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mathematics
calculus

Questions and Answers of Calculus

Graph y = x1/x for x > 0. Show what happens for very small x and very large x. Indicate the maximum value?
Find each limit?(a)(b) (c) (d)
For k ( 0, find limn→( 1k + 2k + ... + nk nk+1 Though this has (/( form, I'Hpital's Rule is not helpful. Think of a Riemann sum?
Let c1, c2 ,...... cn, en be positive constants withAnd let x1, x2, ..........., xn be positive numbers. Take natural logarithms and then use l'Hopital's Rule to show that
Verify the last statement in Problem 46 by calculating each of the following?(a)(b) (c)
Consider f(x) = n2xe-nx?(a) Graph f(x) for n = 1, 2, 3, 4, 5, 6 on [0, 1] in the same graph window?(b) for x > 0. Find limn†’( f(x).(c) Evaluate(d) Guess at
Find the absolute maximum and minimum points (if they exist) for f(x) = (x25 + x3 + 2x)e-x on [0, (]?
In problem 1-24, evaluate each improper integral or show that it1.2. 3. 4. 5.
Find the area of the region under the curve y = 2 / (4x2 - 1) to the right of x = 1. Use partial fractions?
Find the area of the region under the curve y = 1 / (x2 + x) to the right of x = 1?
Suppose that Newton's law for the force of gravity had the form -k/x rather than -k/x2 (see Example 3). Show that it would then be impossible to send anything out of the earth's gravitational field?
If a 1000-pound capsule weighs only 165 pounds on the moon (radius 1080 miles), how much work is done in propelling this capsule out of the moon's gravitational field (see Example 3)?
Suppose that a company expects its annual profits t years from now to be f(t) dollars and that interest is considered to be compounded continuously at an annual rate r. Then the present value of all
Do Problem 29 assuming that f(t) = 100,000 + 10001?
A continuous random variable X has a uniform distribution if it has a probability density function of the form(a) Show that (b) Find the mean g and variance (2 of the uniform distribution. (c) If a =
A random variable X has a Weibull distribution if it has probability density function(a) Show that (Assume ( > 1.) (b) If ( = 3 and p = 2, find the mean (, and the variance (2. (c) If the lifetime of
Sketch the graph of the normal probability density functionAnd show, using calculus, that ( is the distance from the mean ( to the x-coordinate of one of the inflection points?
The Pareto probability density function has the formWhere k and M arc positive constants. (a) Find the value of C that makes f(x) a probability density function. (b) For the value of C found in pan
The Pareto distribution is often used to model income distribution. Suppose that in some economy the income distribution does follow a Pareto distribution with k = 3. Suppose that the mean income is
In electromagnetic theory, the magnetic potential u at a point on the axis of a circular coil is given byWhere A, r, and a are constants. Evaluate u?
There is a subtlety in the definition ofThat is illustrated by the following: Show that (a) Diverges and (b)
Consider an infinitely long wire coinciding with the positive x-axis and having mass density ((x) = (1 + x2)-1, 0 ( x < (. (a) Calculate the total mass of the wire? (b) Show that the wire does not
Give an example of a region in the first quadrant that gives a solid of finite volume when revolved about the x-axis, but gives a solid of infinite volume when revolved about the y-axis?
Let f be a nonnegative continuous function defined on 0 ( xShow that (a) If limx†’( f(x) exists it must be 0; (b) It is possible that limx†’( f(x) does not exist.
We can use a compute to approximateBy taking b very large in Provided we know that the first integral converges. Calculate For p = 2, 1.1, 1.01, 1, and 0.99. This gives no hint that the integral
ApproximateFor a = 10, 50, and 100.
ApproximateFor a = 1, 2, 3, and 4?
In Problems 1-5, evaluate each improper integral or show that it diverges?1.2. 3. 4. 5.
It is often possible to change an improper integral into a proper one by using integration by parts. ConsiderUse integration by parts on the interval [c, 1] where 0 And thus conclude that upon taking
`Use integration by parts and the technique of Problem 33 to transform the improper integralInto a proper integral?
If f(x) tends to infinity at both a and b, then we defineWhere c is any point between a and b, provided of course that both latter integrals converge. Otherwise, we say that the given integral
EvaluateOr show that it diverges. See Problem 35?
EvaluateOr show that it diverges. See Problem 35?
EvaluateOr show that it diverges?
If limx†’0 f(x) = (, we defineProvided both limits exits. Otherwise, we say that Diverges. Show that Diverges for all p.
Suppose that f is continuous on [0, (] except at x = 1, whereHow would you define
Find the area of the region between the curves y = 1/x and y = 1/ (x3 + x) for 0 < x ( 1?
Let R be the region in the first quadrant below the curve y = X-2/3 and to the left of x = 1. (a) Show that the area of R is finite by finding its value. (b) Show that the volume of the solid
Find b so ∫10 that in x dx = 0?
(Comparison Test) If 0 ( f(x) ( g(x) on [a, (), it can be shown that the convergence ofImplies the convergence of And the divergence of Implies the divergence of Use this to show that Converges. On
Use the Comparison Test of Problem 46 to show thatConverges. e-x2 ( e-x on [1, (]?
Use the Comparison Test of Problem 46 to determine whether'Diverges?
Use the Comparison Test of Problem 46 to determine whetherConverges or diverges?
Formulate a comparison test for improper integrals with infinite integrands?
(a) Use Example 2 of Section 8.02 to show that for any positive number n there is a number M such that(b) Use part (a) and Problem 46 to show that Converges?
Using Problem 50, prove thatConverges for n > 0?
(Gamma Function) Let Î“(n) =This integral converges by Problems 51 and 52. Show each of the following (note that the gamma function is defined for every positive real number n): (a)
Evaluate
The Gamma probability density function isWhere a and ( are positive constants. (Both the gamma and the Weibull distributions are used to model lifetimes of people, animals, and equipment.) (a) Find
The Laplace transform, named after the French mathematician Pierre-Simon de Laplace (1749-1827), of a function f(x) is given by L{f(t)}(s) =Laplace transforms are useful for solving differential
By interpreting each of the following integrals as an area and then calculating this area by a y-integration, evaluate:(a)(b)
Suppose that 0Converges. What can you say about p and q?
Recall from Section 0.1 that the converse of an implication P ⇒ Q is Q ⇒ P. and the con-trapositive is not Q ⇒ not P. In Problems 1-8, give the converse and the contrapositive of the given
In Problem 1-4, evaluate the given sum?1.2. 3. 4.
Which of the improper integrals converge?1.2. 3. 4. 5. 6.
In Problem 9-12, evaluate the given sum?1.2. 3. 4.
In Problems 1-3, an explicit formula for an is given. Write the first five terms of {an}, determine whether the sequence converges or diverges and, if it converges, find limx†’( an?1.2. 3.
In Problems 1-4, find an explicit formula an = for each sequence, determine whether the sequence converges or diverges and, if it converges, find limn†’( an?1.2. 3. 4.
In Problems 1-4, write the first four terms of the sequence {an}. Then use Theorem D to show that the sequence converges?1.2. 3. 4.
Assuming that u1 = (3 and un+1 = (3 + un determine a convergent sequence, find limn→( un to four decimal places?
Show that {u"} of Problem 37 is bounded above and in-creasing. Conclude from Theorem D that {u"} converges?
Find limn→( un of Problem 37 algebraically. Let u = limn→( un, Then, since un+1 = (3 + un, u = (3 + u. Now 11-603 square both sides and solve for u?
Use the technique of Problem 39 to find limn→( an of Problem 36?
Assuming that u1 = 0 and un+1 = 1.1un determine a convergent sequence, find limn→( to four decimal places?
Show that {un} of Problem 41 is increasing and bounded above by 2?
Using the definition of limit, prove that limn→( n / (n+1) = 1; that is, for a given ( > 0, find N such that n ( N ⇒ | n / (n + 1) - 1 | < (?
Let S = {x: x is rational and x2 < 2}. Convince yourself that S does not have a least upper bound in the rational numbers, but does have such a bound in the real numbers. In other words, the sequence
The completeness property of the real numbers says that for every set of real numbers that is bounded above, there exists a real number that is a least upper bound for the set. This property is
Prove that if {an} converges and {bn} diverge then {an + bn} diverge?
If {an} and {bn} both diverge, does it follow that {an + bn} diverge?
A famous sequence {fn}, called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200, is defined by the recursion formulaf1 = f2 = 1, fn+2 = fn+1 + fn(a) Find .13
Consider an equilateral triangle containing 1 + 2 + 3 + ... + n = n(n + 1)/2 circles, each of diameter 1 and stacked as indicated in Figure 4 for the case n = 4. FindWhere An is the total area of the
In Problem 1-4, use the fact thatTo find the limits? 1. 2. 3. 4.
1.
In Problems 1-3, indicate whether the given series converges or diverges and give a reason for your conclusion?1.2. 3.
In Problems 1-3, state whether the given series is absolutely convergent, conditionally convergent, or divergent?1.2. 3.
In Problems 1-3, find the convergence set for the power series?1.2. 3.
By differentiating the geometric series 1 / 1 + x = 1 - x + x2 - x3 + x4 - ..., |x| < 1, Find a power series that represents 1 / (1 + x)2. What is the interval of convergence?
Find a power series that represents 1 / (1 + x)3 on the interval (- 1, 1)?
Find the Maclaurin series for sin2 x. for what values of x does the series represent the function?
Find the first five terms of the Taylor series for ex based at the point x = 2?
Write the Maclaurin series for f(x) = sin x + cos x. For what values of x does it represent f?
Determine how large n must be so that using the nth partial sum to approximate the seriesGives an error of no more than 0.00005?
Determine how large n must be so that using the nth partial sum to approximate the seriesGives an error of no more than 0.000005?
Find the Maclaurin polynomial of order 2 for f(x) = cos x and use it to approximate cos 0.1?
Find the Maclaurin polynomial of order 1 for f(x) = x cos x2 and use it to approximate f(0.2)?
Find the Maclaurin polynomial of order 1 for f(x) = x cos x2 and use it to approximate f(0.2)? Discuss.
Find the Taylor polynomial of order 3 based at 2 for g(x) = x3 - 2x2 + 5x - 7, and show that it is an exact representation of g(x)?
Find the Taylor polynomial of order 4 based at 1 for f(x) = 1 / (x + 1)?
Obtain an expression for the error term R4 (x) in Problem 57, and find a bound for it if x = 1.2?
Find the Maclurin polynomial of order 4 for f(x) = sin2 x = ½ (1 - cos 2x), and find a bound for the error R4(x) if |x| ( 0.2. A better bound is obtained if you observe that R4(x) = R5(x) and then
If f(x) = 1n x, then f(n) (x) = (-1)n-1 (n - 1)! / xn. Thus, the Taylor polynomial of order n based at 1 for 1n x is1n x = (x - 1) - 1 / 2 (x - 1)2 + 1 / 3 (x - 1)3 + ...How large would n have to be
Refer to Problem 60. Use the Taylor polynomial of order 5 based at 1 to approximateAnd give a good bound for the error that is made?
In Problems 1-3, determine whether the given series converges or diverges and, if it converges, find its sum.1.2. 3. In ½ + In 2 / 3 + In 3 / 4 + ...
In Problems 1-4, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series?1.2. 3. 4.
In Problems 1-4, write the given decimal as an infinite series then find the sum of the series, and finally, use the result to write the decimal as a ratio of two integers (see Example 2)? 1.
A ball is dropped from a height of 100 feet. Each time it hits the floor, it rebounds to 2/3 its previous height. Find the total distance it travels before coming to rest?
Three people, A, B, and C, divide an apple as follows. First they divide it into fourths, each taking a quarter. Then they divide the leftover quarter into fourths, each taking a quarter, and so on.
Suppose that the government pumps an extra $1 billion into the economy. Assume that each business and individual saves 25% of its income and spends the rest, so of the initial $1 billion, 75% is
Assume that square ABCD (Figure 2) has sides of length 1 and that E, F; G, and 11 are the midpoints of the sides. If the indicated pattern is continued indefinitely, what will be the area of the

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