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mathematics
applied calculus

Questions and Answers of Applied Calculus

Use Euler’s method with n = 6 on the interval 0 ≤ t ≤ 3 to approximate the solution f (t) to y'.ly(20 - y), y(0) = 2.
Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1 1- x 9 et, or cos x.
Use the integral test to determine whether the infinite series is convergent or divergent. 00 2 k=15k - 1
Find the nth Taylor polynomial of 2 2-x at x = 0.
Determine the sums of the following geometric series when they are convergent. + ala て 81] + + ala + 2/3 2+
Determine the third Taylor polynomial of the given function at x = 0. f(x)=√4x+1
Use three repetitions of the Newton–Raphson algorithm to approximate the following:3√11
Explain how the Newton–Raphson algorithm is used to approximate a zero of a function.
Use the integral test to determine whether the infinite series is convergent or divergent. 8 1 k=2 k Vink
Determine the third Taylor polynomial of the given function at x = 0.f (x) = cos(π - 5x)
Determine the sums of the following geometric series when they are convergent. 3 + 615 + 12 24 + 25 125 48 625 +
Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1 1- x 9 et, or cos x.
Determine the third Taylor polynomial of the given function at x = 0. f(x) = 1 x + 2
Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of x2 - x - 5 between 2 and 3
Determine the sums of the following geometric series when they are convergent. 1 1 + 5 54 57 + - -| + 1 1 + 510 + 513
What is the nth partial sum of an infinite series?
Find the third Taylor polynomial of x2 at x = 3.
Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1 1- x 9 et, or cos x.
Use the integral test to determine whether the infinite series is convergent or divergent. k k=2 (k² + 1)³/2
Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of x2 + 3x - 11 between -5 and -6
What is a convergent infinite series? Divergent?
Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1 1- x 9 et, or cos x.
Find the third Taylor polynomial of ex at x = 2.
Determine the sums of the following geometric series when they are convergent. 1 1 + 1 1 3² 33 34 35 | T + 1 36
Determine the third Taylor polynomial of the given function at x = 0. f(x) = V1- x VI
Use the integral test to determine whether the infinite series is convergent or divergent. 1 Σ k=1 (2k + 1)3
Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of sin x + x2 - 1 near x0 = 0
Determine the sums of the following geometric series when they are convergent. 3 7 + + 3²
What is meant by the sum of a convergent infinite series?
Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1 1- x 9 et, or cos x.
Use a second Taylor polynomial at t = 0 to estimate the area under the graph of y = -ln(cos 2t) between t = 0 and t = 1/2.
Determine the third Taylor polynomial of the given function at x = 0.f (x) = xe3x
Use three repetitions of the Newton–Raphson algorithm to approximate the following: The zero of ex + 10x - 3 near x0 = 0
What is a geometric series and when does it converge?
Use a second Taylor polynomial at x = 0 to estimate the value of tan(.1).
Determine the sums of the following geometric series when they are convergent. 61.2.24.048 +.0096 -
Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 of 1 1- x 9 et, or cos x.
Use the integral test to determine whether the infinite series is convergent or divergent. (You may assume that the hypotheses of the integral test are satisfied. 00 1 k=1 (3k)²
What is the sum of a convergent geometric series?
(a) Find the second Taylor polynomial of 1x at x = 9.(b) Use part (a) to estimate √8.7 to six decimal places.(c) Use the Newton–Raphson algorithm with n = 2 and x0 = 3 to approximate the solution
Determine the fourth Taylor polynomial of f (x) = ex at x = 0, and use it to estimate e0.01.
Determine the sums of the following geometric series when they are convergent. 53 55 3 34 + 37 5⁹ 310 + 511 313
Find the sum of the given infinite series if it is convergent. 1 - 3 + 4 9 16 27 81 + 64 256
Sketch the graph of y = x3 + x - 1, and use the Newton– Raphson algorithm (three repetitions) to approximate all x-intercepts.
Sum an appropriate infinite series to find the rational number whose decimal expansion is given. 5.444
Define the Taylor series of f (x) at x = 0.
Find the Taylor series at x = 0 of the given function. Use suitable operations (differentiation, substitution, etc.) on the Taylor series at x = 0 ofx sin x2 1 1- x 9 et, or cos x.
(a) Use the third Taylor polynomial of ln(1 - x) at x = 0 to approximate ln 1.3 to four decimal places.(b) Find an approximate solution of the equation ex = 1.3 using the Newton–Raphson algorithm
Find the sum of the given infinite series if it is convergent. 1 + 2 + 2! + 3! + 4! ...
Determine the fourth Taylor polynomial of f (x) = ln(1 - x) at x = 0, and use it to estimate ln(.9).
Discuss the three possibilities for the radius of convergence of a Taylor series.
Find the sum of the given infinite series if it is convergent. --- + (†) ² + ¸‚ ( 1 ) ² + ¸( 1 ) ³ + + +=+1
Use the Newton–Raphson algorithm with n = 2 to approximate the zero of x2 - 3x - 2 near x0 = 4.
Determine the nth Taylor polynomial for f (x) = ex at x = 0.
Determine the third and fourth Taylor polynomials of cos x at x = π.
Sum an appropriate infinite series to find the rational number whose decimal expansion is given. Show that .999 = 1.
Use properties of convergent series to find 00 1 + 2k 3k k=0
Use the comparison test to determine whether the infinite series is convergent or divergent. 1 k=2 k4 + 5 Σ Compare with 00 k=2k2°
Suppose that the graph of the function f(x) has slope -2 at the point (1, 2). If the Newton–Raphson algorithm is used to find a root of f(x) = 0 with the initial guess x0 = 1, what is x1?
Figure 8 contains the graph of the function f(x) = x2 - 2. The function has zeros at x = √2 and x = -√2. When the Newton–Raphson algorithm is applied to find a zero, what values of x0 lead to
Determine the third and fourth Taylor polynomials of x3 + 3x - 1 at x = -1.
Find the Taylor series of xex2 at x = 0.
Use the second Taylor polynomial of f (x) = √x at x = 9 to estimate √9.3.
Figure 9 contains the graph of the function f (x) = x3 - 12x. The function has zeros at x = -√12, 0, and √12. Which zero of f(x) will be approximated by the Newton–Raphson method starting with
Use the comparison test to determine whether the infinite series is convergent or divergent. 8 1 Σ k=2Vk2 - 1 00 Compare with Σ k=2 k
The hyperbolic cosine of x, denoted by cosh x, is defined byThis function occurs often in physics and probability theory. The graph of y = cosh x is called a catenary.(a) Use differentiation and the
Use the comparison test to determine whether the infinite series is convergent or divergent. 00 Σ k=1 1 =1 2* + k Compare with k=1 24
Determine if the given series is convergent. 00 1 k=1 k³
Compute the value of .12121212 as a geometric series with a = .1212 and r = .0001. Compare your answer with the result of Example 2.
Find ∑∞k=0 (3k + 5k)/7k.
Use the second Taylor polynomial of f (x) = ln x at x = 1 to estimate ln .8.
Compute the total new spending created by a $10-billion federal income tax cut when the population’s marginal propensity to consume is 95%. Compare your result with that of Example 3, and note how
What special occurrence takes place when the Newton–Raphson algorithm is applied to the linear function f (x) = mx + b with m ≠ 0?
Determine all Taylor polynomials of f (x) = x4 + x + 1 at x = 2.
The hyperbolic sine of x is defined byRepeat parts (a) and (b) of Exercise 23 for sinh x.Exercise 23The hyperbolic cosine of x, denoted by cosh x, is defined byThis function occurs often in physics
Compute the effect of a $20-billion federal income tax cut when the population’s marginal propensity to consume is 98%. What is the “multiplier” in this case?
Use the comparison test to determine whether the infinite series is convergent or divergent. ∞0 1/ [Compare Compare with Σ k=1 k3k -] k=13k*
Determine if the given series is convergent. 1 Σ 3k k=]
Use the Taylor series expansion forto find the function whose Taylor series is 1 + 4x + 9x2 + 16x3 + 25x4 + · · ·. X (1 − x)²
Can the comparison test be used withto deduce anything about the first series? 1 Σ k=1 k2 Ink and Σ - k=2k2
What happens when the first approximation, x0, is actually a zero of f (x)?
For what values of p isconvergent? 1 Σ k k=1p
Let p2(x) be the second Taylor polynomial of f (x) = √x at x = 9,(a) Give the second remainder for f (x) at x = 9.(b) Show that(c) Show that the error in using p2(9.3) as an approximation for
Determine the nth Taylor polynomial of f (x) = 1/x at x = 1.
A perpetuity is a periodic sequence of payments that continues forever. The capital value of the perpetuity is the sum of the present values of all future payments.Consider a perpetuity that promises
Apply the Newton–Raphson algorithm to the function f (x) = x3 - 5x with x0 = 1. After observing the behavior, graph the function along with the tangent lines at x = 1 and x = -1, and explain
A patient receives 6 mg of a certain drug daily. Each day the body eliminates 30% of the amount of the drug present in the system. After extended treatment, estimate the total amount of the drug that
Let p2 (x) be the second Taylor polynomial of f (x) = ln x at x = 1,(a) Show that(b) Show that the error in using p2(.8) as an approximation for ln .8 is at most |f) (c) < 4 if c≥.8.
Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1 1- x and et.
Use the Taylor series for ex to show that d dx - et = et.
The following property is true for any two series (with possibly some negative terms): Letbe convergent series whose sums are S and T, respectively. Then,is a convergent series whose sum is S + T.
Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 ofln(1 + x3) 1 1- x and et.
Graph the functionand its fourth Taylor polynomial in the window [-1, 1] by [-1, 5]. Find a number b such that graphs of the two functions appear identical on the screen for x between 0 and b.
Letbe a convergent series with sum S, and let c be a constant. Then,is a convergent series whose sum is c · S. Make a geometric picture to illustrate why this is true when c = 2 and the terms ak are
Graph the functionThe function has 0 as a zero. By looking at the graph, guess at a value of x0 for which x1 will be exactly 0 when the Newton– Raphson algorithm is invoked. Then, test your guess
Draw the graph of f (x) = x4 - 2x2, [-2, 2] by [-2, 2]. The function has zeros at x = -√2, x = 0, and x = √2. By looking at the graph, guess which zero will be approached when you apply the
Find the Taylor series at x = 0 of the given function. Use suitable operations on the Taylor series at x = 0 of 1 1- x and et.
The functionand its seventh Taylor polynomial. Y₁ 1 1- x
A patient receives 2 mg of a certain drug each day. Each day the body eliminates 20% of the amount of drug present in the system. After extended treatment, estimate the total amount of the drug
The Taylor series at x = 0 forFind f(5)(0). ¹(1+ f(x) = In 1 + x 1-x,

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