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mathematics
calculus

Questions and Answers of Calculus

Suppose that F, G and Q are polynomials and for all except when Q(x) = 0. Prove that for F(x) = G(x) all x.
If f is a quadratic function such that f(0) = 1 and is a rational function, find the value of f€™(0)
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent.
Computer algebra systems sometimes need a helping hand from human beings. Ask your CAS to evaluate ∫2x √4x – 1dx. If it doesn’t return an answer, ask it to try ∫2x √22x
Try to evaluate with a computer algebra system. If it doesn€™t return an answer,Make a substitution that changes the integral into one that the CAS can evaluate.
Use a CAS to find an anti-derivative F of f such that F (0) = 0. Graph f and F and locate approximately the x-coordinates of the extreme points and inflection points of
Let l = ∫4 f(x) dx, where is the function whose graph is shown.(a) Use the graph to find L2, R2 and M2.(b) Are these underestimates or overestimates of l?(c) Use the graph to find T2. How does it
The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate ∫2 f(x) dx, where f is the function whose graph is shown. The estimates were 0.7811, 0.8675, 0.8632, and
Estimate ∫1 cos(x2) dx using (a) The Trapezoidal Rule and (b) The Midpoint Rule, each with n = 4. From a graph of the integrand, decide whether your answers are underestimates or
Draw the graph of f(x) = sin (x2/2) in the viewing rectangle [0, 1] by [0, 0.5] and let l = ∫1 f(x) dx. (a) Use the graph to decide whether L2, R2, M2, and T2 underestimate or overestimate
Use(a) The Midpoint Rule and(b) Simpson€™s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the
Use(a) The Trapezoidal Rule,(b) The Midpoint Rule, and(c) Simpson€™s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places)
(a) Find the approximations T10 and M10 for the integral ∫2 e –x2 dx. (b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose so that the
(a) Find the approximations T8 and M2 for ∫1 cos(x2) dx. (b) Estimate the errors involved in the approximations of part (a). (c) How large do we have to choose so that the approximations Tn
(a) Find the approximations T10 and S10 for ∫1 ex and the corresponding errors ET and ES. (b) Compare the actual errors in part (a) with the error estimates given by (3) and (4). (c) How
How large should be to guarantee that the Simpson’s Rule approximation to ∫1 ex2 dx is accurate to within 0.00001?
The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound for by K | f (4)(x) | hand. But computer algebra systems have no
Repeat Exercise 23 for the integral ∫1–1 √4 – x3 dx.
Find the approximations Ln, Rn, Tn, and Mn, for n = 4, 8, and 16. Then compute the corresponding errors EL, ER, ET, and EM. (Round your answers to six decimal places you may wish to use the sum
Find the approximations Tn, Mn, and Sn for n = 6 and 12. Then compute the corresponding errors ET, EM, and ES. (Round your answers to six decimal places you may wish to use the sum command on a
Estimate the area under the graph in the figure by using(a) The Trapezoidal Rule,(b) The Midpoint Rule, and(c) Simpson€™s Rule, each with n = 4.
The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson€™s Rule to estimate the area of the pool.
(a) Use the Midpoint Rule and the given data to estimate the value of the integral ∫3.2.f(x) dx. (b) If it is known that for all – 4 < f”(x) < 1, estimate the error involved in the
A radar gun was used to record the speed of a runner during the first 5 seconds of a race (see the table). Use Simpson€™s Rule to estimate the distance the runner covered during those 5
The graph of the acceleration a(t) of a car measured in is ft/s2 shown. Use Simpson€™s Rule to estimate the increase in the velocity of the car during the 6-second time interval.
Water leaked from a tank at a rate of r(t) liters per hour, where the graph of is as shown. Use Simpson€™s Rule to estimate the total amount of water that leaked out during the first six
The table (supplied by San Diego Gas and Electric) gives the power consumption in megawatts in San Diego County from midnight to 6:00 A.M. on December 8, 1999. Use Simpson€™s Rule to
Shown is the graph of traffic on an Internet service provider€™s T1 data line from midnight to 8:00 A.M.D is the data throughput, measured in megabits per second. Use Simpson€™s
If the region shown in the figure is rotated about the -axis to form a solid, use Simpson€™s Rule with n = 8 to estimate the volume of the solid.
The table shows values of a force function f(x) where is measured in meters and f(x) in newtons. Use Simpson€™s Rule to estimate the work done by the force in moving an object a distance of
The region bounded by the curves y = 3√1 + x3, y = 0, x = 0, and x = 2 is rotated about the -axis. Use Simpson’s Rule with n = 10 to estimate the volume of the resulting solid.
The figure shows a pendulum with length L that makes a maximum angle θ0 with the vertical. Using Newton€™s Second Law it can be shown that the period T (the time for one complete
The intensity of light with wavelength traveling through a diffraction grating with N slits at an angle is given by l(θ) = N2 sin2k/k2, where k = (π Nd sin θ(/λ and is the
Use the Trapezoidal Rule with n = 10 to approximate ∫20 cos (πx) dx. Compare your result to the actual value. Can you explain the discrepancy?
Sketch the graph of a continuous function on [0, 2] for which the Trapezoidal Rule with n = 2 is more accurate than the Midpoint Rule.
Sketch the graph of a continuous function on [0, 2] for which the Trapezoidal Rule with n = 2 is more accurate than the Midpoint Rule. Discuss.
If f is a positive function and f€ (x)
Show that if f is a polynomial of degree 3 or lower, then Simpson’s Rule gives the exact value of ∫b f(x) dx.
Find the area under the curve y = 1/x3 from x = 1 to x = t and evaluate it for t = 10, 100, and 1000. Then find the total area under this curve for x > 1.
(a) Graph the functions f(x) = 1/x1.1 and g(x) = 1/x0.9 in the viewing rectangles [0, 10] by [0, 1] and [0, 100] by [0, 1].(b) Find the areas under the graphs of f and g from x = 1 to x = t and
Sketch the region and find its area (if the area is finite).
(a) If g(x) = (sin2x)/x2, use your calculator or computer to make a table of approximate values of ∫t g(x) dx for t = 2, 5, 10, 100, 1000, and 10,000. Does it appear that ∫∞ g(x) is
(a) If g(x) = 1/√x–1, use your calculator or computer to make a table of approximate values of ∫t g(x) dx for t = 5, 10, 100, 1000, and 10,000. Does it appear that is convergent or
Use the Comparison Theorem to determine whether the integral is convergent or divergent.
Find the values of p for which the integral converges and evaluate the integral for those values of p.
We know from Example 1 that the region R = {(x, y) | x > 1, 0 < y < 1/x} has infinite area. Show that by rotating about the -axis we obtain a solid with finite volume.
Use the information and data in Exercises 29 and 30 of Section 6.4 to find the work required to propel a 1000-kg satellite out of Earth’s gravitational field.
Find the escape velocity vo that is needed to propel a rocket of mass m out of the gravitational field of a planet with mass M and radius R. Use Newton’s Law of Gravitation (see Exercise 29 in
Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed froma photograph. Suppose
A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let F (t) be the fraction of the company’s bulbs that burn out
As we will see in Section 9.4, a radioactive substance decays exponentially: The mass at time is m (t) = m (0) ekr, where m (0) is the initial mass and is a negative constant. The mean life of an
Determine how large the number has to be so that
Estimate the numerical value of ∫∞ e–x2 dx by writing it as the sum of ∫4 e–x2 dx and ∫∞ e–x2 dx. Approximate the first integral by using Simpson’s Rule with n =
If f(t) is continuous for t > 0, the Laplace transform of f is the function F defined byand the domain of is the set consisting of all numbers for which the integral converges. Find the Laplace
Show that if 0 < f (t) < Meat for t > 0, where M and a are constants, then the Laplace transform F(s) exists for s > a.
Suppose that 0 < f (t) < Meat and 0 < f ’(t) < Keat for t < 0, where f’ is continuous. If the Laplace transform of and the Laplace transform of f'(t) is G(s), show that G(s) = sF(s) - f(0) s > a
If ∫∞ f(x) dx- is convergent and a and b are real numbers, show that
Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C.
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its anti-derivative (take C = 0).
Graph the function f(x) = cos2 x sin3 x and use the graph to guess the value of the integral ∫2π f(x) dx Then evaluate the integral to confirm your guess.
(a) How would you evaluate ∫x5 e–2x dx by hand? (Don’t actually carry out the integration.) (b) How would you evaluate ∫x5 e–2x dx using tables? (Don’t actually do it.) (c) Use
Use the Table of Integrals on the Reference Pages to evaluate the integral.
The Chebyshev’s polynomials are defined by Tn(x) = cos (n arcos x) n = 0, 1, 2, 3,, .(a) What are the domain and range of these functions?(b) We know that T0(x) = 1 and T1 (x) = x. Express T2
Verify Formula 33 in the Table of Integrals (a) By differentiation and (b) By using a trigonometric substitution.
Verify Formula 62 in the Table of Integrals.
Is it possible to find a number such that ∫∞ xn dx is convergent?
For what values of is ∫∞ e ax cos x dx convergent? Evaluate the integral for those values of a.
Use(a) The Trapezoidal Rule,(b) The Midpoint Rule, and(c) Simpson€™s Rule with n = 10 to approximate the given integral, round your answers to six decimal places.
Estimate the errors involved in Exercise 63, parts (a) and (b). How large should be in each case to guarantee an error of less than 0.00001?
Use Simpson’s Rule with n = 6 to estimate the area under the curve y = ex/x from x = 1 to x = 4.
The speedometer reading (v) on a car was observed at 1-minute intervals and recorded in the chart. Use Simpson€™s Rule to estimate the distance traveled by the car.
A population of honeybees increased at a rate of r(t) bees per week, where the graph of is as shown. Use Simpson€™s Rule with six subintervals to estimate the increase in the bee population
(a) If f(x) = sin (sin x), use a graph to find an upper bound for | f (4) (x) |. (b) Use Simpson’s Rule with n = 10 to approximate ∫π f(x) dx and use part (a) to estimate the
Suppose you are asked to estimate the volume of a football. You measure and find that a football is 28 cm long. You use a piece of string and measure the circumference at its widest point to be 53
Use the Comparison Theorem to determine whether the integral is convergent or divergent.
Find the area of the region bounded by the hyperbola y2 – x2 = 1 and the line y = 3.
Find the area bounded by the curves y = cos x and y – cos2x between x = 0 and x = π.
Find the area of the region bounded by the curves y = 1/ (2 + √x), y = 1/ (2 – √x), and x = 1.
The region under the curve y = cos2x, 0 < x < π/2, is rotated about the -axis. Find the volume of the resulting solid.
The region in Exercise 75 is rotated about the -axis. Find the volume of the resulting solid.
If f’ is continuous on [0, ∞] and lim x→∞ f(x) = 0, show that ∫∞ f(x) dx = – f(0)
We can extend our definition of average value of a continuous function to an infinite interval by defining the average value of f on the interval [a, ∞] to be(a) Find the average value of y =
Use the substitution u = 1/x to show that
The magnitude of the repulsive force between two point charges with the same sign, one of size 1 and the other of size q, is F = q/4πε0r2 where is the distance between the charges and
Three mathematics students have ordered a 14-inch pizza. Instead of slicing it in the traditional way, they decide to slice it by parallel cuts, as shown in the figure. Being mathematics majors, they
Evaluate ∫ 1 / x7 – x dx. The straightforward approach would be to start with partial fractions, but that would be brutal. Try a substitution.
A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The path followed by the boat is a curve called a
Graph f(x) = sin (ex) and use the graph to estimate the value of such that ∫t+1 f(x) dx is a maximum. Then find the exact value of that maximizes this integral.
The circle with radius 1 shown in the figure touches the curve y = | 2x |twice. Find the area of the region that lies between the two curves.
A rocket is fired straight up, burning fuel at the constant rate of kilograms per second. Let v = v(t) be the velocity of the rocket at time and suppose that the velocity of the exhaust gas is
Use integration by parts to show that, for all x > 0,
Use the arc length formula (3) to find the length of the curve y = 2 – 3x, – 2 < x < 1. Check your answer by noting that the curve is a line segment and calculating its length by the distance
Use the arc length formula to find the length of the curve y = √4 – x2, 0 < x < 2. Check your answer by noting that the curve is a quarter-circle.
Graph the curve and visually estimate its length. Then find its exact length.
Use Simpson€™s rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator.
(a) Graph the curve y = x3√4 - x, 0 < x < 4. (b) Compute the lengths of inscribed polygons with n = 1, 2, and sides. (Divide the interval into equal subintervals.) Illustrate by sketching
Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve x = in (1 – y2) that lies between the points (0, 0) and (In ¾, ½).
Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve y = x4/3 that lies between the points (0, 0) and (1, 1). If your CAS has trouble
Sketch the curve with equation x2/3 + y2/3 = 1 and use symmetry to find its length.
(a) Sketch the curve y3 = x2.(b) Use Formulas 3 and 4 to set up two integrals for the arc length from (0, 0) to (1, 1). Observe that one of these is an improper integral and evaluate both of them.(c)

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