In a sample based on 28 species, the volume fraction of spongy mesophyll was found to be

In Problems 14, state the range for the given functions. Graph each function. f (x) = x2, x R

In Problems 14, state the range for the given functions. Graph each function. f (x) = x2, x [0, 1]

In Problems 14, state the range for the given functions. Graph each function. f (x) = x2, 1 < x 0

In Problems 14, state the range for the given functions. Graph each function. f (x) = x2, 1 2 < x < 1 2

(a) Show that, for x _= 1, x2 1 x 1 = x + 1 (b) Are the functions f (x) = x2 1 x 1 , x _= 1 and g(x) = x + 1, x R equal?

(a) Show that 2|x 1| = _ 2(x 1) for x 1 2(1 x) for x 1 (b) Are the functions f (x) = _ 2 2x for 0 x 1 2x 2 for 1 x 2 and g(x) = 2|x 1|, x [0, 2] equal?

In Problems 712, sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. f (x) = 2x

In Problems 712, sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. f (x) = 3x2

In Problems 712, sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. f (x) = |3x|

In Problems 712, sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. f (x) = 2x + 1

In Problems 712, sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. f (x) = |x|

In Problems 712, sketch the graph of each function and decide in each case whether the function is (i) even, (ii) odd, or (iii) does not show any obvious symmetry. Then use the criteria in Subsection 1.2.1 to check your answers. f (x) = 3x3

Suppose that f (x) = x2, x R and g(x) = 3 + x, x R (a) Show that ( f g)(x) = (3 + x)2, x R (b) Show that (g f )(x) = 3 + x2, x R

Suppose that f (x) = x3, x R and g(x) = 1 x, x R (a) Show that ( f g)(x) = (1 x)3, x R (b) Show that (g f )(x) = 1 x3, x R

Suppose that f (x) = 1 x2, x R and g(x) = 2x, x 0 (a) Find ( f g)(x) together with its domain. (b) Find (g f )(x) together with its domain.

Suppose that f (x) = 1 x + 1 , x _= 1 and g(x) = 2x2, x R (a) Find ( f g)(x). (b) Find (g f )(x). In both (a) and (b), find the domain.

Suppose that f (x) = 3x2, x 3 and g(x) = x, x 0 Find ( f g)(x) together with its domain.

Suppose that f (x) = x4, x 3 and g(x) = _ x + 1, x 3 Find ( f g)(x) together with its domain.

Suppose that f (x) = x2, x 0, and g(x) = x, x 0. Typically, f g _= g f , but this is an example in which the order of composition does not matter. Show that f g = g f .

Suppose that f (x) = x4, x 0. Find g(x) so that f g = g f .

Use a graphing calculator to graph f (x) = x2, x 0, and g(x) = x4, x 0, together. For which values of x is f (x) > g(x), and for which is f (x) < g(x)?

Use a graphing calculator to graph f (x) = x3, x 0, and g(x) = x5, x 0, together. When is f (x) > g(x), and when is f (x) < g(x)?

Graph y = xn, x 0, for n = 1, 2, 3, and 4 in one coordinate system.Where do the curves intersect?

(a) Graph f (x) = x, x 0, and g(x) = x2, x 0, together, in one coordinate system. (b) For which values of x is f (x) g(x), and for which values of x is f (x) g(x)?

(a) Graph f (x) = x2 and g(x) = x3 for x 0, together, in one coordinate system. (b) Show algebraically that x2 x3 for 0 x 1. (c) Show algebraically that x2 x3 for x 1.

Show algebraically that if n m, xn xm for 0 x 1 and xn xm for x 1

(a) Show that y = x2, x R, is an even function. (b) Show that y = x3, x R, is an odd function.

Show that (a) y = xn, x R, is an even function when n is an even integer. (b) y = xn, x R, is an odd function when n is an odd integer.

In Example 5 of this section, we considered the chemical reaction A + B AB Assume that initially only A and B are in the reaction vessel and that the initial concentrations are a = [A] = 3 and b = [B] = 4. (a) We found that the reaction rate R(x), where x is the concentration of AB, is given by R(x) = k(a x)(b x) where a is the initial concentration of A, b is the initial concentration of B, and k is the constant of proportionality. Suppose that the reaction rate R(x) is equal to 9 when the concentration of AB is x = 1. Use this relationship to find the reaction rate R(x). (b) Determine the appropriate domain of R(x), and use a graphing calculator to sketch the graph of R(x).

An autocatalytic reaction uses its resulting product for the formation of a new product, as in the reaction A + X X If we assume that this reaction occurs in a closed vessel, then the reaction rate is given by R(x) = kx(a x) for 0 x a, where a is the initial concentration of A and x is the concentration of X. (a) Show that R(x) is a polynomial and determine its degree. (b) Graph R(x) for k = 2 and a = 6. Find the value of x at which the reaction rate is maximal.

Suppose that a beetle walks up a tree along a straight line at a constant speed of 1 meter per hour.What distance will the beetle have covered after 1 hour, 2 hours, and 3 hours? Write an equation that expresses the distance (in meters) as a function of the time (in hours), and show that this function is a polynomial of degree 1.

Suppose that a fungal disease originates in the middle of an orchard, initially affecting only one tree. The disease spreads out radially at a constant speed of 10 feet per day. What area will be affected after 2 days, 4 days, and 8 days? Write an equation that expresses the affected area as a function of time, measured in days, and show that this function is a polynomial of degree 2.

In Problems 3336, for each function, find the largest possible domain and determine the range. f (x) = 1 1 x

In Problems 3336, for each function, find the largest possible domain and determine the range. f (x) = 2x (x 2)(x + 3)

In Problems 3336, for each function, find the largest possible domain and determine the range. f (x) = x 2 x2 9

In Problems 3336, for each function, find the largest possible domain and determine the range. f (x) = 1 x2 + 1

Compare y = 1 x and y = 1 x2 for x > 0 by graphing the two functions. Where do the curves intersect? Which function is greater for small values of x? for large values of x?

Let n and m be two positive integers with m n. Answer the following questions about y = xn and y = xm for x > 0:Where do the curves intersect?Which function is greater for small values of x? for large values of x?

Let f (x) = 1 x + 1 , x > 1 (a) Use a graphing calculator to graph f (x). (b) On the basis of the graph in (a), determine the range of f (x). (c) For which values of x is f (x) = 2? (d) On the basis of the graph in (a), determine how many solutions f (x) = a has, where a is in the range of f (x).

Let f (x) = 2x 3 + x , x 0 (a) Use a graphing calculator to graph f (x). (b) Find the range of f (x). (c) For which values of x is f (x) = 1? (d) Based on the graph in (a), explain in words why, for any value a in the range of f (x), you can find exactly one value x 0 such that f (x) = a. Determine x by solving f (x) = a.

Let f (x) = 3x 1 + x , x 0 (a) Use a graphing calculator to graph f (x). (b) Find the range of f (x). (c) For which values of x is f (x) = 2? (d) On the basis of the graph in (a), explain in words why, for any value a in the range of f (x), you can find exactly one value x 0 such that f (x) = a. Determine x by solving f (x) = a.

Use a graphing calculator to investigate the Monod growth function r (N) = aN k + N , N 0 where a and k are positive constants. (a) Graph r (N) for (i) a = 5 and k = 1, (ii) a = 5 and k = 3, and (iii) a = 8 and k = 1. Place all three graphs in one coordinate system. (b) On the basis of the graphs in (a), describe in words what happens when you change a. (c) On the basis of the graphs in (a), describe in words what happens when you change k.

The Monod growth function r (N) describes growth as a function of nutrient concentration N. Assume that r (N) = 5 N 1 + N , N 0 Find the percentage increase when the nutrient concentration is doubled from N = 0.1 to N = 0.2. Compare this result with what you find when you double the nutrient concentration from N = 10 to N = 20. This is an example of .

The Monod growth function r (N) describes growth as a function of nutrient concentration N. Assume that r (N) = a N k + N , N 0 where a and k are positive constants. (a) What happens to r (N) as N increases? Use this relationship to explain why a is called the saturation level. (b) Show that k is the half-saturation constant; that is, show that if N = k, then r (N) = a/2.

Let f (x) = x2 4 + x2 , x 0 (a) Use a graphing calculator to graph f (x). (b) On the basis of your graph in (a), find the range of f (x). (c) What happens to f (x) as x gets larger?

The function f (x) = xn bn + xn , x 0 where n is a positive integer and b is a positive real number, is used in biochemistry to model reaction rates as a function of the concentration of some reactants. (a) Use a graphing calculator to graph f (x) for n = 1, 2, and 3 in one coordinate system when b = 2. (b) Where do the three graphs in (a) intersect? (c) What happens to f (x) as x gets larger? (d) For an arbitrary positive value of b, show that f (b) = 1/2. On the basis of this demonstration and your answer in (c), explain why b is called the half-saturation constant.

In Problems 4750, use a graphing calculator to sketch the graphs of the functions. y = x3/2, x 0

In Problems 4750, use a graphing calculator to sketch the graphs of the functions. y = x1/3, x 0

In Problems 4750, use a graphing calculator to sketch the graphs of the functions. y = x1/4, x > 0

In Problems 4750, use a graphing calculator to sketch the graphs of the functions. y = 2x7/8, x > 0

(a) Graph y = x1/2, x > 0, and y = x1/2, x 0, together, in one coordinate system. (b) Show algebraically that x1/2 x1/2 for 0 < x 1. (c) Show algebraically that x1/2 x1/2 for x 1.

(a) Graph y = x5/2, x 0, and y = x1/2, x 0, together, in one coordinate system. (b) Show algebraically that x5/2 x1/2 for 0 x 1. (: Show that x1/2/x1/2 = x 1 for 0 < x 1.) (c) Show algebraically that x5/2 x1/2 for x 1.

In a sample based on 46 species, leaf area was found to be proportional to (stem diameter)1.84. On the basis of your graph, as stem diameter increases, does leaf area increase or decrease?

In a sample based on 28 species, the volume fraction of spongy mesophyll was found to be proportional to (leaf thickness) 0.49. (The spongy mesophyll is part of the internal tissue of a leaf blade.) On the basis of your graph, as leaf thickness increases, does the volume fraction of spongy mesophyll increase or decrease?

In a sample of 60 species of trees, wood density was found to be proportional to (breaking strength)0.82. On the basis of your graph, does breaking strength increase as wood density increases? or as wood density decreases?

Suppose that a cube of length L and volume V has mass M and that M = 0.35V. How does the length of the cube depend on its mass?

Assume that a population size at time t is N(t) and that N(t) = 2t , t 0 (a) Find the population size for t = 0, 1, 2, 3, and 4. (b) Graph N(t) for t 0.

Assume that a population size at time t is N(t) and that N(t) = 40 2t , t 0 (a) Find the population size at time t = 0. (b) Show that N(t) = 40et ln 2, t 0 (c) How long will it take until the population size reaches 1000? [: Find t so that N(t) = 1000.]

The half-life of C14 is 5730 years. If a sample of C14 has a mass of 20 micrograms at time t = 0, how much is left after 2000 years?

The half-life of C14 is 5730 years. If a sample of C14 has a mass of 20 micrograms at time 0, how long will it take until (a) 10 grams and (b) 5 grams are left?

After 7 days, a particular radioactive substance decays to half of its original amount. Find the decay rate of this substance.

After 5 days, a particular radioactive substance decays to 37% of its original amount. Find the half-life of this substance.

Polonium 210 (Po210) has a half-life of 140 days. (a) If a sample of Po210 has a mass of 300 micrograms, find a formula for the mass after t days. (b) How long would it take this sample to decay to 20% of its original amount? (c) Sketch the graph of the amount of mass left after t days.

The half-life of C14 is 5730 years. Suppose that wood found at an archeological excavation site contains about 35% as much C14 (in relation to C12) as does living plant material. Determine when the wood was cut.

The half-life of C14 is 5730 years. Suppose that wood found at an archeological excavation site is 15,000 years old. How much C14 (based on C12 content) does the wood contain relative to living plant material?

The age of rocks of volcanic origin can be estimated with isotopes of argon 40 (Ar40) and potassium 40 (K40). K40 decays into Ar40 over time. If a mineral that contains potassium is buried under the right circumstances, argon forms and is trapped. Since argon is driven off when the mineral is heated to very high temperatures, rocks of volcanic origin do not contain argon when they are formed. The amount of argon found in such rocks can therefore be used to determine the age of the rock. Assume that a sample of volcanic rock contains 0.00047% K40. The sample also contains 0.000079% Ar40. How old is the rock? (The decay rate of K40 to Ar40 is 5.335 1010/yr.)

Hall (1964) investigated the change in population size of the zooplankton species in Base Line Lake, Michigan. The population size N(t) at time t was modeled by the equation N(t) = N0ert where N0 denotes the population size at time 0. The constant r is called the intrinsic rate of growth. (a) Plot N(t) as a function of t if N0 = 100 and r = 2. Compare your graph against the graph of N(t) when N0 = 100 and r = 3. Which population grows faster? (b) The constant r is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length 1, and you find that at the beginning there were 200 individuals and after one unit of time there were 250 individuals. Determine r. [: Consider the ratio N(t+1)/N(t).]

Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function L(x) = L(1 ekx ) for x 0, where L(x) is the length of the fish at age x and k and L are positive constants. (a) Use a graphing calculator to graph L(x) for L = 20, for (i) k = 1 and (ii) k = 0.1. (b) For k = 1, find x so that the length is 90% of L. Repeat for 99% of L. Can the fish ever attain length L? Interpret the meaning of L. (c) Compare the graphs obtained in (a). Which growth curve reaches 90% of L faster? Can you explain what happens to the curve of L(x) when you vary k (for fixed L)?

Which of the following functions is one to one (use the horizontal line test)? (a) f (x) = x2, x 0 (b) f (x) = x2, x R (c) f (x) = 1 x , x > 0 (d) f (x) = ex , x R (e) f (x) = 1 x2 , x _= 0 (f) f (x) = 1 x2 , x > 0

(a) Show that f (x) = x3 1, x R, is one to one, and find its inverse together with its domain. (b) Graph f (x) and f 1(x) in one coordinate system, together with the line y = x, and convince yourself that the graph of f 1(x) can be obtained by reflecting the graph of f (x) about the line y = x.

(a) Show that f (x) = x2 + 1, x 0, is one to one, and find its inverse together with its domain. (b) Graph f (x) and f 1(x) in one coordinate system, together with the line y = x, and convince yourself that the graph of f 1(x) can be obtained by reflecting the graph of f (x) about the line y = x.

(a) Show that f (x) = x, x 0, is one to one, and find its inverse together with its domain. (b) Graph f (x) and f 1(x) in one coordinate system, together with the line y = x, and convince yourself that the graph of f 1(x) can be obtained by reflecting the graph of f (x) about the line y = x.

(a) Show that f (x) = 1/x3, x > 0, is one to one, and find its inverse together with its domain. (b) Graph f (x) and f 1(x) in one coordinate system, together with the line y = x, and convince yourself that the graph of f 1(x) can be obtained by reflecting the graph of f (x) about the line y = x.

The reciprocal of a function f (x) can be written as either 1/ f (x) or [ f (x)]1. The point of this problem is to make clear that a reciprocal of a function has nothing to do with the inverse of a function. As an example, let f (x) = 2x +1, x R. Find both [ f (x)]1 and f 1(x), and compare the two functions. Graph all three functions together.

Find the inverse of f (x) = 3x , x R, together with its domain, and graph both functions in the same coordinate system.

Find the inverse of f (x) = 5x , x R, together with its domain, and graph both functions in the same coordinate system.

Find the inverse of f (x) = ( 1 4 )x , x R, together with its domain, and graph both functions in the same coordinate system.

Find the inverse of f (x) = ( 1 3 )x , x R, together with its domain, and graph both functions in the same coordinate system.

Find the inverse of f (x) = 2x , x 0, together with its domain, and graph both functions in the same coordinate system.

Find the inverse of f (x) = ( 1 2 )x , x 0, together with its domain, and graph both functions in the same coordinate system.

Simplify the following expressions: (a) 25 log2 x (b) 34 log3 x (c) 55 log1/5 x (d) 42 log2 x (e) 23 log1/2 x (f) 4log1/2 x

Simplify the following expressions: (a) log4 16x (b) log2 16x (c) log3 27x (d) log1/2 4x (e) log1/2 8x (f) log3 9x

Simplify the following expressions: (a) ln x2 + ln x3 (b) ln x4 ln x2 (c) ln(x2 1) ln(x + 1) (d) ln x1 + ln x3

Simplify the following expressions: (a) e3 ln x (b) eln(x2+1) (c) e2 ln(1/x) (d) e2 ln x

Write the following expressions in terms of base e, and simplify: (a) 3x (b) 4x21 (c) 2x1 (d) 34x+1

Write the following expressions in terms of base e: (a) log2(x2 1) (b) log3(5x + 1) (c) log(x + 2) (d) log2(2x2 1)

Show that the function y = (1/2)x can be written in the form y = ex , where is a positive constant. Determine .

Show that if 0 < a < 1, then the function y = ax can be written in the form y = ex , where is a positive constant. Write in terms of a.

Assume that two DNA sequences of common origin, each of length 300 nucleotides, differ from each other by 47 nucleotides. Use the Jukes and Cantor correction of Example 15 to find an estimate for the number K of substitutions per site.

Acommunity measure that takes both species abundance and species richness into account is the Shannon diversity index H. To calculate H, the proportion pi of species i in the community is used. Assume that the community consists of S species. Then H = (p1 ln p1 + p2 ln p2 + + pS ln pS) (a) Assume that S = 5 and that all species are equally abundant; that is, p1 = p2 = = p5. Compute H. (b) Assume that S = 10 and that all species are equally abundant; that is, p1 = p2 = = p10. Compute H. (c) A measure of equitability (or evenness) of the species distribution can be measured by dividing the diversity index H by ln S. Compute H/ ln S for S = 5 and S = 10. (d) Show that, in general, if there are N species and all species are equally abundant, then H ln S = 1

In Problems 9196, for each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. y = sin x and y = 2 sin x

In Problems 9196, for each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. y = sin x and y = sin(2x)

In Problems 9196, for each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. y = cos x and y = 2 cos x

In Problems 9196, for each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. y = cos x and y = cos(2x)

In Problems 9196, for each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. y = tan x and y = 2 tan x

In Problems 9196, for each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. y = tan x and y = tan(2x)

Let f (x) = 3 sin(4x), x R Find the amplitude and the period of f (x).

Let f (x) = 2 sin _ x 2 _ , x R Find the amplitude and the period of f (x).

Let f (x) = 4 sin(2x), x R Find the amplitude and the period of f (x).

Let f (x) = 3 2 sin _ 3 x _ , x R Find the amplitude and the period of f (x).

Let f (x) = 4 cos _ x 4 _ , x R Find the amplitude and the period of f (x).

Let f (x) = 7 cos(2x), x R Find the amplitude and the period of f (x).

Let f (x) = 3 cos _ x 5 _ , x R Find the amplitude and the period of f (x).

Let f (x) = 2 3 cos _ 3x _ , x R Find the amplitude and the period of f (x).

Use the fact that sec x = 1 cos x to explain why the maximum domain of y = sec x consists of all real numbers except odd integer multiples of /2.

Use the fact that csc x = 1 sin x to explain why the maximum domain of y = csc x consists of all real numbers except integer multiples of .