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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Suppose that cos θ = 1/3.(a) Show that if 0 ≤ θ < π/2, then sin θ = 2 √2/3 and tanθ = 2√2.(b) Find sin θ and tan θ if 3π/2 ≤ θ < 2π.
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(n) = 4n + 1/6n − 5 -1- OF
Find the equation of the line.Line passing through (−1, 4) and (2, 6)
Assume that 0 ≤ θ < π/2.Find sin θ and tan θ if cos θ = 5/13.
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 + 6/x − 4)x
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
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
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
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
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
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)
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
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
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).
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
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
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
Calculate the composite functions ƒ ◦ g and g ◦ ƒ, and determine their domains.ƒ(t) = √t, g(t) = 1 − t3

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