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mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Find the equation of the line.Line passing through (−1, 4) and (−1, 6)
Assume that 0 ≤ θ < π/2.Find cos θ and tanAssume that 0 ≤ θ < π/2.Find cos θ and tan θ if sin θ = 3/5.θ if sin θ = 3/5.
Investigate the behavior of the function as n or x grows large by making a table of function values and plotting a graph (see Example 4). Describe the behavior in words.ƒ(x) = (x tan 1/x)x -1- OF FIGURE 13 horizontally by a factor of 2 EXAMPLE 4 Sketch the graph of f(x) = 3 cos (2(x + 7)) over [0,
Find the equation of the line.Line of slope 6 through (9, 1)
Assume that 0 ≤ θ < π/2.Find sin θ, secθ, and cot θ if tan θ = 2/7.
Find the equation of the line.Line of slope −3/2 through (4, −12)
Investigate the behavior of the function as n or x grows large by making a table of function values and plotting a graph (see Example 4). Describe the behavior in words.ƒ(x) = (x tan 1/x)x2 OF FIGURE 13 horizontally by a factor of 2 EXAMPLE 4 Sketch the graph of f(x) = 3 cos (2(x+)) over [0,
Assume that 0 ≤ θ < π/2.Find cos 2θ if sin θ = 1/5.
The graph of ƒ(θ) = A cos θ + Bsin θ is a sinusoidal wave for any constants A and B. Confirm this for (A, B) = (1, 1), (1, 2), and (3, 4) by plotting ƒ.
Assume that 0 ≤ θ < π/2.Find sin 2θ and cos 2θ if tan θ =√2.
Find the maximum value of ƒ for the graphs produced in Exercise 23. Can you guess the formula for the maximum value in terms of A and B?Data From Exercise 23The graph of ƒ (θ) = A cos θ + Bsin θ is a sinusoidal wave for any constants A and B. Confirm this for (A, B) = (1, 1), (1, 2), and (3,
Find the intervals on which ƒ(x) = x(x + 2)(x − 3) is positive by plotting a graph.
Find cos θ and sin θ if tan θ = 4 and π ≤ θ < 3π/2.
Does the following table of market data suggest a linear relationship between price and number of homes sold during a one-year period? Explain. Price (thousands of $) No. of homes sold 180 195 127 118 220 240 103 91
Find the set of solutions to the inequality (x2 − 4)(x2 − 1) < 0 by plotting a graph.
Does the following table of revenue data for a computer manufacturer suggest a linear relation between revenue and time? Explain. Year Revenue (billions of $) 2005 2009 13 18 2011 2014 15 11
Find cos θ if cot θ = 4/3 and sinθ < 0.
Let ƒ1(x) = x and define a sequence of functions by ƒn+1(x) = 1/2 (ƒn(x) + x/ƒn(x)). For example,ƒ2(x) = 1/2 (x + 1). Use a computer algebra system to compute ƒn(x) for n = 3, 4, 5 and plot y = ƒn(x) together with y = √x for x ≥ 0. What do you notice?
Set P0(x) = 1 and P1(x) = x. The Chebyshev polynomials (useful in approximation theory) are defined inductively by the formula Pn+1(x) = 2xPn(x) − Pn−1(x).(a) Show that P2(x) = 2x2 − 1.(b) Compute Pn(x) for 3 ≤ n ≤ 6 using a computer algebra system or by hand, and plot y = Pn(x) over
Find the values of sin θ, cosθ, and tanθ for the angles corresponding to the eight points on the unit circles in Figures 22(A) and (B). (A) (0.3965, 0.918) -x D X (B) (0.3965, 0.918)
Suppose that a cell phone plan that is offered at a price of P dollars per month attracts C customers, where C(P) is a linear demand function for $100 ≤ P ≤ $500. Assume C(100) = 1,000,000 and C(500) = 100,000.(a) Determine the demand function C(P).(b) What is the slope of the graph of C(P)?
Refer to Figure 23(A). Express the functions sin θ, tanθ, and cscθ in terms of c. 0 (A) с
Suppose that Internet domain names are sold at a price of $P per month for $2 ≤ P ≤ $100. The number of customers C who buy the domain names is a linear function of the price. Assume that 10,000 customers buy a domain name when the price is $2 per month, and 1000 customers buy when the price is
Refer to Figure 23(B). Compute cos ψ, sinψ, cot ψ, and cscψ. 4 1 0.3 (B)
Express cos (θ + π/2) and sin (θ + π/2) in terms of cos θ and sin θ. Find the relation between the coordinates (a, b) and (c, d) in Figure 24. (c,d) y (a, b) 1 x
Find the roots of ƒ (x) = x4 − 4x2 and sketch its graph. On which intervals is ƒ decreasing?
Let h(z) = −2z2 + 12z + 3. Complete the square and find the maximum value of h.
Use addition formulas and the values of sin θ and cos θ for θ = π/3 , π/4 to compute sin 7π/2 and cos 7π/12 exactly.
Let ƒ (x) be the square of the distance from the point (2, 1) to a point (x, 3x + 2) on the line y = 3x + 2. Show that ƒ is a quadratic function, and find its minimum value by completing the square.
Use addition formulas and the values of sin θ and cos θ for θ = π/3 , π/4 to compute sin π/12 and cos π/12 exactly.
Prove that x2 + 3x + 3 ≥ 0 for all x.
Sketch the graph over [0, 2π].ƒ(θ) = 2 sin 4θ
Sketch the graph by hand.y = t4
Sketch the graph by hand.y = t5
Sketch the graph over [0, 2π].ƒ(θ) = cos(2(θ − π/2)
Sketch the graph by hand.y = sin θ/2
Sketch the graph over [0, 2π].ƒ(θ) = cos(2θ − π/2)
Sketch the graph by hand.y = |x − 3|
Sketch the graph over [0, 2π].ƒ(θ) = sin (2(θ − π/2) + π) + 2
Show that the parabola y = x2 consists of all points P such that d1 = d2, where d1 is the distance from P to (0, 1/4) and d2 is the distance from P to the line y = −1/4 (Figure 15). -4 |-|- y d₁ y=x² P=(x, x²) d₂ X
Is there a function that is both even and odd?
Which of (a)–(d) are true for a = −3 and b = 2?(a) a < b (b) |a| < |b| (c) ab > 0(d) 3a < 3b
The definition of cos θ using right triangles applies when (choose the correct answer):(a) 0 < θ < π/2(b) 0 < θ < π (c) 0 < θ < 2π
Find the equation of the line with the given description.Slope −5, passes through (0, 0)
Find the equation of the line with the given description.Horizontal, passes through (0, −2)
Find the equation of the line with the given description.Passes through (−1, 4) and (2, 7)
Find the equation of the line with the given description.Passes through (1, 4) and (12, −3)
Find the equation of the line with the given description.Horizontal, passes through (8, 4)
Find the equation of the line with the given description.Slope 3, x-intercept 6
In Exercise identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = x2/x + sin x
Find the equation of the line with the given description.Parallel to y = 3x − 4, passes through (1, 1)
Write the inequality in the form a < x < b.|3x − 4| < 2
Is ƒ(x) = 2x2 a transcendental function?
Show that if ƒ and g are linear, then so is ƒ + g. Is the same true of ƒ g?
Sketch the graph of y = x2 − 6x + 8 by plotting the roots and the minimum point.
Determine the domain of the function. g(t) = t2/3
Determine the domain of the function.h(z) = z3 + z−3
Determine the domain of the function.ƒ(x) = 1/x2 + 4
Determine the domain of the function.G(u) = 1/u2 − 4
Determine the domain of the function.ƒ(x) = √x/x2 − 9
Determine the domain of the function.ƒ(x) = x−4 + (x − 1)−3
Determine the domain of the function.F(s) = sin(s/s + 1)
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = 4x3 + 9x2 − 8
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = x−4
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = √x
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = √1 − x2
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = 2x
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = 2x3 + 3x/9 − 7x2
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = 3x − 9x−1/2 / 9 − 7x2
This exercise combined with Exercise 46 shows that for all whole numbers k, there exists a polynomial P satisfying Eq. (1). The solution requires the Binomial Theorem and proof by induction.(a) Show that δ(xk+1) = (k + 1) xk + · · · , where the dots indicate terms involving smaller powers of
Give an example of numbers a and b such that a < b and |a| > |b|
Which numbers satisfy |a| = a? Which satisfy |a| = −a? What about |−a| = a?
Are there numbers a and b such that |a + b| > |a| + |b|?
What are the coordinates of the point lying at the intersection of the lines x = 9 and y = −4?
What is the radius of the circle with equation (x − 7)2 + (y − 8)2 = 9?
The equation ƒ(x) = 5 has a solution if (choose one):(a) 5 belongs to the domain of ƒ.(b) 5 belongs to the range of ƒ.
What kind of symmetry does the graph have if ƒ(−x) = −ƒ(x)?
Explain why both ƒ(x) = x3 + 1 and g(x) = 1/x3 + 1 are rational functions.
Is y = |x| a polynomial function? What about y = |x2 + 1|?
Explain why both ƒ(x) = x/1 − x4 and g(x) = x/√1 − x4 are algebraic functions.
We have ƒ(x) = (x + 1)1/2, g(x) = x−2 + 1, h(x) = 2x, and k(x) = x2 + 1. Identify which of the functions satisfies each of the following.(a) Transcendental(b) Polynomial(c) Rational but not polynomial(d) Algebraic but not rational
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = sin(x2)
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = x / √x + 1
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = x2 + 3x−1
Identify each of the following functions as polynomial, rational, algebraic, or transcendental.ƒ(x) = sin(3x)
Show that ƒ(x) = x2 + 3x−1 and g(x) = 3x3 − 9x + x−2 are rational functions—that is, quotients of polynomials.
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(x) = √x, g(x) = x + 1
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(x) = 1/x , g(x) = x−4
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(x) = 1/√x , g(x) = x2
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(x) = |x|, g(θ) = sin θ
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(θ) = cos θ, g(x) = x3 + x2
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(x) = 1/x2 + 1 , g(x) = x−2
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(t) = 1/√t , g(t) = −t2
The volume V and surface area of a sphere [Figure 6(A)] are expressed in terms of radius r by V(r) = 43 πr3 and S (r) = 4πr2, respectively. Determine r(V), the radius as a function of volume. Then determine S (V), the surface area as a function of volume, by computing the composite S ◦ r(V).
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(t) = √t, g(t) = 1 − t3
Draw the graphs of each of the piecewise-defined functions. f(x) = [x² + 3 when x < 0 when x > 0
A tetrahedron is a polyhedron with four equilateral triangles as its faces [Figure 6(B)]. The volume V and surface area of a tetrahedron are expressed in terms of the side-length L of the triangles by V(L) = √2L3/12 and S (L) = √3L2, respectively. Determine L(V), the side length as a function
Draw the graphs of each of the piecewise-defined functions. f(x) = [x+1 1-x when x < 0 when x ≥ 0
Draw the graphs of each of the piecewise-defined functions. f(x) = (x² -x² when x < 0 when x ≥ 0
Draw the graphs of each of the piecewise-defined functions. f(x) = (²x-2 (2x-2 when x < 0 when x > 0
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![OF FIGURE 13 horizontally by a factor of 2 EXAMPLE 4 Sketch the graph of f(x) = 3 cos (2(x+)) over [0, 27].](https://dsd5zvtm8ll6.cloudfront.net/images/question_images/1698/1/5/2/7976537c15d275781698152797826.jpg)
