program optional.) Given (x) 4x 2 , construct a function ...
 7.7.7.17: Public health researchers have developed a new measure of overall p...
 7.17: Describe any apparent symmetries and asymptotes for each of the fun...
 7.7.2.17: Construct formulas to represent the following relationships. a. The...
 7.7.5.18: When installing Christmas lights on the outside of your house, you ...
 7.7.6.18: (Graphing program optional.) Are (x) x 3 and g(x) x 3 reflections o...
 7.18: (Requires a calculator that can compute fractional powers.) Because...
 7.7.2.18: a. An insulation blanket for a cylindrical hot water heater is sold...
 7.7.5.19: The rate of vibration (frequency) of a string under constant tensio...
 7.7.6.19: If and , construct the following functions. a. (2x) and 2(x) b. 2(x...
 7.19: a. Given the following graph, what kind of function is Y with respe...
 7.7.2.19: The volume, V, of a cylinder, with radius r and height h, is given ...
 7.7.5.20: The weight of a body is inversely proportional to the square of the...
 7.7.6.20: (Graphing program required.) Begin with a function . In each part d...
 7.20: (Requires a graphing program that can convert the scale on either a...
 7.7.2.20: If you were designing cylinders, what are two different ways you co...
 7.7.5.21: (Requires technology to generate a bestfit power function.) Oil is...
 7.7.6.21: (Graphing program optional.) Determine which of the graphs of the f...
 7.21: According to Adrian Bejans Constructal Theory, the optimal cruising...
 7.7.2.21: Given : a. Calculate h(2) and compare this value with h(6). b. Calc...
 7.7.5.22: (Requires technology to generate a bestfit power function.) Recall...
 7.22: In 2229, construct a function or functions with the specified prope...
 7.7.2.22: a. Let . Describe the difference between (3x) and 3(x). b. Let . De...
 7.7.5.23: (Requires a graphing program.) The newest Coke minicans hold only ...
 7.23: In 2229, construct a function or functions with the specified prope...
 7.24: In 2229, construct a function or functions with the specified prope...
 7.25: In 2229, construct a function or functions with the specified prope...
 7.26: In 2229, construct a function or functions with the specified prope...
 7.27: In 2229, construct a function or functions with the specified prope...
 7.28: In 2229, construct a function or functions with the specified prope...
 7.29: In 2229, construct a function or functions with the specified prope...
 7.30: Is each of the statements in 3040 true or false? If a statement is ...
 7.31: Is each of the statements in 3040 true or false? If a statement is ...
 7.32: Is each of the statements in 3040 true or false? If a statement is ...
 7.33: Is each of the statements in 3040 true or false? If a statement is ...
 7.34: Is each of the statements in 3040 true or false? If a statement is ...
 7.35: Is each of the statements in 3040 true or false? If a statement is ...
 7.36: Is each of the statements in 3040 true or false? If a statement is ...
 7.37: Is each of the statements in 3040 true or false? If a statement is ...
 7.38: Is each of the statements in 3040 true or false? If a statement is ...
 7.39: Is each of the statements in 3040 true or false? If a statement is ...
 7.40: Is each of the statements in 3040 true or false? If a statement is ...
Solutions for Chapter 7: POWER FUNCTIONS
Full solutions for Explorations in College Algebra  5th Edition
ISBN: 9780470466445
Solutions for Chapter 7: POWER FUNCTIONS
Get Full SolutionsSince 166 problems in chapter 7: POWER FUNCTIONS have been answered, more than 8721 students have viewed full stepbystep solutions from this chapter. Explorations in College Algebra was written by and is associated to the ISBN: 9780470466445. Chapter 7: POWER FUNCTIONS includes 166 full stepbystep solutions. This textbook survival guide was created for the textbook: Explorations in College Algebra, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).