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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Evaluate the limit of the following sequences. an 75"-1 99n + 5" sin n 8n
Evaluate the limit of the following sequences. an tan 10n 10n + 4
a. Find the value of the seriesb. For what value of a does the series 3k k=1 (3k+1 - 1)(3* - 1)*
Evaluate the limit of the following sequences. an = COS (0.99") + 7" + 9" 63"
Evaluate the limit of the following sequences. an || 4" + 5n! n! + 2"
Evaluate the limit of the following sequences. an 6" + 6" 62 3n 3″ 100 + n'
Evaluate the limit of the following sequences. an n8 + n² n² + n³ In n 8
Evaluate the limit of the following sequences. an ייד n' 5n 7
For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.Euler’s method time step y'(t) = 2 - y, y(0) = 1; Δt =
Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to
Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to
A differential equation of the form y' (t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y yo is an equilibrium solution of the equation provided f(y0) = 0
Solve the following initial value problems. When possible, give the solution as an explicit function of t. y' (t) = y + 3 5t + 6' y(2) = 0
For the following separable equations, carry out the indicated analysis.a. Find the general solution of the equation.b. Find the value of the arbitrary constant associated with each initial
For the following separable equations, carry out the indicated analysis.a. Find the general solution of the equation.b. Find the value of the arbitrary constant associated with each initial
Use a calculator or computer program to carry out the following steps.a. Approximate the value of y(T) using Euler’s method with the given[0, T].b. Using the exact solution (also given), find the
Solve the following initial value problems. When possible, give the solution as an explicit function of t. y' (t) 3y(y + 1) t y(1) = 1
A differential equation of the form y' (t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y yo is an equilibrium solution of the equation provided f(y0) = 0
Use a calculator or computer program to carry out the following steps.a. Approximate the value of y(T) using Euler’s method with the given[0, T].b. Using the exact solution (also given), find the
Solve the following initial value problems. When possible, give the solution as an explicit function of t. y' (t) = cos²t 2y y(0) = -2
Use a calculator or computer program to carry out the following steps.a. Approximate the value of y(T) using Euler’s method with the given[0, T].b. Using the exact solution (also given), find the
Use a calculator or computer program to carry out the following steps.a. Approximate the value of y(T) using Euler’s method with the given[0, T].b. Using the exact solution (also given), find the
For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.Euler’s method time step y'(t) = t + y, y(0) = 4; Δt =
Solve the following initial value problems. When possible, give the solution as an explicit function of t. e'y' (t) = In²t t y(1) = In 2
Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to
A community of hares on an island has a population of 50 when observations begin (at t = 0). The population is modeled by the initial value problema. Find and graph the solution of the initial value
Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to
Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to
Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to
For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.Euler’s method time step y' (t) = -y, y(0) = -1; At = 0.2
Determine whether the following equations are separable. If so, solve the initial value problem. y' (t) = cos² y, y(1) = ㅠ |- 4
For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.Euler’s method time step y'(t) = 2y, y(0) = 2; At = 0.5
Consider the following logistic equations, for t ≥ 0. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed
Consider the following logistic equations, for t ≥ 0. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed
Consider the following logistic equations, for t ≥ 0. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed
Consider the following logistic equations, for t ≥ 0. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed
Consider the following differential equations. A detailed direction field is not needed.a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).b. In what regions are
Consider the following differential equations. A detailed direction field is not needed.a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).b. In what regions are
Determine whether the following equations are separable. If so, solve the initial value problem. y'(t) = e, y(0) = 1
Determine whether the following equations are separable. If so, solve the initial value problem. sec ty' (t) = 1, y(0) = 1
Consider the following differential equations. A detailed direction field is not needed.a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).b. In what regions are
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. u'(x) = ²x-u
Determine whether the following equations are separable. If so, solve the initial value problem. 2yy' (t) = 3t², y(0) = 9
Consider the following differential equations. A detailed direction field is not needed.a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).b. In what regions are
Determine whether the following equations are separable. If so, solve the initial value problem. ty' (t) = 1, y(1) = 2, t > 0
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. xu'(x) = u²4, x > 0
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. y' (t)e¹/2 = y² + 4
Use the window [-2, 2] × [-2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. (t² + 1)³yy' (t) = (y² + 4)
Use the window [-2, 2] × [-2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. x²y'(x) = y², x > 0 2
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 3 y'(t) csct = 2
Which of the differential equations a–d corresponds to the following direction field? Explain your reasoning. a. y'(t) = 0.5(y+1)(t -1) b. y'(t) = -0.5(y+1)(t-1) c. y'′(t) = 0.5(y-1)|(t + 1) d.
Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions y(0) = A lead to solutions
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. dw dx = Vw (3x + 1), x > 0
Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions y(0) = A lead to solutions
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. y' (t) = e/2 sin t
Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions y(0) = A lead to solutions
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. dy dx = y(x² + 1)
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. dy dt || 31² y
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. ety' (t) = 5
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. t³y' (t) = 1
Is the equationseparable? t²y' (t) = t + 4 2 y²
Give the antiderivatives of xp. For what values of p does your answer apply?
Explain the Mean Value Theorem with a sketch. THEOREM 4.9 Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on (a, b), then there is at least one point c in (a,
Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle’s Theorem. h(x) =
Find all the antiderivatives of the following functions. Check your work by taking derivatives. H(z) = -62-7
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and minimum values (if they exist). Graph the function to confirm your conclusions. g(x) =
Evaluate the following limits using l’Hôpital’s Rule. lim X-0 3 sin 4x 5x
Find all the antiderivatives of the following functions. Check your work by taking derivatives. f(x) = et
Assume that ƒ twice differentiable at c and that ƒ has a local maximum at c. Explain why ƒ"(c) ≤ 0.
Give the antiderivatives of 1/x, for x > 0.
Evaluate the following limits using l’Hôpital’s Rule. lim x-2π x sin x + x² 47² x - 2π
Find all the antiderivatives of the following functions. Check your work by taking derivatives. h(y) = y ¹
a. Write the equation of the line that represents the linear approximation to the following functions at the given point a.b. Graph the function and the linear approximation at a.c. Use the linear
Find all the antiderivatives of the following functions. Check your work by taking derivatives. G(s) = 1 s² + 1
Evaluate the following limits using l’Hôpital’s Rule. lim U→π/4 tan u - cot u u - π/4
Evaluate the following limits using l’Hôpital’s Rule. tan 4z lim Z-0 tan 7z
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find or approximate the point(s) that are guaranteed to exist by the Mean Value
Find all the antiderivatives of the following functions. Check your work by taking derivatives. F(t) = π ㅠ
Evaluate the following limits. lim x→0 sin² 3x 2 X
Evaluate the following limits. lim x-0 1 - - cos 3x 8x²
Determine the following indefinite integrals. Check your work by differentiation. √(3x5 (3x5 - 5x) dx
Determine the following indefinite integrals. Check your work by differentiation. 4 [(4√x - +13) d dx Vx,
Determine the following indefinite integrals. Check your work by differentiation / 13 (3u 2 - 4u² + 1) du
Evaluate the following limits. e sin x 1 lim x0x4 + 8x³ + 12x²
Evaluate the following limits. lim XT COS X + 1 (x - ㅠ)2
Determine the following indefinite integrals. Check your work by differentiation. [(5 (5s + 3)² ds
Evaluate the following limits. lim x-0 sin x - x 7x³ 3
Evaluate the following limits. lim x→0 ex - x - 1 5x²
Determine the following indefinite integrals. Check your work by differentiation. 5 2 +41² dt
Determine the following indefinite integrals. Check your work by differentiation. 5m (12m³ 10m) dm
Evaluate the following limits. lim 004-X el/x - 1 1/x
Determine the following indefinite integrals. Check your work by differentiation. Jo (3x1/3 + 4x 1/3 + 6) dx
Find the intervals on which f is increasing and decreasing.ƒ(x) = x4/3
Determine the following indefinite integrals. Check your work by differentiation. [6. 6 √x dx
Find the intervals on which f is increasing and decreasing. f(x) = x²√9 – x² on (−3, 3) -
Evaluate the following limits. x³ x²- rẻ X 3 – 5x – 3 lim x1x² + 2x³ = x² - 4x - 2 3
Evaluate the following limits. lim X-00 tan ¹x x - π/2 1/x
Determine the following indefinite integrals. Check your work by differentiation. [(3² (3x + 1)(4x) dx

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