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Introduction to graph theory 2nd edition Douglas B. West - Solutions

What is unusual about the domain of the composite function f ◦ g for the functions f (x) = x1/2 and g(x) = −1−|x|?
Calculate the composite functions f ◦ g and g ◦ f, and determine their domains. 31. f(θ) = cos θ, g(x) = x3 + x2 33. f(t) = 1/√t, g(t) = −t2
The population (in millions) of a country as a function of time t (years) is P(t) = 30.20.1t . Show that the population doubles every 10 years. Show more generally that for any positive constants a and k, the function g(t) = a2kt doubles after 1/k years.
We define the first difference δf of a function f (x) by δf (x) = f (x + 1) − f (x). 37. Show that if f (x) = x2, then δf (x) = 2x + 1. Calculate δf for f (x) = x and f (x) = x3. 39. Show that for any two functions f and g, δ(f + g) = δf + δg and δ(cf ) = cδ(f ), where c is any constant.
we define the first difference Î´f of a function f (x) by Î´f (x) = f (x + 1) ˆ’ f (x).41. First show thatsatisfies Î´P = (x + 1). Then apply Exercise 40 to conclude that
Determine the domain of the function.7.9. (x) = xˆ’4 + (x ˆ’ 1)ˆ’3 11. g(y) = 10 ˆšy+yˆ’1
How is it possible for two different rotations to define the same angle?
Find the lengths of the arcs subtended by the angles Î¸ and Ï† radians in Figure 20.
Fill in the following table of values:
Show that if tanθ = c and 0 ≤ θ < π/2, then cos θ = 1/ √1 + c2. Draw a right triangle whose opposite and adjacent sides have lengths c and 1.
Assume that 0 ≤ θ < π/2. 19. Find sin θ and tan θ if cos θ = 5/13. 21. Find sin θ, sec θ, and cot θ if tan θ = 2/7. 23. Find cos 2θ if sinθ = 1/5.
Give two different positive rotations that define the angle π/4.
Find cos θ and tan θ if sin θ = 0.4 and π/2 ≤ θ < π.
Find cosθ = 4/3 and sin θ < 0.
Find the values of sin Î¸, cos Î¸, and tan Î¸ for the angles corresponding to the eight points in Figure 23(A) and (B).
Give a negative rotation that defines the angle π/3.
Refer to Figure 24(B). Compute cosÏˆ, sin Ïˆ, cot Ïˆ, and cscÏˆ.
Use the addition formula to compute cos (π/3+ π/4) exactly.
Sketch the graph over [ 0,2Ï€].35. 2 sin 4Î¸37.39. How many points lie on the intersection of the horizontal line y = c and the graph of y = sin x for 0 ‰¤ x
Find the angle between 0 and 2π equivalent to 13π/4.
Solve for 0 ≤ θ 2π 41. Sin 2θ + sin 3θ = 0 43. cos 4θ + cos 2θ = 0
What is the unit circle definition of sin θ?
Refer to Figure 24(B). Compute cosÏˆ, sin Ïˆ, cot Ïˆ, and cscÏˆ.
How does the periodicity of sin θ and cos θ follow from the unit circle definition?
Fill in the remaining values of (cos Î¸, sin Î¸) for the points in Figure 22.
Convert from radians to degrees: (a) 1 (b) π/3 (c) 5/12 (d) - 3π/4
Use figure 22 to find all angles between 0 and 2Ï€ satisfying the given condition.9. cos Î¸ = 1 11. tan Î¸ = -1 13. sin x = 2ˆš3/2
A ball dropped from a state of rest at time t = 0 travels a distance s(t) = 4.9t2 m in t seconds.(a) How far does the ball travel during the time interval [2, 2.5]?(b) Compute the average velocity over [2, 2.5].(c) Compute the average velocity for the time intervals in the table and estimate the
Estimate the instantaneous rate of change at the point indicated 11. P(x) = 3x2 − 5; x = 2 13. y(x) = 1/x+2; x = 2
Estimate the instantaneous rate of change at the point indicated. 15. f (x) = 3x ; x = 0 17. f(x) = sin x; x = π / 6
The height (in centimeters) at time t (in seconds) of a small mass oscillating at the end of a spring is h(t) = 8 cos(12πt). (a) Calculate the mass's average velocity over the time intervals [0, 0.1] and [3, 3.5]. (b) Estimate its instantaneous velocity at t = 3.
Assume that the period T (in seconds) of a pendulum (the time required for a complete back-and-forth cycle) is T = 3/2ˆšL, where L is the pendulum's length (in meters).(a) What are the units for the rate of change of T with respect to L? Explain what this rate measures.(b) Which
An advertising campaign boosted sales of Crunchy Crust frozen pizza to a peak level of S0 dollars per month. A marketing study showed that after t months, monthly sales declined toDo sales decline more slowly or more rapidly as time increases? Answer by referring to a sketch of the graph of g(t)
The graphs in Figure 13 represent the positions s of moving particles as functions of time t. Match each graph with a description:(a) Speeding up(b) Speeding up and then slowing down(c) Slowing down(d) Slowing down and then speeding up
The fungus Fusarium exosporium infects a field of flax plants through the roots and causes the plants to wilt. Eventually, the entire field is infected. The percentage f (t) of infected plants as a function of time t (in days) since planting is shown in Figure 15.(a) What are the units of the rate
If an object in linear motion (but with changing velocity) covers Δs meters in Δt seconds, then its average velocity is v0 = Δs / Δt m/s. Show that it would cover the same distance if it traveled at constant velocity v0 over the same time interval. This justifies our calling Δs / Δt the
Let v = 20 √T as in Example 2. Estimate the instantaneous rate of change of v with respect to T when T = 300K.
Which graph in Figure 16 has the following property: For all x, the average rate of change over [0, x] is greater than the instantaneous rate of change at x. Explain.
Let Q(t) = t2. As in the previous exercise, find a formula for the average rate of change of Q over the interval [1, t] and use it to estimate the instantaneous rate of change at t = 1. Repeat for the interval [2, t] and estimate the rate of change at t = 2.
Find a formula for the average rate of change of f (x) = x3 over [2, x] and use it to estimate the instantaneous rate of change at x = 2.
The rate of change of atmospheric temperature with respect to altitude is equal to the slope of the tangent line to a graph. Which graph? What are possible units for this rate?
Compute the stone's average velocity over the time interval [0.5, 2.5] and indicate the corresponding secant line on a sketch of the graph of h(t). A stone is tossed vertically into the air from ground level with an initial velocity of 15 m/s. Its height at time t is h(t) = 15t − 4.9t2 m.
With an initial deposit of $100, the balance in a bank account after t years is f (t) = 100(1.08)t dollars. (a) What are the units of the rate of change of f (t)? (b) Find the average rate of change over [0, 0.5] and [0, 1]. (c) Estimate the instantaneous rate of change at t = 0.5 by computing the
Figure 8 shows the estimated number N of Internet users in Chile, based on data from the United Nations Statistics Division.(a) Estimate the rate of change of N at t = 2003.5.(b) Does the rate of change increase or decrease as t increases? Explain graphically.(c) Let R be the average rate of change
Fill in the tables and guess the value of the limit.1.3.
Verify each limit using the limit definition. For example, in Exercise 9, show that |3x ˆ’ 12| can be made as small as desired by taking x close to 4.13.15.
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ˆž or - ˆž.17.19.
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ˆž or - ˆž.21.23.
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ˆž or - ˆž.25.27.
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ˆž or - ˆž.29.31.
Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are ˆž or - ˆž.33.35.
The greatest integer function is defined by [x] = n, where n is the unique integer such that n ‰¤ x (a)(b)
Determine the one-sided limits numerically or graphically. If infinite, state whether the one sided limits are ˆž or ˆ’ˆž, and describe the corresponding vertical asymptote. In Exercise 46, [x] is the greatest integer function defined in Exercise 37.39.41.
Can f (x) approach a limit as x → c if f (c) is undefined? If so, give an example.
Determine the one-sided limits numerically or graphically. If infinite, state whether the one sided limits are ˆž or ˆ’ˆž, and describe the corresponding vertical asymptote. In Exercise 46, [x] is the greatest integer function defined in Exercise 37.43.45.
Determine the one-sided limits at c = 2, 4 of the function f (x) in Figure 12. What are the vertical asymptotes of f (x)?
Sketch the graph of a function with the given limits.49.51.
Determine the one-sided limits of the function f (x) in Figure 14, at the points c = 1, 3, 5, 6.
Plot the function and use the graph to estimated the value of the limit.55.57.
Can you tell whether f (x) exists from a plot of f (x) for x > 5? Explain.
Let n be a positive integer. For which n are the two infinite one-sided limits
In some cases, numerical investigations can be misleading. Plot f (x) = cos Ï€/x.(b) Show, by evaluating f (x) at x = ±1/2 ,±1/4 ,±1/6 , . . . , that you might be able to trick your friends into believing that the limit exists and is equal to L = 1.(c) Which sequence of evaluations might
Investigatenumerically for several values of n. Then guess the value in general.
Investigatefor (m, n) equal to (2, 1), (1, 2), (2, 3), and (3, 2). Then guess the value of the limit in general and check your guess for two additional pairs.
Plot the graph of f (x) = 2x - 8 / x - 3.(a) Zoom in on the graph to estimate(b) Explain whyf (2.99999) ‰¤ L ‰¤ f (3.00001)Use this to determine L to three decimal places.
Verify each limit using the limit definition. For example, in Exercise 9, show that |3x ˆ’ 12| can be made as small as desired by taking x close to 4.9.11.
State the Sum Law and Quotient Law.
Evaluate the limit using the Basic Limit Laws and the limits1. 3. 5.
Use the Quotient Law to prove that if f (x) exists and is nonzero, then
Evaluate the limit assuming that27. 29.
Can the Quotient Law be applied to evaluateExplain.
Give an example where (f (x) + g(x)) exists but neither f(x) nor g(x) exists.
Suppose that t g(t) = 12. Show that g(t) exists and equals 4.
(a) f (0) = 0.
(c) Illustrate (a) and (b) with the function f (x) = x2.
Referring to Figure 14, state whether f (x) is left-or right-continuous (or neither) at each point of discontinuity. Does f (x) have any removable discontinuities?
f(x) = [1 / 2x]
f(x) = x + 1 / 4x - 2
Solve the following
g(t) = tan 2t
f (x) = tan(sin x)
f(x) = 1 / 2x - 2-x
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section. 35. f(x) = 2 sin x + 3 cos x 37. f(x) = √x sin x
Show that the functionis continuous for x ‰ 1, 2. Then compute the right- and left-hand limits at x = 1, 2, and determine whether f (x) is left-continuous, right-continuous, or continuous at these points (Figure 17).
Are the following true or false? If false, state a correct version. (a) f (x) is continuous at x = a if the left- and right-hand limits of f (x) as x → a exist and are equal. (b) f (x) is continuous at x = a if the left- and right-hand limits of f (x) as x → a exist and equal f (a). (c) If the
Sketch the graph of f (x). At each point of discontinuity, state whether f is left- or right continuous.51.53.
Show that the functionhas a removable discontinuity at x = 4.
Find the value of the constant (a, b, or c) that makes the function continuous.57.59.
In Figure 16, determine the one-sided limits at the points of discontinuity. Which discontinuity is removable and how should f be redefined to make it continuous at this point?
Define g(t) = 21/(t−1) for t ≠ 0. Answer the following questions, using a plot if necessary. (a) Can g(1) be defined so that g(t) is continuous at t = 1? (b) How should g(1) be defined so that g(t) is left-continuous at t = 1?
Draw the graph of a function on [0, 5] with the given properties.65. f(x) has a removable discontinuity at x = 1, a jump discontinuity at x = 2, and
Evaluate using substitution67.69.
Use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous. 7. f(x) = x + sin x 9. f(x) = 3x + sin x 11. f(x) = 1 / x2 + 1
Suppose that f (x) and g(x) are discontinuous at x = c. Does it follow that f (x) + g(x) is discontinuous at x = c? If not, give a counterexample. Does this contradict Theorem 1 (i)?
In 2009, the federal income tax T (x) on income of x dollars (up to $82,250) was determined by the formulaSketch the graph of T (x). Does T (x) have any discontinuities? Explain why, if T (x) had a jump discontinuity, it might be advantageous in some situations to earn less money.
Give an example of functions f (x) and g(x) such that f (g(x)) is continuous but g(x) has at least one discontinuity.
Show that f (x) is a discontinuous function for all x where f (x) is defined as follows:Show that f (x)2 is continuous for all x.
Let f (x) = 5x2. Show that f (3 + h) = 5h2 + 30h + 45. Then show thatand compute f '(3) by taking the limit as h †’ 0.
Refer to Figure 12.11. Determine f €²(a) for a = 1, 2, 4, 7.13. Which is larger, f €²(5.5) or f €²(6.5)?
15. F(x) = 7x - 9 17. G(t) = 8 - 3t

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