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mathematics
calculus

Questions and Answers of Calculus

The first appearance in print of l€™ Hospital€™s Rule was in the book Analyse des Infiniment Petites published by the Marquis de l€™ Hospital in 1696. This was the first
The figure shows a sector of a circle with central angle θ. Let A (θ) be the area of the segment between the chord PR and the arc PR. Let B (θ) is the area of the triangle PQR. Find
If f€™ is continuous, f€™ (2) = 0, and f€™ (2) = 7, evaluate
For what values of and is the following equation true?
If f€™ is continuous, use l€™ Hospital€™s Rule to show thatExplain the meaning of this equation with the aid of a diagram.
If f€™€™ is continuous, show that
Let(a) Use the definition of derivative to compute f€™(0).(b) Show that f has derivatives of all orders that are defined on R.
Let(a) Show that f is continuous at 0.(b) Investigate graphically whether f is differentiable at by zooming in several times toward the point (0, 1) on the graph of f.(c) Show that f is not
The figure shows a beam of length L embedded in concrete walls. If a constant load W is distributed evenly along its length, the beam takes the shape of the deflection curve where E and are positive
Coulomb€™s Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the
Find an equation of the slant asymptote. Do not sketch the curve.
Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote.
Show that the curve y = x – tan–1x has two slant asymptotes: y = x + π/2 and y = x – π/2. Use this fact to help sketch the curve.
Show that the curve y = √x2 + 4x has two slant asymptotes: y = x + 2 and y = – x – 2. Use this fact to help sketch the curve.
Show that the lines y = (b/a) x and y = – (b/a) x are slant asymptotes of the hyperbola (x2/a2) – (y2/b2) = 1.
Let f(x) = (x3 + a)/x. Show that lim x →±∞ [f (x) – x2] = 0. This shows that the graph of f approaches the graph of y = x2, and we say that the curve is asymptotic to the parabola y =
Discuss the asymptotic behavior f(x) = (x4 + 1)/x of in the same manner as in Exercise 68. Then use your results to help sketch the graph of f.
Use the asymptotic behavior of f(x) = cos x + 1/x2 to sketch its graph without going through the curve-sketching procedure of this section.
Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f and f€™€™ to estimate the intervals of increase and decrease,
Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points, and use
(a) Graph the function. (b) Use l’ Hospital’s Rule to explain the behavior as x → 0. (c) Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values.
Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the
If f is the function considered in Example 3, use a computer algebra system to calculate f and then graph it to confirm that all the maximum and minimum values are as given in the example. Calculate
If f is the function of Exercise 16, find f ‘and f’’ and use their graphs to estimate the intervals of increase and decrease and concavity of f.
Use a computer algebra system to graph f and to find f€™ and f€™€™. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values,
(a) Graph the function.(b) Explain the shape of the graph by computing the limit as x → 0+ or as → ∞.(c) Estimate the maximum and minimum values and then use calculus to find the
In Example 4 we considered a member of the family of functions that f(x) = sin(x + sin cx) occur in FM synthesis. Here we investigate the function with c = 3. Start by graphing f in the viewing
Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and
The family of functions f (t) = C (e-at – e-bt), where a, b, and are positive numbers and, has been used to b > a model the concentration of a drug injected into the blood at time t = 0. Graph
Investigate the family of curves given by f(x) = Xe-ex, where is a real number. Start by computing the limits as x → ± ∞. Identify any transitional values of c where the basic shape
Investigate the family of curves given by the equation f(x) = x4 + cx2 + x. Start by determining the transitional value of at which the number of inflection points changes. Then graph several members
(a) Investigate the family of polynomials given by the equation f(x) = cx4 – cx2 + 1. For what values of does the curve have minimum points?(b) Show that the minimum and maximum points of every
(a) Investigate the family of polynomials given by the equation f(x) = 2x3 + cx2 + 2x. For what values of does the curve have maximum and minimum points?(b) Show that the minimum and maximum points
Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum.(a) Make a table of values, like the following one, so that the sum of the numbers in the first two
Find two numbers whose difference is 100 and whose product is a minimum.
Find two positive numbers whose product is 100 and whose sum is a minimum.
Find a positive number such that the sum of the number and its reciprocal is as small as possible.
Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.
Find the dimensions of a rectangle with area 1000m2 whose perimeter is as small as possible.
Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the
Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the
A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to
A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.
If 1200 cm of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
A rectangular storage container with an open top is to have a volume of 10 m . The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides
Do Exercise 12 assuming the container has a lid that is made from the same material as the sides.
(a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square.(b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.
Find the point on the line y = 4x + 7 that is closest to the origin.
Find the point on the line 6x + y = 9 that is closest to the point (-3, 1).
Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).
Find, correct to two decimal places, the coordinates of the point on the curve y = tan x that is closest to the point (1, 1).
Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.
Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.
Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.
Find the dimensions of the rectangle of largest area that has its base on the -axis and its other two vertices above the –axis and lying on the parabola y = 8 – x2.
Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.
Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs.
A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.
A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.
A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible surface area of such a cylinder.
A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle. See Exercise 52 on page 24.) If the perimeter of
The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions of the poster with the
A poster is to have an area of 180 in with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?
A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle, how should the wire be cut so that the total area enclosed is (a)
A cylindrical can without a top is made to contain V cm3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can.
A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the
A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.
A cone-shaped paper drinking cup is to be made to hold 27cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.
A cone with height is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h = 1/3H.
For a fish swimming at a speed relative to the water, the energy expenditure per unit time is proportional to v3. It is believed that migrating fish try to minimize the total energy required to swim
In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end. It is believed that bees form their cells in such a way as to minimize the surface area
A boat leaves a dock at 2:00 P.M. and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 P.M. At what time were the two boats
Solve the problem in Example 4 if the river is 5 km wide and point B is only 5 km downstream from A.
A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite on the other side of the lake in the shortest possible A time. She can
The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources,
Find an equation of the line through the point (3, 5) that cuts off the least area from the first quadrant.
Let and be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point (a, b).
At which points on the curve y = 1 + 40x3 – 3x5 does the tangent line have the largest slope?
Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.
The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the
A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B, and C is minimized (see the figure). Express L as a function of x = | AP |
The graph shows the fuel consumption of a car (measured in gallons per hour) as a function of the speed of the car. At very low speeds the engine runs inefficiently, so initially decreases as the
Let v1 the velocity of light in air and v2 the velocity of light in water.According to Fermat€™s Principle, a ray of light will travel from a point A in the air to a point B in the water by
Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show
The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words,
A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be
An observer stands at a point P, one unit away from a track. Two runners start at the point in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum
A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle θ. How should θ be chosen so that the gutter will
Where should the point P be chosen on the line segment AB so as to maximize the angle θ?
A painting in an art gallery has height and is hung so that its lower edge is a distance above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the
Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W.
The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as
Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than
Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line ℓ parallel to the line joining the light sources and at a distance d meters from
A manufacturer keeps precise records of the cost C(x) of making items and produces the graph of the cost function shown in the figure.(a) Explain why C (0) > 0.(b) What is the significance of the
The graph of a cost function C is given.(a) Draw a careful sketch of the marginal cost function.(b) Use the geometric interpretation of the average cost c(x) as a slope (see Figure 1) to draw a
The average cost of producing units of a commodity is c(x) = 21.4 – 0.002x. Find the marginal cost at a production level of 1000 units. In practical terms, what is the meaning of your answer?
The figure shows graphs of the cost and revenue functions reported by a manufacturer.(a) Identify on the graph the value of for which the profit is maximized.(b) Sketch a graph of the profit
For each cost function (given in dollars), find.(a) The cost, average cost, and marginal cost at a production level of 1000 units;(b) The production level that will minimize the average cost; and(c)
A cost function is given.(a) Find the average cost and marginal cost functions.(b) Use graphs of the functions in part (a) to estimate the production level that minimizes the average cost.(c) Use

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