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mathematics
precalculus

Questions and Answers of Precalculus

Evaluate the integral. sin 0 d0 T/6
Evaluate the integral. -2/3 dx 8, х
Evaluate the integral (x dx
Evaluate the integral. ( (1 – 8v³ + 16v') dv
Evaluate the integral. |(- +3) dt
Evaluate the integral. 100 dx
Evaluate the integral. •3 (x² + 2x - 4) dx
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. V1 + t² dt sin x y =
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. п/4 0 tan 0 d0 y = (х
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. cx4 У cos²0 d0 —
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. * 3x+2 dt .3 1 + t³ 1
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. 2 h(x) dz z* + 1
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. In t dt h(x)
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. R(y) = t° sin t dt
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. 1 + sec t dt F(x): х 1 + sec t dt = 1 + sec t dt х
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(s) = [ (1 – t²)° dt 15
Let g(x) = ∫ x0 f (g) dt, where f is the function whose graph is shown.(a) Evaluate g(0) and g(6).(b) Estimate t(x) for x = 1, 2, 3, 4, and 5.(c) On what interval is t increasing? (d) Where
FindChoose xi* to be the geometric mean of xi-1 and xi (that is, xi* = √xi-1xi) and use the identity fx² dx. -2
Prove Property 3 of integrals.
Which of the integralsis larger? Why? ГО.5 o*cos x dx ГО.5 °cos(x²) dx, fº S
Use Property 8 to estimate the value of the integral. х3 dx
Iff(x) = cos x ................... 0 < x < 3π/4Evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does
The table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003.(a) By using an argument similar to that in Example 4, estimate the
Evaluate the upper and lower sums for f (x) = - 1 sin x, 0 < x < π, with n = 2, 4, and 8. Illustrate with diagrams like Figure 14.
An arc PQ of a circle subtends a central angle θ as in the figure. Let A(θ) be the area between the chord PQ and the arc PQ. Let B(θ) be the area between the tangent lines PR, QR, and the arc.
Use the guidelines of Section 4.5 to sketch the curve.y = 4x - tan x, -π/2 < x < π/2
Use the guidelines of Section 4.5 to sketch the curve.y = ex sin x, - π < x < π
Use the guidelines of Section 4.5 to sketch the curve.y = √1 - x + √1 + x
Use the guidelines of Section 4.5 to sketch the curve.y = (x - 1)3/x2
Use the guidelines of Section 4.5 to sketch the curve.y = 1/x2 - 1/(x - 2)2
Use the guidelines of Section 4.5 to sketch the curve.y = -2x3 - 3x2 + 12x + 5
Use the guidelines of Section 4.5 to sketch the curve.y = 2 - 2x - x3
Sketch the graph of a function that satisfies the given conditions. 0 for 0 < x < 3, f"(x) < 0 for x > 3, f'(x) < 0 for 0 < x < 2, lim f(x) = -2 " style="" class="fr-fic fr-dib"> f is odd, , f'(x) >
Sketch the graph of a function that satisfies the given conditions. f(0) = 0, f'(-2) = f'(1) = f'(9) = 0, lim f(x) = 0, lim f(x) = х>6 f'(x) < 0 on (-∞, -2), (1, 6), and (9, 0), f'(x) > 0 on (-2,
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.limx→0 x/ex = 1
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If limx→0 f(x) = 1 and limx→0 g(x) = ∞,
Use Newton’s method to find the absolute maximum value of the function f (x) = x cos x, 0 < x < π, correct to six decimal places.
Consider the situation in Exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land ($400,000/km). You may suspect that in some instances,
An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with a plane, then the magnitude of the force iswhere
In the theory of relativity, the energy of a particle iswhere m0 is the rest mass of the particle, λ is its wave length, and h is Planck’s constant. Sketch the graph of E as a function of λ. What
Find the limit. Use 1’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If 1’Hospital’s Rule doesn’t apply, explain why. lim (tan 2x)*
Find dy/dx by implicit differentiation.ey sin x = x + xy
Find dy/dx by implicit differentiation.y cos x = x2 + y2
Find dy/dx by implicit differentiation.xey = x - y
Show that each sequence is arithmetic. Find the common difference and write out the first four terms.{Sn} = {ln 3n}
Show that each sequence is arithmetic. Find the common difference and write out the first four terms. {t,} 3 4
Show that each sequence is arithmetic. Find the common difference and write out the first four terms.
Show that each sequence is arithmetic. Find the common difference and write out the first four terms.{an} = {4 – 2n}
Show that each sequence is arithmetic. Find the common difference and write out the first four terms.{cn} = {6 – 2n}
Show that each sequence is arithmetic. Find the common difference and write out the first four terms.{bn} = {3n + 1}
Show that each sequence is arithmetic. Find the common difference and write out the first four terms.{an} = {2n – 5}
Show that each sequence is arithmetic. Find the common difference and write out the first four terms.{Sn} = { n – 5}
Show that each sequence is arithmetic. Find the common difference and write out the first four terms.{Sn} = {n + 4}
True or False .The sum Sn of the first n terms of an arithmetic sequence {an} whose first term is a1 can be found using the formula Sn = n/2(a1 + an).
If the fifth term of an arithmetic sequence is 12 and the common difference is 5, then the sixth term of the sequence is ______.
True or False.For an arithmetic sequence whose first term is and whose common difference is d, the nth term is determined by the formula an = a1 + nd.
In a(n)_________sequence, the difference between successive terms is a constant.
If is a point on the terminal side of the angle at a distance r from the origin, then_________.
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.sin x - cos x = x
Use a graphing utility to solve equation. Express the solution(s) rounded to two decimal places.sin x + cos x = x
Make up three quadratic equations: one having two distinct solutions, one having no real solution, and one having exactly one real solution.
Explain the benefits of evaluating the discriminant of a quadratic equation before attempting to solve it.
Describe three ways you might solve a quadratic equation. State your preferred method; explain why you chose it.
Factor 2x2 – x – 3.
Show that any vector field of the formF(x, y, z) = f (y, z) i + g(x, z) j + h(x, y) kis incompressible.
Show that any vector field of the formF(x, y, z) = f (x) i + g(y) j + h(z) kwhere f , g, h are differentiable functions, is irrotational.
Is there a vector field G on R3 such that curl G = (x, y, z)? Explain.
Is there a vector field G on R3 such that curl G = (x sin y, cos y, z - xy)? Explain.
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f .F(x, y, z) = ex sin yz i + zex cos yz j + yex cos yz k
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f .F(x, y, z) = eyz i + xzeyz j + xyeyz k
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f .F(x, y, z) = i + sin z j + y cos z k
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f .F(x, y, z) = z cos y i + xz sin y j + x cos y k
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f .F(x, y, z) = xyz4 i + x2z4 j + 4x2yz3 k
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f .F(x, y, z) = y2z3 i + 2xyz3 j + 3xy2z2 k
The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is indepen- dent of z and its z-component is 0.)(a) Is div F positive, negative, or
The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is indepen- dent of z and its z-component is 0.)(a) Is div F positive, negative, or
The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is indepen- dent of z and its z-component is 0.)(a) Is div F positive, negative, or
Find (a) the curl and (b) the divergence of the vector field.F(x, y, z) = (arctan(xy), arctan(yz), arctan(zx))
Find (a) the curl and (b) the divergence of the vector field.F(x, y, z) = (ex sin y, ey sin z , ez sin x)
Find (a) the curl and (b) the divergence of the vector field.F(x, y, z) = √x/1 + z i + √y/1 + x j + √z/1 + y k
Find (a) the curl and (b) the divergence of the vector field.F(x, y, z) = ln(2y + 3z) i + ln(x + 3z) j + ln(x + 2y) k
Find (a) the curl and (b) the divergence of the vector field.F(x, y, z) = sin yz i + sin zx j + sin xy k
Find (a) the curl and (b) the divergence of the vector field.F(x, y, z) = xyez i + yzex k
Find (a) the curl and (b) the divergence of the vector field.F(x, y, z) = x3yz2 j + y4z3 k
Find (a) the curl and (b) the divergence of the vector field.F(x, y, z) = xy2z2 i + x2yz2 j + x2y2z k
Use Green’s Theorem to prove the change of variables formula for a double integral (Formula 15.9.9) for the case where f (x, y) = 1:Here R is the region in the xy-plane that corresponds to the
Complete the proof of the special case of Green’s Theorem by proving Equation 3.
Calculate ʃC F • dr, where F(x, y) = (x2 + y, 3x - y2) and C is the positively oriented boundary curve of a region D that has area 6.
Use Exercise 25 to find the moment of inertia of a circular disk of radius a with constant density about a diameter. (Compare with Example 15.4.4.)
Use Exercise 22 to find the centroid of the triangle with vertices (0, 0), (a, 0), and (a, b), where a > 0 and b > 0.
Use Exercise 22 to find the centroid of a quarter-circular region of radius a.
Let D be a region bounded by a simple closed path C in the xy-plane. Use Green’s Theorem to prove that the coordinates of the centroid (x, y) of D are where A is the area of D.
If a circle C with radius 1 rolls along the outside of the circle x2 + y2 = 16, a fixed point P on C traces out a curve called an epicycloid, with parametric equations x = 5 cos t - cos 5t, y = 5 sin
Use one of the formulas in (5) to find the area under one arch of the cycloid x = t - sin t, y = 1 - cos t.
A particle starts at the origin, moves along the x-axis to (5, 0), then along the quarter-circle x2 + y2 = 25, x > 0, y > 0 to the point (0, 5), and then down the y-axis back to the origin. Use
Use Green’s Theorem to find the work done by the force F(x, y) = x(x + y) i + xy2 j in moving a particle from the origin along the x-axis to (1, 0), then along the line segment to (0, 1), and then
Verify Green’s Theorem by using a computer algebra system to evaluate both the line integral and the double integral.P(x, y) = 2x - x3y5, Q(x, y) = x3y8, C is the ellipse 4x2 + y2 = 4
Verify Green’s Theorem by using a computer algebra system to evaluate both the line integral and the double integral.P(x, y) = x3y4, Q(x, y) = x5y4, C consists of the line segment from (-π/2, 0)
Use Green’s Theorem to evaluate ʃC F • dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = (y - cos y, x sin y), C is the circle (x - 3)2 + (y + 4)2 = 4 oriented
Use Green’s Theorem to evaluate ʃC F • dr. (Check the orientation of the curve before applying the theorem.)F(x, y) = (e-x + y2, e-y + x2), C consists of the arc of the curve y = cos x from

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