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mathematics
calculus

Questions and Answers of Calculus

The velocity of a wave of length L in deep water is where and is known positive constants.What is the length of the wave that gives the minimum velocity?
A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?
A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at %12, average attendance at a game has been 11,000. A market survey indicates that for each
A manufacturer determines that the cost of making units of a commodity is C(x) = 1800 + 25x – 0.2x2 + 0.001x3 and the demand function is p(x) = 48.2 – 0.03x.(a) Graph the cost and revenue
Use Newton’s method to find the root of the equation x5 – x4 + 3x2 – 3x – 2 = 0 in the interval [1, 2] correct to six decimal places.
Use Newton’s method to find all roots of the equation sin x = x2 – 3x + 1 correct to six decimal places.
Use Newton’s method to find the absolute maximum value of the function f (t) = cos t + t – t2 correct to eight decimal places.
(a) If f(x) = 0,1ex + sin x, - 4 < x < 4 use a graph of f to sketch a rough graph of the anti-derivative F of f that satisfies.(b) Find an expression for F(x).(c) Graph F using the expression in part
Investigate the family of curves given by f(x) = x4 + x3 + cx2. In particular you should determine the transitional value of at which the number of critical numbers changes and the transitional value
A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 m/s. Will it burst?
In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make.
A rectangular beam will be cut from a cylindrical log of radius 10 inches.(a) Show that the beam of maximal cross-sectional area is a square.(b) Four rectangular planks will be cut from the four
If a projectile is fired with an initial velocity at an angle of inclination from the horizontal, then its trajectory, neglecting air resistance, is the parabola(a) Suppose the projectile is fired
A light is to be placed atop a pole of height feet to illuminate a busy traffic circle, which has a radius of 40 ft. The intensity of illumination at any point P on the circle is directly
Water is flowing at a constant rate into a spherical tank. Let V (t) be the volume of water in the tank and H (t) be the height of the water in the tank at time t.(a) What are the meanings of V’
Show that, for x > 0, x/1 + x2 < tan–1 x < x
Sketch the graph of a function f such that f’(x) < 0 for all |x, f” (x) > 0 for | x | > 1, f”(x) < 0 for | x | < 1, and lim x→±∞ [f(x) + x] = 0
If a rectangle has its base on the -axis and two vertices on the curve y = e–x2, show that the rectangle has the largest possible area when the two vertices are at the points of inflection of the
Show that | sin x – cos x | < √2 for all x.
Show that, for all positive values of and y,
Show that x2y2 (4 – x2) (4 – y2) < 16 for all numbers and such that | x | < 2 and | y | < 2.
Let a and be positive numbers. Show that not both of the numbers a (1 – b) and b (1 – a) can be greater than ¼.
Find the point on the parabola y = 1 – x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.
Find the highest and lowest points on the curve x2 + xy + y2 = 12.
Sketch the set of all points (x, y) such that | x + y | < ex.
The line y = mx + b intersects the parabola y = x2 in points A and B (see the figure). Find the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB.
A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until chambers, including the sphere, are
Determine the values of the number for which the function f has no critical number: f(x) = (a2 + a – 6) cos 2x + (a – 2) x + cos 1
Sketch the region in the plane consisting of all points (x, y) such that 2xy < | x – y | < x2 + y2
Let ABC be a triangle with < BAC = 120o and | AB | · | AC | = 1.(a) Express the length of the angle bisector AD in terms of x = | AB |.(b) Find the largest possible value of | AD |.
ABCD is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from B to D with center. The piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the
For which positive numbers does the curve y = ax intersect the line y = x?
For what value of is the following equation true?
Let f(x) = a1 sin x + a2 sin 2x + . . . + an, where a1, a2, . . .an, are real numbers and is a positive integer. If it is given that for all | f(x) < | sin x |, show that | a1 – 2a2 + . . . + na n
An arc PQ of a circle subtends a central angle as in the figure. Let A (θ) be the area between the chord PQ and the arc PQ. Let be the area between the tangent lines PR, QR, and the arc. Find
The speeds of sound c1 an upper layer and c12 lower layer of rock and the thickness of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater
For what values of is there a straight line that intersects the curve y = x4 + cx3 + 12x2 – 5x + 2 in four distinct points?
One of the problems posed by the Marquis de l€™ Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a
Given a sphere with radius r, find the height of a pyramid of minimum volume whose base is a square and whose base and triangular faces are all tangent to the sphere. What if the base of the pyramid
Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer
(a) By reading values from the given graph of f, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x = 0 to x = 10. In each case sketch
(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 12.(i) L6 (sample points are left endpoints)(ii) R6 (sample points are right
(a) Estimate the area under the graph of f(x) = 1/x from x = 1 to x =5 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate
(a) Estimate the area under the graph of f(x) = 25 - x2 fromx = 0 to x = 5 using five approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an
(a) Estimate the area under the graph of f(x) = 1 + x2 from x = – 1 to x = 2 using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the
(a) Graph the function f(x) = e –x2, – 2 < x < 2.(b) Estimate the area under the graph of f using four approximating rectangles and taking the sample points to be(i) Right endpoints (ii)
With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of n, using looping. (On a TI use
Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if xi is a left or right endpoint. (For instance, in Maple use left
(a) If f(x) = sin (sin x), 0 < x < π/2 use the commands discussed in Exercise 9 to find the left and right sums for n = 10, 30, and 50. (b) Illustrate by graphing the rectangles in part
The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she
Speedometer readings for a motorcycle at 12-second intervals are given in the table.(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning
Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at 2-hour time intervals are shown in the table. Find lower and upper estimates for
Use these data to estimate the height above Earth€™s surface of the space shuttle Endeavour, 62 seconds after liftoff.
The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.
The velocity graph of a car accelerating from rest to a speed of 120km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period.
Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
(a) Use Definition 2 to find an expression for the area under the curve y = x3 from 0 to 1 as a limit.(b) The following formula for the sum of the cubes of the first n integers is proved in Appendix
(a) Express the area under the curve y = x5 from 0 to 2 as a limit.(b) Use a computer algebra system to find the sum in your expression from part (a).(c) Evaluate the limit in part (a).
Find the exact area of the region under the graph of y = e–x from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the
Find the exact area under the cosine curve y = cos x from x – 0 to x = b, where 0 < b < π/2. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what
(a) Let An be the area of a polygon with equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2π/n, show that An = ½nr2 sin
Evaluate the Riemann sum for f(x) = 2 – x2, 0 < x < 2, with four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.
If f(x) = in x – 1, 1 < x < 4, evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum
If f(x) = √x – 2, 1 < x < 6, find the Riemann sum with n = 5 correct to six decimal places, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a
(a) Find the Riemann sum for f(x) = x – 2 sin 2x, 0 < x < 3, with six terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the
The graph of a function f is given. Estimate f(x) dx using four subintervals with(a) Right endpoints,(b) Left endpoints, and(c) Midpoints.
The graph of is shown. Estimate f-3 g(x) dx with six subintervals using(a) Right endpoints,(b) Left endpoints, and(c) Midpoints.
A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for f25 f(x) dx.
The table gives the values of a function obtained from an experiment. Use them to estimate ∫60 f(x) dx using three equal subintervals with(a) Right endpoints,(b) Left endpoints, and(c) Midpoints.
Use the Midpoint Rule with the given value of to approximate the integral, round the answer to four decimal places.
If you have a CAS that evaluates midpoint approximations and graphs the corresponding rectangles (use middle sum and middle box commands in Maple), check the answer to Exercise 11 and illustrate with
With a programmable calculator or computer (see the instructions for Exercise 7 in Section 5.1), compute the left and rightRiemann sums for the function f(x) = sin(x2) on the interval [0, 1] with n =
Use a calculator or computer to make a table of values of right Riemann sums Rn for the integral ∫π0 sin x dx with n = 5, 10, 50, and 100. What value do these numbers appear to be
Use a calculator or computer to make a table of values of left and right Riemann sums Ln and Rn for the integral ∫2 e–x2 dx with n = 5, 10, 50, and 100. Between what two numbers must the
Express the limit as a definite integral on the given interval.
Use the form of the definition of the integral given in Equation 3 to evaluate the integral.
(a) Find an approximation to the integral ∫4 (x2 – 3x) dx using a Riemann sum with right endpoints and n = 8. (b) Draw a diagram like Figure 3 to illustrate the approximation in part
Express the integral as a limit of Riemann sums. Do not evaluate the limit.
Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit.
The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.
The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral.
Evaluate the integral by interpreting it in terms of areas.
In Example 2 in Section 5.1 we showed that ∫1 x2 dx = 1/3. Use this fact and the properties of integrals to evaluate ∫1 (5 – 6x2) dx.
Use the properties of integrals and the result of Example 3 to evaluate ∫3 (2ex – 1)dx.
Use the result of Example 3 to evaluate ∫3 ex+2 dx.
Use the result of Exercise 27 and the fact that ∫π/2 cos x dx = 1. (From Exercise 25 in Section 5.1), together with the properties of integrals, to evaluate ∫π/2 (2 cos x –
Write as a single integral in the form ∫b f (x) dx:
Use the properties of integrals to verify the inequality without evaluating the integrals.
Use Property 8 to estimate the value of the integral.
Use properties of integrals, together with Exercises 27 and 28, to prove the inequality.
Explain exactly what is meant by the statement that “differentiation and integration are inverse processes.”
Let g(x) = ∫x f(t) dt, where f is the function whose graph is shown.(a) Evaluate g(x) for x = 0, 1, 2, 3, 4, 5, and 6.(b) Estimate g(7).(c) Where does have a maximum value? Where does it have a
Let g(x) ∫x f(t) dt, where f is the function whose graph is shown.(a) Evaluate g(0), g(1), g(2), g(3), and g(6).(b) On what interval is increasing?(c) Where does have a maximum value?(d) Sketch a
Let g(x) = ∫x-3 f(t) dt, where f is the function whose graph is shown.(a) Evaluate and g(€“ 3) and g(3).(b)Estimate g(€“ 2), g(€“ 1), and g(0).(c) On what interval
Sketch the area represented by g(x). Then find g€™(x) in two ways;(a) By using Part 1 of the Fundamental Theorem and.(b) By evaluating the integral using Part 2 and then differentiating.
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.
Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.
The Fresnel function was defined in Example 3 and graphed in Figures 7 and 8.(a) At what values of does this function have local maximum values?(b) On what intervals is the function concave

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