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mathematics
applied calculus

Questions and Answers of Applied Calculus

Use the Taylor series for cos x to show that cos(-x) = cos x.
Show that the seriesis convergent, and determine its sum. y6 + y8 00 Σ k=0 104
Use Exercise 30 to show that the seriesis convergent. Then, use the comparison test to show that the seriesis convergent.Exercise 30Show that the seriesis convergent, and determine its sum. 3 Σ k=1
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.
A patient receives M mg of a certain drug each day. Each day the body eliminates 25% of the amount of drug present in the system. Determine the value of the maintenance dose M such that after many
The Taylor series at x = 0 for f (x) = sec x is 1 + x² + 2x² + 7/201 5 61 6 + Find f(4)(0). ..
The Taylor series at x = 0 for f (x) = tan x is 1.3 2 17 + zx + 3x + ? 3x + +X Find / (49(0).
The Taylor series at x = 0 foris 1 + x + 2x2 + 2x3 + 2x4 +· · ·. Find f (4)(0), where 1 + x² 1- x
(a) Find the Taylor series of cos 2x at x = 0, either by direct calculation or by using the known series for cos x.(b) Use the trigonometric identityto find the Taylor series of sin2 x at x = 0.
A patient receives M mg of a certain drug daily. Each day, the body eliminates a fraction q of the amount of the drug present in the system. After extended treatment, estimate the total amount of the
(a) Find the Taylor series of cos 3x at x = 0.(b) Use the trigonometric identityto find the fourth Taylor polynomial of cos3 x at x = 0. cos³x = (cos 3x + 3 cos x)
Graph the function Y1 = ex and its fourth Taylor polynomial in the window [0, 3] by [-2, 20]. Find a number b such that graphs of the two functions appear identical on the screen for x between 0 and
The infinite series a1 + a2 + a3 +. . . . has partial sums given by Sn = 3 -5/n.(a) Find ∑10k=1ak.(b) Does the infinite series converge? If so, to what value does it converge?
Graph the function Y1 = cos x and its second Taylor polynomial in the window ZDecimal. Find an interval of the form [- b, b] over which the Taylor polynomial is a good fit to the function. What is
The infinite series a1 + a2 + a3 +. . . . . has partial sums given by Sn = n-1/n.(a) Find ∑10k=1ak.(b) Does the infinite series converge? If so, to what value does it converge?
Find an infinite series that converges to the value of the given definite integral. L'e-² do е dx 0 J
Let f (x) = x - 2x3 + 4x5 - 8x7 + 16x9 -····.(a) Find the Taylor series expansion of 1f (x) dx at x = 0.(b) Find a simple formula for 1f (x) dx not involving a series.
Find an infinite series that converges to the value of the given definite integral. L' tets dx
Determine the sums of the following infinite series: 00 k=1 | ادر 3 2k
Find the Taylor series expansion at x = 0 of the given antiderivative. Sexz dx
Find an infinite series that converges to the value of the given definite integral. S 2 sin x² dx
Determine the sums of the following infinite series: 00 j=0 (-1)/ 3j
Determine the sums of the following infinite series: 25-2 j=1
Find the Taylor series expansion at x = 0 of the given antiderivative. tet³ dx
Use the decompositionto find the Taylor series of 1 + x 1- x 1 1- x + 1- x
Determine the sums of the following infinite series: k=0 6 k
Determine the sums of the following infinite series: 00 7 k=0 10k
Find the Taylor series expansion at x = 0 of the given antiderivative. 1 1+x3 zdx
Determine the sums of the following infinite series: 00 (−1)k 3k +1 sk k=0
Let f (x) = 1 + x2 + x4 + x6 +····.(a) Find the Taylor series expansion of f′(x) at x = 0.(b) Find the simple formula for f′(x) not involving a series.
Suppose that the Federal Reserve (the Fed) buys $100 million of government debt obligations from private owners. This creates $100 million of new money and sets off a chain reaction because of the
Let a and r be given nonzero numbers.(a) Show thatand from this conclude that, for r ≠ 1,(c) Use the result of part (a) to explain why the geometric series diverges for |r| < 1.(d) Explain why
Show that the infinite seriesdiverges. - — + + + + +
(a) Use the Taylor series for ex at x = 0 to show that ex > x2/2 for x > 0.(b) Deduce that e-x < 2/x2 for x > 0.(c) Show that xe-x approaches 0 as x → ∞.
What is the exact value of the infinite geometric series whose partial sum appears at the second entry in Fig. 2? NORMAL FLOAT AUTO REAL RADIAN MP sum(seq(1/X.X.1.99,1)) 5.177377518 sum(sea(2/32x.x.
Suppose that the Federal Reserve creates $100 million of new money and the banks lend 85% of all new money they receive. However, suppose that out of each loan, only 80% is redeposited into the
Suppose that, when you die, the proceeds of a life insurance policy will be deposited into a trust fund that will earn 8% interest, compounded continuously. According to the terms of your will, the
The calculator screen computes a partial sum of an infinite series. Write out the first five terms of the series and determine the exact value of the infinite series. NORMAL FLOAT AUTO REAL RADIAN
What is the exact value of the infinite geometric series whose partial sum appears at the first entry in Fig. 3?
The calculator screen computes a partial sum of an infinite series. Write out the first five terms of the series and determine the exact value of the infinite series. NORMAL FLOAT AUTO REAL RADIAN
Show that ex > x3/6 for x > 0, and from this, deduce that x2e-x approaches 0 as x → ∞.
Find the expected value and variance for each random variable whose probability density function is given. When computing the variance, use formula (5). Let X be a continuous random variable whose
The sum of the first n odd numbers is n2; that is,Verify this formula for n = 5, 10, and 25. Σ(2x − 1) = n?. - t= X=1
Suppose that, when you die, the proceeds of a life insurance policy will be deposited into a trust fund that will earn 8% interest, compounded continuously. According to the terms of your will, the
Convince yourself that the equation is correct by summing up the first 999 terms of the infinite series and comparing the sum with the value on the right. || 8 - ੩.
Suppose that, when you die, the proceeds of a life insurance policy will be deposited into a trust fund that will earn 8% interest, compounded continuously. According to the terms of your will, the
Make a small probability table for a discrete random variable X and use it to define E(X ), Var (X ), and the standard deviation of X.
Consider a circle with circumference 1. An arrow (or spinner) is attached at the center so that, when flicked, it spins freely. Upon stopping, it points to a particular point on the circumference of
Find the value of k that makes the given function a probability density function on the specified interval.f (x) = kx2, 0 ≤ x ≤ 2
A person shooting at a target has five successive hits and then a miss. If x is the probability of success on each shot, the probability of having five successive hits followed by a miss is x5(1 -
The men hired by a certain city police department must be at least 69 inches tall. If the heights of adult men in the city are normally distributed, with μ = 70 inches and σ = 2 inches, what
Compute the cumulative distribution function corresponding to the density function f (x) = 1/5, 2 ≤ x ≤ 7.
The number of people arriving during a 5-minute interval at a supermarket checkout counter is Poisson distributed with λ = 8.(a) What is the probability that exactly eight people arrive during a
The number of babies born each day in a certain hospital is Poisson distributed with λ = 6.9.(a) During a particular day, are 7 babies more likely to be born than 6 babies?(b) What is the
Show that E(X) = B - ∫AB F (x) dx, where F (x) is the cumulative distribution function for X on A ≤ x ≤ B.
Let Z be a standard normal random variable. Find the number a such that Pr (a ≤ Z ) = .40.
The gestation period (length of pregnancy) of a certain species is approximately normally distributed with a mean of 6 months and standard deviation of 1/2 month.(a) Find the percentage of births
The time (in minutes) required to complete a certain subassembly is a random variable X with the density function f (x) = 1/21x2, 1 ≤ x ≤ 4.(a) Use f (x) to compute Pr (2 ≤ X ≤ 3).(b) Find
The number of accidents occurring each month at a certain intersection is Poisson distributed with λ = 4.8.(a) During a particular month, are five accidents more likely to occur than four
Suppose that the life span of a certain automobile tire is normally distributed, with μ = 25,000 miles and σ = 2000 miles.(a) Find the probability that a tire will last between 28,000 and 30,000
An experiment consists of selecting a point at random from the region inside the square in Fig. 10(a). Let X be the maximum of the coordinates of the point.Show that the cumulative distribution
The density function for a continuous random variable X on the interval 1 ≤ x ≤ 4 is f (x) = 4/9x - 1/9x2.(a) Use f (x) to compute Pr (3 ≤ X ≤ 4).(b) Find the corresponding cumulative
An experiment consists of selecting a point at random from the region inside the triangle in Fig. 10(b). Let X be the sum of the coordinates of the point.Show that the cumulative distribution
An experiment consists of selecting a point at random from the region inside the square in Fig. 10(a). Let X be the maximum of the coordinates of the point.Find the corresponding density function of
Scores on a school’s entrance exam are normally distributed, with μ = 500 and σ = 100. If the school wishes to admit only the students in the top 40%, what should be the cutoff grade?
If the amount of milk in a gallon container is a normal random variable, with μ = 128.2 ounces and σ = .2 ounce, find the probability that a random container of milk contains less than 128 ounces.
It is useful in some applications to know that about 68% of the area under the standard normal curve lies between -1 and 1.(a) Verify this statement.(b) Let X be a normal random variable with
The Chebyshev inequality says that for any random variable X with expected value μ and standard deviation σ,(a) Take n = 2. Apply the Chebyshev inequality to an exponential random variable.(b) By
The amount of weight required to break a certain brand of twine has a normal density function, with μ = 43 kilograms and σ = 1.5 kilograms. Find the probability that the breaking weight of a piece
An experiment consists of selecting a point at random from the region inside the triangle in Fig. 10(b). Let X be the sum of the coordinates of the point.Find the corresponding density function of X.
(a) Show that about 95% of the area under the standard normal curve lies between -2 and 2.(b) Let X be a normal random variable with expected value μ and variance σ2. Compute Pr (μ - 2σ ≤ X ≤
Do the same as in Exercise 29 with a normal random variable.Exercise 29The Chebyshev inequality says that for any random variable X with expected value μ and standard deviation σ, Pr(uno ≤X ≤
A student with an eight o’clock class at the University of Maryland commutes to school by car. She has discovered that along each of two possible routes her traveling time to school (including the
Upon examination of a slide, 10% of the cells are found to be undergoing mitosis (a change in the cell leading to division). Compute the length of time required for mitosis; that is, find the number
Which route should the student in Exercise 29 take if she leaves home at 7:26 a.m.?Exercise 29A student with an eight o’clock class at the University of Maryland commutes to school by car. She has
A certain type of bolt must fit through a 20-millimeter test hole or else it is discarded. If the diameters of the bolts are normally distributed, with μ = 18.2 millimeters and σ = .8 millimeters,
In a certain cell population, cells divide every 10 days, and the age of a cell selected at random is a random variable X with the density function f (x) = 2ke-kx, 0 ≤ x ≤ 10, k = (ln 2)/10.Find
A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen
The Math SAT scores of a recent freshman class at a university were normally distributed, with μ = 535 and σ = 100.(a) What percentage of the scores were between 500 and 600?(b) Find the minimum
Define and give an example of each of the following types of functions.(a) Quadratic function (b) Polynomial function(c) Rational function (d) Power function
Express f (x) + g(x) as a rational function. Carry out all multiplications. f(x) = 2 x - 3' g(x) = 1 x + 2
Consider a circle of radius r. Write an expression for the area. Write an equation expressing the fact that the circumference is 15 centimeters.
Use the quadratic formula to solve the equations.z2 - √2z - 5/4 = 0
Let X be the time to failure (in years) of a computer chip, and suppose that the chip has been operating properly for a years. Then, it can be shown that the probability that the chip will fail
A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen
A random variable X has a density function f (x) = 1/3, 0 ≤ x ≤ 3. Find b such that Pr (0 ≤ X ≤ b) = .6.
A small volume of blood is selected and examined under a microscope, and the number of white blood cells is counted. Suppose that for healthy people the number of white blood cells in such a specimen
Recall that the median of an exponential density function is that number M such that Pr (X ≤ M ) = 1/2. Show that M = ( ln 2)/k. (We see that the median is less than the mean.)
The computations of the expected value and variance of an exponential random variable relied on the fact that, for any positive number k, be-kb and b2 e-kb approach 0 as b gets large. That is,The
A random variable X has a density function f (x) = 2/3x on 1 ≤ x ≤ 2. Find a such that Pr (a ≤ X ) = 1/3.
A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9.Determine the formula for pn, the
If the lifetime (in weeks) of a certain brand of lightbulb has an exponential density function and 80% of all lightbulbs fail within the first 100 weeks, find the average lifetime of a lightbulb.
random variable X has a cumulative distribution function F (x) = 1/4x2 on 0 ≤ x ≤ 2. Find b such that Pr (X ≤ b) = .09.
A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9.Determine the average number of
A random variable X has a cumulative distribution function F (x) = (x - 1)2 on 1 ≤ x ≤ 2. Find b such that Pr (X ≤ b) = 1/4.
A pair of dice is rolled until a 7 or an 11 appears, and the number of rolls preceding the final roll is observed. The probability of rolling 7 or 11 is 2/9.What is the probability that at least
Use the integral routine to convince yourself that ∫∞-∞ x2 f(x) dx = 1, where f (x) is the standard normal density function. Conclude that the standard deviation of the standard normal random
Assign variables to the dimensions of the geometric object. Rectangle with height=3 width
Let X be a continuous random variable with values between A = 1 and B = ∞, and with the density function f (x) = 4x-5.(a) Verify that f (x) is a probability density function for x ≥ 1.(b) Find
Let X be a continuous random variable with the density function f (x) = 2(x + 1)-3, x ≥ 0.(a) Verify that f (x) is a probability density function for x ≥ 0.(b) Find the cumulative distribution

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