New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
The Heaviside function (named after Oliver Heaviside, 1850–1925) is defined by:The Heaviside function can be used to “turn on” another function at a specific value in the domain, as seen in the four examples here. For each of the following, sketch the graph of ƒ.(a)ƒ(x) = H(x)x2(b) ƒ(x) =
Let ƒ(x) = x/|x|.(a) What are the domain and range of ƒ?(b) Sketch the graph of ƒ.(c) Express ƒ as a piecewise-defined function where each of the “pieces” is a constant.
The population (in millions) of Calcedonia 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.
Find all values of c such that ƒ(x) = x + 1/x2 + 2cx + 4 has domain R.
We define the first difference δ ƒ of a function ƒ by δ ƒ(x) = ƒ(x + 1) − ƒ(x).Show that if ƒ (x) = x2, then δ ƒ(x) = 2x + 1. Calculate δ ƒ for ƒ(x) = x and ƒ(x) = x3.
We define the first difference δ ƒ of a function ƒ by δ ƒ(x) = ƒ(x + 1) − ƒ(x).Show that δ(10x) = 9 · 10x and, more generally, that δ(bx) = (b − 1)bx.
We define the first difference δ ƒ of a function ƒ by δ ƒ(x) = ƒ(x + 1) − ƒ(x).Show that for any two functions ƒ and g, δ(ƒ + g) = δ ƒ + δg and δ(c ƒ) = cδ(ƒ), where c is any constant.
We define the first difference δ ƒ of a function ƒ by δ ƒ(x) = ƒ(x + 1) − ƒ(x).Suppose we can find a function P such that δP(x) = (x + 1)k and P(0) = 0. Prove that P(1) = 1k , P(2) = 1k + 2k, and, more generally, for every whole number n, P(n) = 1 + 2k + ... +n't
We define the first difference δ ƒ of a function ƒ by δ ƒ(x) = ƒ(x + 1) − ƒ(x).Show that ifthen δP = (x + 1). Then apply Exercise 46 to conclude thatData From Exercise 46We define the first difference δ ƒ of a function ƒ by δ ƒ(x) = ƒ(x + 1) − ƒ(x).Suppose we can find a function
We define the first difference δ ƒ of a function ƒ by δ ƒ(x) = ƒ(x + 1) − ƒ(x).Calculate δ(x3), δ(x2), and δ(x). Then find a polynomial P of degree 3 such that δP = (x + 1)2 and P(0) = 0. Conclude that P(n) = 12 + 22 + · · · + n2.
Find the slope, the y-intercept, and the x-intercept of the line with the given equation.y = 3x + 12
Find the slope, the y-intercept, and the x-intercept of the line with the given equation.y = 4 − x
Find the slope, the y-intercept, and the x-intercept of the line with the given equation.4x + 9y = 3
Find the slope, the y-intercept, and the x-intercept of the line with the given equation.y − 3 = 1/2(x − 6)
Find the slope of the line.y = 3x + 2
Find the slope of the line.y = 3(x − 9) + 2
Find the slope of the line.3x + 4y = 12
Find the slope of the line.3x + 4y = −8
Find the equation of the line with the given description.Slope 3, y-intercept 8
Find the equation of the line with the given description.10. Slope −2, y-intercept 3
Find the equation of the line with the given description.Slope 3, passes through (7, 9)
Find the equation of the perpendicular bisector of the segment joining (1, 2) and (5, 4) (Figure 12). The midpoint Q of the segment joining (a, b) and (c, d) is (a + c/2 , b + d/2). Perpendicular bisector (1, 2) (5, 4) → X
Show that if a, b ≠ 0, then the line with x-intercept x = a and y-intercept y = b has equation (Figure 13) X a y + b = 1 ||
Find y such that (3, y) lies on the line of slope m = 2 through (1, 4).
Determine whether there exists a constant c such that the line cx − 2y = 4:(a) Has slope 4 (b) Passes through (1, −4)(c) Is horizontal (d) Is vertical
Suppose that the number of Bob’s Bits computers that can be sold when its price is P (in dollars) is given by a linear function N(P), where N(1000) = 10,000 and N(1500) = 7500.(a) Determine N(P).(b) What is the slope of the graph of N(P), including units? Describe what the slope represents.(c)
Suppose that the demand for Colin’s kidney pies is linear in the price P. Further, assume that he can sell 100 pies when the price is $5.00 and 40 pies when the price is $10.00.(a) Determine the demand N (number of pies sold) as a function of the price P (in dollars).(b) What is the slope of the
In each case, identify the slope and give its meaning with the appropriate units.(a) The function N = −70t + 5000 models the enrollment at Maple Grove College during the fall of 2018, where N represents the number of students and t represents the time in weeks since the start of the semester.(b)
In each case, identify the slope and give its meaning with the appropriate units.(a) The function N = 3.9T − 178.8 models the number of times, N, that a cricket chirps in a minute when the temperature is T° Celsius.(b) The function V = 47, 500d gives the volume (V, in gallons) of molasses in the
Find an expression for the velocity v as a linear function of t that matches the following data: t (s) v (m/s) 0 39.2 2 58.6 4 78 6 97.4
Materials expand when heated. Consider a metal rod of length L0 at temperature T0. If the temperature is changed by an amount ΔT, then the rod’s length approximately changes by ΔL = αL0ΔT, where α is the thermal expansion coefficient and ΔT is not an extreme temperature change. For steel,
Do the points (0.5, 1), (1, 1.2), (2, 2) lie on a line?
Show that ƒ is linear of slope m if and only if ƒ(x + h) − ƒ(x) = mh (for all x and h).That is to say, prove the following two statements:(a) ƒ is linear of slope m implies that ƒ(x + h) − ƒ(x) = mh (for all x and h).(b) ƒ (x + h) − ƒ(x) = mh (for all x and h) implies that ƒ is
Find the roots of the quadratic polynomials:(a) ƒ(x) = 4x2 − 3x − 1 (b) ƒ(x) = x2 − 2x − 1
Complete the square and find the minimum or maximum value of the quadratic function.y = x2 + 2x + 5
Complete the square and find the minimum or maximum value of the quadratic function.y = x2 − 6x + 9
Complete the square and find the minimum or maximum value of the quadratic function.y = −9x2 + x
Complete the square and find the minimum or maximum value of the quadratic function.y = x2 + 6x + 2
Complete the square and find the minimum or maximum value of the quadratic function.y = 2x2 − 4x − 7
Complete the square and find the minimum or maximum value of the quadratic function.y = −4x2 + 3x + 8
Complete the square and find the minimum or maximum value of the quadratic function.y = 3x2 + 12x − 5
Complete the square and find the minimum or maximum value of the quadratic function.y = 4x − 12x2
If the alleles A and B of the cystic fibrosis gene occur in a population with frequencies p and 1 − p (where p is between 0 and 1), then the frequency of heterozygous carriers (carriers with both alleles) is 2p(1 − p). Which value of p gives the largest frequency of heterozygous carriers?
Let ƒ(x) = x2 + x − 1.(a) Show that the lines y = x + 3, y = x − 1, and y = x − 3 intersect the graph of ƒ in two, one, and zero points, respectively.(b) Sketch the graph of f and the three lines from (a).(c) Describe the relationship between the graph of ƒ and the lines y = x + c as c
Let ƒ(x) = x2 + 2x − 21.(a) Show that the lines y = 3x − 25, y = 6x − 25, and y = 9x − 25 intersect the graph of ƒ in zero, one, and two points, respectively.(b) Sketch the graph of f and the three lines from (a).(c) Describe the relationship between the graph of ƒ and the lines y = cx
If objects of weights x and w1 are suspended from the balance in Figure 14(A), the cross-beam is horizontal if bx = aw1. If the lengths a and b are known, we may use this equation to determine an unknown weight x by selecting w1 such that the cross-beam is horizontal. If a and b are not known
Let a, b > 0. Show that the geometric mean √ab is not larger than the arithmetic mean (a + b)/2. Consider (a1/2 − b1/2)2.
Find numbers x and y with sum 10 and product 24. Find a quadratic polynomial satisfied by x.
Show, by completing the square, that the parabolacan be obtained from y = ax2 by a vertical and horizontal translation. y = ax² +bx+c
Complete the square and use the result to derive the quadratic formula for the roots of ax2 + bx + c = 0.
Prove Viete’s Formulas: The quadratic polynomial with α and β as roots is x2 + bx + c, where b = −α − β and c = αβ.
What is the slope of the line y = −4x − 9?
When is the line ax + by = c parallel to the y-axis? To the x-axis?
Suppose y = 3x + 2. What is Δy if x increases by 3?
What is the minimum of ƒ(x) = (x + 3)2 − 4?
What is the result of completing the square for ƒ(x) = x2 + 1?
Describe how the parabolas y = ax2 − 1 change as a changes from −∞ to ∞.
Describe how the parabolas y = x2 + bx change as b changes from −∞ to ∞.
Find the equation of the line with the given description.Perpendicular to 3x + 5y = 9, passes through (2, 3)
Find the equation of the line with the given description.Vertical, passes through (−4, 9)
Find an equation of the line with x-intercept x = 4 and y-intercept y = 3.
Show that if ƒ and g are linear functions such that ƒ (0) = g(0) and ƒ (1) = g(1), then ƒ = g.
Show that if |a − 5| < 12 and |b − 8| < 12, then|(a + b) − 13| < 1. Use the triangle inequality (|a + b| ≤ |a| + |b|).
Suppose that |x − 4| ≤ 1.(a) What is the maximum possible value of |x + 4|?(b) Show that |x2 − 16| ≤ 9.
Assume that p is a function that is defined for x > 0 and satisfies p(a/b) = p(b) − p(a). Prove that ƒ(x) = p(2 − x/2 + x) is an odd function.
Which of the following equations is incorrect? (a) 3².35 = 37 (c) 3².2³ = 1 (b) (√5)4/3 = 52/3 (d) (2-2)-² = 16
Rewrite as a whole number (without using a calculator): (a) 70 (c) 過 (e) 8-1/3.85/3 (b) 10² (2-² +5-²) (d) 274/3 (f) 3.4¹/4-12-2-3/2
Use the binomial expansion formula to expand (2 − x)7.
Use the binomial expansion formula to expand (x + 1)9.
Which of (a)–(d) are true for a = 4 and b = −5?(a) −2a < −2b (b) |a| < −|b| (c) ab < 0(d) 1/a < 1/b
Express the interval in terms of an inequality involving absolute value.[−2, 2]
Express the interval in terms of an inequality involving absolute value.(−4, 4)
Express the interval in terms of an inequality involving absolute value.[−4, 0]
Express the interval in terms of an inequality involving absolute value.[−1, 8]
Express the interval in terms of an inequality involving absolute value.(−2.4, 1.9)
Write the inequality in the form a < x < b.|x| < 8
Write the inequality in the form a < x < b.|x − 12| < 8
Write the inequality in the form a < x < b.|2x + 1| < 5
Express the set of numbers x satisfying the given condition as an interval.|x| < 4
Express the set of numbers x satisfying the given condition as an interval.|x| ≤ 9
Express the set of numbers x satisfying the given condition as an interval.|x + 7| < 2
Express the set of numbers x satisfying the given condition as an interval.|3x + 5| < 1
Describe the set as a union of finite or infinite intervals.{x : |2x + 4| > 3}
Describe the set as a union of finite or infinite intervals.{x : |x2 − 1| > 2}
Describe the set as a union of finite or infinite intervals.{x : |x2 + 2x| > 2}
Describe {x : x/x + 1 < 0} as an interval. Consider the sign of x and x + 1 individually.
Describe {x : x2 + 2x < 3} as an interval. Plot y = x2 + 2x − 3.
Describe the set of real numbers satisfying |x − 3| = |x − 2| + 1 as a half-infinite interval.
Show that if a > b, and a, b ≠ 0, then b−1 > a−1, provided that a and b have the same sign. What happens if a > 0 and b < 0?
Which x satisfies both |x − 3| < 2 and |x − 5| < 1?
Prove that |x| − |y| ≤ |x − y|. Apply the triangle inequality to y and x − y.
Express r1 = 0.27 as a fraction. 100r1 − r1 is an integer. Then express r2 = 0.2666 . . . as a fraction.
Represent 1/7 and 4/27 as repeating decimals.
Plot each pair of points and compute the distance between them:(a) (1, 4) and (3, 2) (b) (2, 1) and (2, 4)
Plot each pair of points and compute the distance between them:(a) (0, 0) and (−2, 3) (b) (−3, −3) and (−2, 3)
Find all points in the xy-plane with integer coordinates located at a distance 5 from the origin. Then find all points with integer coordinates located at a distance 5 from (2, 3).
Give an example of a function whose domain D has three elements and whose range R has two elements. Does a function exist whose domain D has two elements and whose range R has three elements?
Find the domain and range of the function.g(t) = t4
Find the domain and range of the function.ƒ(x) = x3
Find the domain and range of the function.g(t) = √2 − t
Showing 8200 - 8300
of 8339
First
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
Step by Step Answers
Get 50% OFF
Claim Now
