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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

What is the maximum area of a rectangle inscribed in a right triangle with legs of length 3 and 4 as in Figure 12? The sides of the rectangle are parallel to the legs of the triangle. 5 3 4
Find the dimensions of the rectangle of maximum area that can be inscribed in a circle of radius r = 4 (Figure 13).
In the setting of Examples 2 and 3, let r denote the speed along the road, and h denote the speed along the highway.(a) Show that the travel–time function T(x) has a critical point at x = 30/√(h/r)2 − 1 and explain why this indicates that if r ≥ h there is no critical point.(b) Explain why
In the setting of Examples 2 and 3, replace 30 and 50 with general distances D and L, respectively. Also, let r denote the speed along the road, and h denote the speed along the highway. Show that the travel–time function T(x) has a critical point at x = √D (h/r)2 − 1.Example 2Example 3
Find the dimensions x and y of the rectangle inscribed in a circle of radius r that maximizes the quantity xy2.
In the article “Do Dogs Know Calculus?” the author Timothy Pennings explained how he noticed that when he threw a ball diagonally into Lake Michigan along a straight shoreline, his dog Elvis seemed to pick the optimal point in which to enter the water so as to minimize his time to reach the
A four-wheel-drive vehicle is transporting an injured hiker to the hospital from a point that is 30 km from the nearest point on a straight road. The hospital is 50 km down that road from that nearest point. If the vehicle can drive at 30 kph over the terrain and at 120 kph on the road, how far
Find the point P on the parabola y = x2 closest to the point (3, 0) (Figure 15). y P y = x² 3 >X
Find the point on the line y = x closest to the point (1, 0). It is equivalent and easier to minimize the square of the distance.
Among all positive numbers a, b whose sum is 8, find those for which the product of the two numbers and their difference is largest.
A right circular cone (Figure 18) has volume V = π/3 r2 h and surface area S = πr √r2 + h2. Find the dimensions of the cone with surface area 1 and maximal volume. h r
Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 1 (Figure 19). -1 X (x, y) y
Find the radius and height of a cylindrical can of total surface area A whose volume is as large as possible. Does there exist a cylinder of surface area A and minimal total volume?
A poster of area 6000 cm2 has blank margins of width 10 cm on the top and bottom and 6 cm on the sides. Find the dimensions that maximize the printed area.
Find the angle θ that maximizes the area of the trapezoid with a base of length 4 and sides of length 2, as in Figure 20. 2 Ө 4 Ө 2
According to postal regulations, a carton is classified as “oversized” if the sum of its height and girth (perimeter of its base) exceeds 108 in. Find the dimensions of a carton with a square base that is not oversized and has maximum volume.
In his work Nova stereometria doliorum vinariorum (New Solid Geometry of a Wine Barrel), published in 1615, astronomer Johannes Kepler stated and solved the following problem: Find the dimensions of the cylinder of largest volume that can be inscribed in a sphere of radius R. Hint: Show that an
A landscape architect wishes to enclose a rectangular garden of area 1000 m2 on one side by a brick wall costing $90/m and on the other three sides by a metal fence costing $30/m. Which dimensions minimize the total cost?
Find the maximum area of a rectangle inscribed in the region bounded by the graph of y = 4 − x/2 + x and the axes (Figure 21). 2 y = 4-x 2 + x 4 -X
The amount of light reaching a point at a distance r from a light source A of intensity IA is IA/r2. Suppose that a second light source B of intensity IB = 4IA is located 10 m from A. Find the point on the segment joining A and B where the total amount of light is at a minimum.
Find the maximum area of a triangle formed by the axes and a tangent line to the graph of y = (x + 1)−2 with x > 0.
Find the maximum area of a rectangle circumscribed around a rectangle of sides L and H. Hint: Express the area in terms of the angle θ (Figure 22). L Ө H
A contractor is engaged to build steps up the slope of a hill that has the shape of the graph of y = x2(120 − x)/6400 for 0 ≤ x ≤ 80 with x in meters (Figure 23). What is the maximum vertical rise of a stair if each stair has a horizontal length of 1/3 m? 40 20 20 40 60 80 ➤X
Find the equation of the line through P = (4, 12) such that the triangle bounded by this line and the axes in the first quadrant has minimal area.
A spherical cap (Figure 24) of radius r and height h has volume V = πh2(r − 1/3h) and surface area S = 2πrh. Prove that the hemisphere encloses the largest volume among all spherical caps of fixed surface area S. T h!
Let P = (a, b) lie in the first quadrant. Find the slope of the line through P such that the triangle bounded by this line and the axes in the first quadrant has minimal area. Then show that P is the midpoint of the hypotenuse of this triangle.
Find the isosceles triangle of smallest area (Figure 25) that circumscribes a circle of radius 1 (from Thomas Simpson’s The Doctrine and Application of Fluxions, a calculus text that appeared in 1750) 7 I 0/
A box of volume 72 m3 with a square bottom and no top is constructed out of two different materials. The cost of the bottom is $40/m2 and the cost of the sides is $30/m2. Find the dimensions of the box that minimize total cost.
Find the dimensions of a cylinder of volume 1 m3 of minimal cost if the top and bottom are made of material that costs twice as much as the material for the side.
Your task is to design a rectangular industrial warehouse consisting of three separate spaces of equal size as in Figure 26. The wall materials cost $500 per linear meter and your company allocates $2,400,000 for that part of the project involving the walls.(a) Which dimensions maximize the area of
According to a model developed by economists E. Heady and J. Pesek, if fertilizer made from N pounds of nitrogen and P lb of phosphate is used on an acre of farmland, then the yield of corn (in bushels per acre) isA farmer intends to spend $30/acre on fertilizer. If nitrogen costs 25 cents/lb and
Experiments show that the quantities x of corn and y of soybean required to produce a hog of weight Q satisfy Q = 0.5x1/2y1/4. The unit of x, y, and Q is the cwt, an agricultural unit equal to 100 lb. Find the values of x and y that minimize the cost of a hog of weight Q = 2.5 cwt if corn costs
All units in a 100-unit apartment building are rented out when the monthly rent is set at r = $900/month. Suppose that one unit becomes vacant with each $10 increase in rent and that each occupied unit costs $80/mon in maintenance. Which rent r maximizes monthly profit?
An 8-billion-bushel corn crop brings a price of $2.40/bushel. A commodity broker uses the rule of thumb: If the crop is reduced by x percent, then the price increases by 10x cents. Which crop size results in maximum revenue and what is the price per bushel? Hint: Revenue is equal to price times
The rectangular plot in Figure 27 has size 100 m × 200 m. Pipe is to be laid from A to a point P on side BC and from there to C. The cost of laying pipe along the side of the plot is $45/m and the cost through the plot is $80/m (since it is underground). (a) Let f(x) be the total cost, where x is
The monthly output of a Spanish light bulb factory is P = 2LK2 (in millions), where L is the cost of labor and K is the cost of equipment (in millions of euros). The company needs to produce 1.7 million units per month. Which values of L and K would minimize the total cost L + K?
Brandon is on one side of a river that is 50 m wide and wants to reach a point 200 m downstream on the opposite side as quickly as possible by swimming diagonally across the river and then running the rest of the way. Find the best route if Brandon can swim at 1.5 m/s and run at 4 m/s.
When a light beam travels from a point A above a swimming pool to a point B below the water (Figure 28), it chooses the path that takes the least time. Let v1 be the velocity of light in air and v2 the velocity in water (it is known that v1 > v2). Prove Snell’s Law of Refraction: A h₁ sin
A small blood vessel of radius r branches off at an angle θ from a larger vessel of radius R to supply blood along a path from A to B. According to Poiseuille’s Law, the total resistance to blood flow is proportional towhere a and b are as in Figure 29. Show that the total resistance is
A box (with no top) is to be constructed from a piece of cardboard with sides of length A and B by cutting out squares of length h from the corners and folding up the sides (Figure 30).Find the value of h that maximizes the volume of the box if A = 15 and B = 24. What are the dimensions of this
A box (with no top) is to be constructed from a piece of cardboard with sides of length A and B by cutting out squares of length h from the corners and folding up the sides (Figure 30).Which values of A and B maximize the volume of the box if h = 10 cm and AB = 900 cm2? A В
The force F (in Newtons) required to move a box of mass m kg in motion by pulling on an attached rope (Figure 33) iswhere θ is the angle between the rope and the horizontal, ƒ is the coefficient of static friction, and g = 9.8 m/s2. Find the angle θ that minimizes the required force F, assuming
Which value of h maximizes the volume of the box if A = B?
In the setting of Exercise 67, show that for any ƒ the minimal force required is proportional to 1/√1 + ƒ2.Data From Exercise 67 The force F (in Newtons) required to move a box of mass m kg in motion by pulling on an attached rope (Figure 33) iswhere θ is the angle between the rope and the
Ornithologists have found that the power (in joules per second) consumed by a certain pigeon flying at velocity v m/s is described well by the function P(v) = 17v−1 + 10−3v3 joules/s. Assume that the pigeon can store 5 × 104 joules of usable energy as body fat.(a) Show that at velocity v, a
The problem is to put a “roof” of side s on an attic room of height h and width b. Find the smallest length s for which this is possible if b = 27 and h = 8 (Figure 35). b h S
Redo Exercise 70 for arbitrary b and h.Data From Exercise 70The problem is to put a “roof” of side s on an attic room of height h and width b. Find the smallest length s for which this is possible if b = 27 and h = 8 (Figure 35). b h S
Find the maximum length of a pole that can be carried horizontally around a corner joining corridors of widths a = 24 and b = 3 (Figure 36). a iz b
A basketball player stands d feet from the basket. Let h and α be as in Figure 38. Using physics, one can show that if the player releases the ball at an angle θ, then the initial velocity required to make the ball go through the basket satisfies 2² = 16d cos² (tan tan a)
Redo Exercise 72 for arbitrary widths a and b.Data From Exercise 72Find the maximum length of a pole that can be carried horizontally around a corner joining corridors of widths a = 24 and b = 3 (Figure 36). a iz b
Find the minimum length ℓ of a beam that can clear a fence of height h and touch a wall located b ft behind the fence (Figure 37). b h l x
Are concerned with determining the thickness d of a layer of soil that lies on top of a rock formation. Geologists send two sound pulses from point A to point D separated by a distance s. The first pulse travels directly from A to D along the surface of the earth. The second pulse travels down to
Are concerned with determining the thickness d of a layer of soil that lies on top of a rock formation. Geologists send two sound pulses from point A to point D separated by a distance s. The first pulse travels directly from A to D along the surface of the earth. The second pulse travels down to
Are concerned with determining the thickness d of a layer of soil that lies on top of a rock formation. Geologists send two sound pulses from point A to point D separated by a distance s. The first pulse travels directly from A to D along the surface of the earth. The second pulse travels down to
In this exercise, we use Figure 42 to prove Heron’s principle of Example 7 without calculus. By definition, C is the reflection of B across the line MN (so that BC is perpendicular to MN and BN = CN). Let P be the intersection of AC and MN. Use geometry to justify the following:Example 7 =
State whether ƒ(2) and ƒ(4) are local minima or local maxima, assuming that Figure 15 is the graph of ƒ. তिक ++x 1 2 3 /4 5 6
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x2 − 4x3
Determine the intervals on which the function is concave up or down and find the points of inflection. f(x) = 3 X- 1 + x
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = 1/3 x3 + x2 + 3x
Figure 16 shows the graph of the derivative ƒ' of a function ƒ. Find the critical points of ƒ and determine whether they are local minima, local maxima, or neither. -2 -1 6 -2 0.5 y = f'(x) 2 3
Determine the intervals on which the function is concave up or down and find the points of inflection. w(t) = -1 t
Sketch the graph of a function ƒ whose derivative ƒ' has the given description. f'(x) > 0 for x > 3 and f'(x) < 0 for x < 3
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = 4 − 2x2 + 1/6x4
The position of an ambulance in kilometers on a straight road over a period of 4 hours is given by the graph in Figure 15.(a) Describe the motion of the ambulance.(b) Explain what the fact that this graph is concave up tells us about the speed of the ambulance. y 100- 4
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = 7x4 − 6x2 + 1
Sketch the graph of a function ƒ whose derivative ƒ' has the given description. f'(x) > 0 for x < 1 and f'(x) < 0 for x > 1
The position of a bicyclist on a straight road in kilometers over a period of 4 h is given by the graph in Figure 16, where inflection points occur when t = 0.5 and t = 2.(a) Describe the motion of the bicyclist.(b) Explain what the concavity of the graph over various intervals tells us about the
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x5 + 5x
Sketch the graph of a function ƒ whose derivative ƒ' has the given description.ƒ'(x) is negative on (1, 3) and positive everywhere else.
The growth of a sunflower during the first 100 days after sprouting is modeled well by the logistic curve y = h(t) shown in Figure 17. Estimate the growth rate at the point of inflection and explain its significance. Then make a rough sketch of the first and second derivatives of h.
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x5 − 15x3
Assume that Figure 18 is the graph of ƒ. Where do the points of inflection of ƒ occur, and on which interval is ƒ concave down? a b c d e f Do 8 -X
Sketch the graph of a function ƒ whose derivative ƒ' has the given description.ƒ'(x) makes the sign transitions +, −, +, −.
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x4 − 3x3 + 4x
Find all critical points of ƒ and use the First Derivative Test to determine whether they are local minima or maxima.ƒ(x) = 4 + 6x − x2
Repeat Exercise 22 but assume that Figure 18 is the graph of the derivative ƒ'.Data From Exercise 22Assume that Figure 18 is the graph of ƒ. Where do the points of inflection of ƒ occur, and on which interval is ƒ concave down? a b c d e f Do 8 -X
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x2(x − 4)2
Figure 19 shows the derivative ƒ' on [0, 1.2]. Locate the points of inflection of ƒ and the points where the local minima and maxima occur. Determine the intervals on which ƒ has the following properties:(a) Increasing (b) Decreasing(c) Concave up (d) Concave down y = f'(x) 0.17
Find all critical points of ƒ and use the First Derivative Test to determine whether they are local minima or maxima.ƒ(x) = x3 − 12x − 4
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x7 − 14x6
Find all critical points of ƒ and use the First Derivative Test to determine whether they are local minima or maxima. f(x) = zł x + 1
Leticia has been selling solar-powered laptop chargers through her Web site, with monthly sales as recorded below. In a report to investors, she states, “Sales reached a point of inflection when I started using pay-per-click advertising.” In which month did that occur? Explain. Month 1 2 Sales
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x6 − 9x4
Find all critical points of ƒ and use the First Derivative Test to determine whether they are local minima or maxima.ƒ(x) = x3 + x−3
Find the critical points and apply the Second Derivative Test (or state that it fails).ƒ(x) = x3 − 12x2 + 45x
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x − 4√x
Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point yields a local min or max (or neither).y = −x2 + 7x − 17
Find the critical points and apply the Second Derivative Test (or state that it fails).ƒ(x) = x4 − 8x2 + 1
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = √x + √16 − x
Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point yields a local min or max (or neither).y = 5x2 + 6x − 4
Find the critical points and apply the Second Derivative Test (or state that it fails).ƒ(x) = 3x4 − 8x3 + 6x2
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x(8 − x)1/3
Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point yields a local min or max (or neither).y = x3 − 12x2
Find the critical points and apply the Second Derivative Test (or state that it fails).ƒ'(x) = x5 − x3
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = (x2 − 4x)1/3
Find the critical points and apply the Second Derivative Test (or state that it fails). f(x) = x²2²8x x + 1
Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point yields a local min or max (or neither).y = x(x − 2)3
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = (2x − x2)1/3
Find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point yields a local min or max (or neither).y = 3x4 + 8x3 − 6x2 − 24x
Find the critical points and apply the Second Derivative Test (or state that it fails). f(x) = 1 x²-x+ 2

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