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mathematics
calculus 6th edition

Questions and Answers of Calculus 6th edition

Find the surface area of a sphere of radius a.
Find the area of the part of the paraboloid z = x2 + y2 that lies under the plane z = 9.
Solve the equation y" + y' – 2y = x2.
A spring with a mass of 2 kg has natural length 0.5 m. A force of 25.6 N is required to maintain it stretched to a length of 0.7 m. If the spring is stretched to a length of 0.7 m and then released
Solve the differential equation or initial-value problem using the method of undetermined coefficients.y" + 3y' + 2y = x2
Use power series to solve the equation y" + y = 0.
Solve the differential equation.y" – 2y' – 15y = 0
Solve the equation y" + y' – 6y = 0.
Solve y" + 4y = e3x.
Suppose that the spring of Example 1 is immersed in a fluid with damping constant c = 40. Find the position of the mass at any time t if it starts from the equilibrium position and is given a push to
Solve the differential equation or initial-value problem using the method of undetermined coefficients.y" + 9y = e3x
Solve 3 d²y dx² dy dx - y = 0.
Solve y" – 2xy' + y = 0.
Solve the differential equation.y" + 4y' + 4y = 0
Find the charge and current at time in the circuit of Figure 7 ifand the initial charge and current are both 0.Figure 7 R = 40 0, L = 1 H, C = 16 Xx 10-4F, E(t) = 100 cos 10t,
Solve y" + y' – 2y = sin x.
Solve the differential equation or initial-value problem using the method of undetermined coefficients.y" – 2y' = sin 4x
Solve the equation 4y" + 12y' + 9y = 0.
Solve the differential equation.y" + 16y = 0
Solve y" – 4y = xex + cos2x.
Solve the differential equation or initial-value problem using the method of undetermined coefficients.y" + 6y' + 9y = 1 + x
Solve y" + y = sin x.
Solve the initial-value problem y"+y' - 6y = 0 y(0) = 1 y'(0) = 0
Solve the differential equation.d2y/dx2 – 4 dy/dx + 5y = e2x
Solve the differential equation.9y" – 12y´ + 4y = 0
Determine the form of the trial solution for the differential equation y" – 4y' + 13y = e2x cos 3x.
Solve the differential equation.d2y/dx2 + dy/dx – 2y = x2
Solve the differential equation or initial-value problem using the method of undetermined coefficients.y" + 2y' + y = xe–x
Solve the initial-value problem y" + y = 0 y(0) = 2 y'(0) = 3
Solve the differential equation.25y" + 9y = 0
Solve the equation y" + y = tan x, 0 < x < π/2.
Solve the boundary-value problem y" + 2y' + y = 0 y(0) = 1 y(1) = 3
Solve the differential equation or initial-value problem using the method of undetermined coefficients.y" + y = ex + x3, y(0) = 2. y'(0) = 0
Solve the differential equation.y' = 2y"
Solve the differential equation. d²y dy +2 -y=0 dt² dt
Solve the differential equation or initial-value problem using the method of undetermined coefficients.y" – 4y = ex cos x, y(0) = 1, y'(0) = 2
Solve the differential equation. 8. d²y dt² + 12 dy dt + 5y = 0
Solve the differential equation. 100 d²P dt² + 200 dP dt + 101P = 0
Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients.kky" + 9y = e2x + x2 sin x
Solve the initial-value problem.9y" + y = 3x + e–x, y(0) = 1, y'(0) = 2
Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients.y" + 9y' = xe–x cos πx
Graph the two basic solutions of the differential equation and several other solutions. What features do the solutions have in common?d2ydx2 + 4 dy/dx + 20y = 0
Graph the two basic solutions of the differential equation and several other solutions. What features do the solutions have in common?5 d2y/dx2 – 2 dy/dx – 3y = 0
Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients.y" + 9y' = 1 + xe9x
Graph the two basic solutions of the differential equation and several other solutions. What features do the solutions have in common?9 d2y/dx2 + 6 dy/dx + y = 0
Solve the initial-value problem.kk2y" + 5y' + 3y = 0, y(0) = 3, y'(0) = –4
Solve the initial-value problem.y" + 3y = 0, y(0) = 1, y'(0) = 3
Solve the initial-value problem.4y" – 4y' + y = 0, y(0) = 1, y'(0) = –1.5
Solve the initial-value problem.2y" + 5y' – 3y = 0, y(0) = 1, y'(0) = 4
Solve the initial-value problem.y" + 16y = 0, y(π/4) = –3, y'(π/4) = 4
Solve the differential equation using (a) Undetermined coefficients (b) Variation of parameters.y" – 2y' + y = e2x
Solve the initial-value problem.y" – 2y' + 5y = 0, y(π) = 0, y'(π) = 2
Solve the initial-value problem.y" + 2y' + 2y = 0, y(0) = 2, y'(0) = 1
Solve the initial-value problem.y" + 12y' + 36y = 0, y(1) = 0, y'(1) = 1
Solve the boundary-value problem, if possible.4y" + y = 0, y(0) = 3, y(π) = –4
Solve the boundary-value problem, if possible.y" + 2y' = 0, y(0) = 1, y(1) = 2
Solve the boundary-value problem, if possible.y" – 3y' + 2y = 0, y(0) = 1, y(3) = 0
Solve the boundary-value problem, if possible.y" + 100y = 0, y(0) 2, y(π) = 5
Solve the boundary-value problem, if possible.y" – 6y' + 25y = 0, y(0) = 1, y(π) = 2
Solve the boundary-value problem, if possible.y" – 6y' + 9y = 0, y(0) = 1, y(1) = 0
Solve the boundary-value problem, if possible.y" + 4y' + 13y = 0, y(0) = 2, y(π/2) = 1
Solve the boundary-value problem, if possible.9y" – 18y' + 10y = 0, y(0) = 0, y(π) = 1
How large should we take n in order to guarantee that the Trapezoidal and Midpoint Rule approximations for ∫12(1/x) dx are accurate to within 0.0001?
Use Simpson's Rule with n = 10 to approximate ∫12 (1/x) dx.
Calculate ∫01 tan–1 x dx.
Figure 9 shows data traffic on the link from the United States to SWITCH the Swiss academic and research network, on February 10, 1998. D(t) is the data through- put, measured in megabits per second
Find (3-√√3/2 0 .3 1-3 (4.x² + 9)³/2 dx.
Prove the reduction formulawhere n ≥ 2 is an integer. sin"x dx 1 n cos x sin" ¹x + - 1 - ¹- J sin¹-²x dx n n-
Determine whetherconverges or diverges. *TT/2 Jo sec x dx
Evaluate 4x² 3x + 2 4x² - 4x + 3 dx.
Find tan5θ sec7θ dθ.
How large should we take n in order to guarantee that the Simpson's Rule approximation for ∫12(1/x) dx is accurate to within 0.0001?
(a) Use Simpson's Rule with n = 10 to approximate the integral e01 ex2 dx. (b) Estimate the error involved in this approximation.
Evaluate if possible. *3 dx Jo x - 1
Write out the form of the partial fraction decomposition of the function x³ + x² + 1 x(x - 1)(x² + x + 1)(x² + 1)³
Find ∫tan3x dx.
Evaluate ∫01 ln x dx
Evaluate 1-x+ 2x²x³ x(x² + 1)² - dx.
Find ∫sec3x dx.
Show that is convergent. Jo exdx
Evaluate √x + 4 X dx.
Evaluate ∫sin 4x cos 5x dx.
Find the length of the arc of the semi cubical parabola y2 = x3 between the points (1, 1) and (4, 8). (See Figure 5.)Figure 5 y 0 (1, 1) (4,8) y² = x³ X
The curve y = √4 – x2, –1 ≤ x ≤ 1, is an arc of the circle x2 + y2 = 4. Find the area of the surface obtained by rotating this arc about the x-axis. (The surface is a portion of a sphere of
A dam has the shape of the trapezoid shown in Figure 2. The height is 20 m, and the width is 50 m at the top and 30 m at the bottom. Find the force on the dam due to hydrostatic pressure if the water
The demand for a product, in dollars, isFind the consumer surplus when the sales level is 500. p 1200 0.2x -0.0001x² =
Let f(x) = 0.006x(10 – x) for 0 ≤ x ≤ 10 and f(x) = 0 for all other values of x. (a) Verify that f is a probability density function. (b) Find P(4 ≤ X ≤ 8).
Find the length of the arc of the parabola y2 = x from (0, 0) to (1, 1).
Find the unit normal and binormal vectors for the circular helix r(t) = cos ti+ sin tj + tk
Find the equations of the normal plane and osculating plane of the helix in Example 6 at the point P(0, 1, π/2).Data from Example 6Find the unit normal and binormal vectors for the circular helix
A projectile is fired with muzzle speed 150 m/s and angle of elevation 45° from a position 10 m above ground level. Where does the projectile hit the ground, and with what speed?
A particle moves with position function r(t) = 〈t2, t2, t3〉. Find the tangential and normal components of acceleration.
Show that does not exist. x2 - 2 lim (x,y) - (0,0) x2 +
Find and graph the osculating circle of the parabola y = x2 at the origin.
Find the tangent plane to the elliptic paraboloid z = 2x2 + y2 at point (1, 1, 3).
If f(x, y) = x3 + x2y3 – 2y2, find fx(2, 1) and fy(2, 1).
If z = x2y + 3xy4, where x = sin 2t and y = cos t, find dz/dt when t = 0.
For each of the following functions, evaluate f(3, 2) and find the domain. (a) f(x, y) = √x + y +1 x - 1 (b) f(x, y) = x ln(y² - x)
If does exist? f(x, y) xy x² + y² 2
Find an equation of the tangent plane to the given surface at the specified point.Z = 4x2 – y2 + 2y, (–1, 2, 4)

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