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mathematics
calculus 4th

Questions and Answers of Calculus 4th

In Exercises 17 and 18, let y(t) be a solution of the logistic equationwhere A > 0 and k > 0.(a) Differentiate Eq. (9) with respect to t and use the Chain Rule to show that dt = ky(1 A
Find the general solution of the first-order linear differential equation.ex y'= 1 + 2exy
As in Example 3, a 1-kg ball is launched upward at 30 m/s and is acted on by gravity and air resistance. (a) Show that the ball's velocity is zero at time t* = (b) Show that y(t) = example). 30k -
Use Separation of Variables to find the general solution.2y' + 5y = 4
Let y(t) be the solution of (x3 + 1) dy/dt = y satisfying y(0) = 1. Compute approximations to y(0.1), y(0.2), and y(0.3) using Euler’s Method with time step h = 0.1.
Use Euler’s Method to approximate the given value of y(t) with the time step h indicated. dy y(0.7); = dt : 2y, y(0) = 3, h = 0.1
In Exercises 17 and 18, let y(t) be a solution of the logistic equationwhere A > 0 and k > 0.Let y = A/1 − e−kt/B be the general nonequilibrium solution to Eq. (9). If y(t) has a vertical
Find the general solution of the first-order linear differential equation.e2xy'= 1 − exy
Use Separation of Variables to find the general solution. dy dt : 8 √y =
Solve using the method of integrating factors. dy dt -=y+t², y(0) = 4
A 500-g ball is launched upward at 60 m/s and is acted on by gravity and air resistance that can be expressed in the form −kv, where v is the ball’s velocity.(a) Determine the ball’s velocity
Use Euler’s Method to approximate the given value of y(t) with the time step h indicated. y(3.3); = ? - y, y(3) = 1, h = 0.05 dy dt
Use Separation of Variables to find the general solution. V1 – x²y' = ху
Find the general solution of the first-order linear differential equation.y' − (ln x)y = xx
Solve using the method of integrating factors. dy dx || y 2x - x, y(1) = 1
Use Euler’s Method to approximate the given value of y(t) with the time step h indicated. y(3); dy dt = √t+y, y(2.7) = 5, h = 0.05
Find the general solution of the first-order linear differential equation.y' + y = cos x
Solve using the method of integrating factors. dy dt =y-3t, y(-1) = 2
Use Separation of Variables to find the general solution.y' = y2(1 − x2)
Sam borrows $10000 from a bank at an interest rate of 9% and pays back the loan continuously at a rate of N dollars per year. Let P(t) denote the amount still owed at time t.(a) Explain why P(t)
Use Euler’s Method to approximate the given value of y(t) with the time step h indicated. y(2); = t siny, y(1) = 2, h = 0.2 dy dt
Solve the Initial Value Problem.y' + 3y = e2x, y(0) = −1
April borrows $18000 at an interest rate of 5% to purchase a new automobile. At what rate (in dollars per year) must she pay back the loan, if the loan must be paid off in 5 years? Set up the
Use Euler’s Method to approximate the given value of y(t) with the time step h indicated. y(5.2); = t - secy, y(4) = -2, h = 0.2 dy dt
Solve using the method of integrating factors.y' + 2y = 1 + e−x, y(0) = −4
Solve the Initial Value Problem.xy' + y = ex, y(1) = 3
Use Separation of Variables to find the general solution.(ln y)y' − ty = 0
Solve using the appropriate method.x2y' = x2 + 1, y(1) = 10
Consider a series circuit (Figure 4) consisting of a resistor of R ohms, an inductor of L henries, and a variable voltage source of V(t) volts (time t in seconds). The current through the circuit
Let N(t) be the fraction of the population who have heard a given piece of news t hours after its initial release. According to one model, the rate N'(t) at which the news spreads is equal to k times
Write out the first four terms of the Maclaurin series of ƒ(x) if f(0) = 2, f'(0) = 3, f"(0) = 4, f"(0) = 12
According to one hypothesis, the growth rate dV/dt of a cell’s volume V is proportional to its surface area a. Since V has cubic units such as cm3 and A has square units such as cm2, we may assume
Consider the geometric series 00 Σα. n=0
What is T3 centered at a = 3 for a function ƒ such that ƒ(3) = 9, ƒ'(3) = 8, ƒ"(3) = 4, and ƒ"'(3) = 12?
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 n=1 1 5n
Give an example of a series such that ∑an converges but ∑ ΙanΙ diverges.
For the series if the partial sums S N are increasing, then (choose the correct conclusion)(a) {an} is an increasing sequence.(b) {an} is a positive sequence. 00 Σ n=1 an
What is a4 for the sequence an = n2 − n?
What role do partial sums play in defining the sum of an infinite series?
Let an = n − 3/n! and bn = an+3. Calculate the first three terms in each sequence.(a) a2n(b) bn(c) anbn (d) 2an+1 – 3an
Suppose that ∑anxn converges for x = 5. Must it also converge for x = 4? What about x = −3?
Determine ƒ(0) and ƒ"'(0) for a function ƒ with Maclaurin seriesT(x) = 3 + 2x + 12x2 + 5x3 + · · ·
The dashed graphs in Figure 7 are Taylor polynomials for a function f . Which of the two is a Maclaurin polynomial? (A) y = f(x) 2 + 3 +x +4 + 2 (B) y = f(x)
Use the Ratio Test to show thathas radius of convergence R = 2. Then determine whether it converges at the endpoints R = ±2. √n2n 4x I=U 3
Write out the first four terms of the Maclaurin series of ƒ(x) if f(3) 1, f'(3) = 2, f"(3) = 12, f"" (3)= 3
Which of the following statements is equivalent to Theorem 1? THEOREM 1 Absolute Convergence Implies Convergence Iflan converges. then an also converges.
Consider the p-series (a) In the Ratio Test, what do the terms Ιan+1/anΙ equal?(b) What can be concluded from the Ratio Test? 00 Ση'. n=1
What are the hypotheses of the Integral Test?
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ n=1 (−1)"-1n 5η
Prove that using the limit definition. 2n-1 2 lim n→∞0 3n+2 || 2/3
Which of the following sequences converge to zero? (a) n² n2+1 (b) 27 (c) (글)"
Suppose that ∑an(x − 6)n converges for x = 10. At which of the points (a)–(d) must it also converge?(a) x = 8 (b) x = 11(c) x = 3 (d) x = 0
Show that the power series (a)–(c) have the same radius of convergence. Then show that (a) Diverges at both endpoints, (b) Converges at one endpoint but diverges at the other, and (c) Converges at
Determine ƒ(−2) and ƒ(4)(−2) for a function with Taylor seriesT(x) = 3(x + 2) + (x + 2)2 − 4(x + 2)3 + 2(x + 2)4 + · · ·
Find the Maclaurin series and find the interval on which the expansion is valid. f(x) = 1 1 + 10x
Indicate whether or not the reasoning in the following statement is correct: is an alternating series, it must converge. 00 Since (-1)" √n n=1
Which test would you use to determine whether converges? 00 Ση n=1 -3.2 n
For which value of x does the Maclaurin polynomial Tn satisfy Tn(x) = ƒ(x), no matter what ƒ is?
Is the Ratio Test conclusive for Σ. n=1 - ? Is it conclusive for Σ n! n=1 1 n + 1 -?
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ n=1 1 n
Let an be the nth decimal approximation to √2. That is, a1 = 1, a2 = 1.4, a3 = 1.41, and so on. What is lim an? n→∞0
Compute the limit (or state that it does not exist) assuming that lim an = 2. n→∞0
What is the radius of convergence of F(3x) if F(x) is a power series with radius of convergence R = 12?
Find the Maclaurin series and find the interval on which the expansion is valid. f(x) = 1² 1-x³
Let Tn be the Maclaurin polynomial of a function ƒ satisfying |ƒ(4)(x)| ≤ 1 for all x. Which of the following statements follow from the Error Bound? (a) |T4 (2) (b) |T3 (2) (c) |T3 (2) f(2)|
What is the easiest way to find the Maclaurin series for the function ƒ(x) = sin(x2)?
Suppose that bn is positive, decreasing, and tends to 0, and let What can we say about |S − S100| if a101 = 10−3? Is S larger or smaller than S100? 00 S = [(-1)n-¹ bn. n=1
Which test would you use to determine whether converges? 00 n=1 1 2n + √n Vn
Is the Root Test conclusive for -? Is it conclusive for for Σ (1 + ²) "² ? n n=1 2" n=1
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ n=0 3η + 2 5n3 + 1
Which of the following sequences is defined recursively? (a) an √4+n = (b) bn = √4+bn-1
Indicate whether or not the reasoning in the following statement is correct: because 12 tends to zero. 00 n=1 1 n² : 0
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 1 Σ n=2 n3/2 In n
Compute the limit (or state that it does not exist) assuming that lim an = 2. n→∞0
Ralph hopes to investigate the convergence of by comparing it with Is Ralph on the right track? 00 n=1 e-n n
The power serieshas radius of convergence R = 1. What is the power series expansion of F'(x) and what is its radius of convergence? 00 F(x) = Enx" n=1
Find the Taylor series for ƒ centered at c = 3 if ƒ'(3) = 4 and ƒ'(x) has a Taylor expansion f'(x) = = Σ n=1 (x - 3) n
Indicate whether or not the reasoning in the following statement is correct: converges because 00 1 Σ √n h=1
Let T(x) be the Maclaurin series of ƒ(x). Which of the following guarantees that ƒ(2) = T(2)?(a) T(x) converges for x = 2.(b) The remainder Rk(2) approaches a limit as k→∞.(c) The remainder
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ n=1 n n2 + 1
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = cos 3x
Compute the limit (or state that it does not exist) assuming that lim an = 2. n→∞0
Compute the limit (or state that it does not exist) assuming that lim an = 2. n→∞0
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. n=1 2n n
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = sin(2x)
Does there exist an N such that S N > 25 for the series 00 n=1 2-"? Explain.
Compute the limit (or state that it does not exist) assuming that lim an = 2. n→∞0
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = sin(x2)
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 n=1 2n n100
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ 3n² n=1 43
Compute the limit (or state that it does not exist) assuming that lim an = 2. n→∞0
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = e4x
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 n=1 10″ 2n²
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = ln(1 − x2)
Determine the limit of the sequence or show that the sequence diverges.an = √n + 5 − √n + 2
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 n=1 en n!
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = (1 − x)−1/2
Determine the limit of the sequence or show that the sequence diverges. an 3n3 - n 1 - 2n³
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 00 Σ n=1 en nn
Find the Maclaurin series and find the interval on which the expansion is valid.ƒ(x) = tan−1(x2)

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