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Mathematics LibreTexts

4.E: Applications of Integration (Exercises)

 

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4.1: Newton's Method

Terms and Concepts

1. T/F: Given a function \(f(x)\), Newton's Method produces an exact solution to \(f(x)=0\).

2. T/F: In order to get a solution to \(f(x) =0\) accurate to \(d\) places after the decimal, at least \(d+1\) iterations of Newton's Method must be used.

Problems

In Exercises 3-7, the roots of \(f(x)\) are known or are easily found. Use 5 iterations of Newton's Method with the given initial approximation to approximate the root. Compare it to the known value of the root.

3. \(f(x) =\cos x,\, x_0 =1.5\)

4. \(f(x) =\sin x,\, x_0 =1\)

5. \(f(x) =x^2+x-2,\, x_0 =0\)

6. \(f(x) =x^2-2,\, x_0 =1.5\)

7. \(f(x) =\ln x,\, x_0 =2\)

In Exercises 8-11, use Newton's Method to approximate all roots of the given functions accurate to 3 places after the decimal. If an interval is given, find only the roots that lie in that interval. Use technology to obtain good initial approximations.

8. \(f(x) =x^3+5x^2-x-1\)

9. \(f(x) =x^4+2x^3-7x^2-x+5\)

10. \(f(x) =x^{17}-2x^{13}-10x^8+10\text{ on }(-2,2)\)

11. \(f(x) =x^2\cos x +(x-1)\sin x \text{ on }(-3,3)\)

In exercises 12-15, use Newton's Method to approximate when the given functions are equal, accurate to 3 places after the decimal. Use technology to obtain good initial approximations.

12. \(f(x)=x^2,\,g(x)=\cos x\)

13. \(f(x)=x^2-1,\,g(x)=\sin x\)

14. \(f(x)=e^{x^2},\,g(x)=\cos x\)

15. \(f(x)=x,\,g(x)=\tan x\text{ on }[-6,6]\)

16. Why does Newton's Method fail in finding the root of \(f(x)=x^3-3x^2+x+3\text{ when }x_0=1\)?

17. Why does Newton's Method fail in finding the root of \(f(x)=-17x^4+130x^3-301x^2+156x+156\text{ when }x_0=1\)?

 

4.2: Related Rates

Terms and Concepts

1. T/F: Implicit differentiation is often used when solving "related rates" type problems.

2. T/F: A study of related rates is part of the standard police officer training.

Problems

3. Water flows onto a flat surface at a rate of 5cm\(^3\)/s forming a circular puddle 10mm deep. How fast is the radius growing when the radius is:
(a) 1 cm?
(b) 10 cm?
(c) 100 cm?

4. A circular balloon is inflated with air flowing at a rate of 10cm\(^3\)/s. How fast is the radius of the balloon increasing when the radius is:
(a) 1 cm?
(b) 10 cm?
(c) 100 cm?

5. Consider the traffic situation introduced in Example 100. How fast is the “other car” traveling if the officer and the other car are each 1/2 mile from the intersection, the other car is traveling due west, the officer is traveling north at 50mph, and the radar reading is −80mph?

6. Consider the traffic situation introduced in Example 100. Calculate how fast the “other car” is traveling in each of the following situations.
(a) The officer is traveling due north at 50mph and is 1/2 mile from the intersection, while the other car is 1 mile from the intersection traveling west and the radar reading is −80mph?
(b) The officer is traveling due north at 50mph and is 1 mile from the intersection, while the other car is 1/2 mile from the intersection traveling west and the radar reading is −80mph?

7. An F-22 aircraft is flying at 500mph with an elevation of 10,000ft on a straight–line path that will take it directly over an anti–aircraft gun.
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How fast must the gun be able to turn to accurately track the aircraft when the plane is:
(a) 1 mile away? 
(b) 1/5 mile away?
(c) Directly overhead?

8. An F-22 aircraft is flying at 500mph with an elevation of 100ft on a straight–line path that will take it directly over an anti–aircraft gun as in Exercise 7 (note the lower elevation here).

How fast must the gun be able to turn to accurately track the aircraft when the plane is:
(a) 1000 feet away?
(b) 100 feet away?
(c) Directly overhead?

9. A 24ft. ladder is leaning against a house while the base is pulled away at a constant rate of 1 ft/s.
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At what rate is the top of the ladder sliding down the side of the house when the base is:
(a) 1 foot from the house?
(b) 10 feet from the house? 
(c) 23 feet from the house?
(d) 24 feet from the house?

10. A boat is being pulled into a dock at a constant rate of 30ft/min by a winch located 10ft above the deck of the boat.
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At what rate is the boat approaching the dock when the boat is:
(a) 50 feet out?
(b) 15 feet out?
(c) 1 foot from the dock?
(d) What happens when the length of rope pulling in the boat is less than 10 feet long?

11. An inverted cylindrical cone, 20ft deep and 10ft across at the top, is being filled with water at a rate of 10ft\(^3\)/min. At what rate is the water rising in the tank when the depth of the water is:
(a) 1 foot?
(b) 10 feet?
(c) 19 feet?

How long will the tank take to fill when starting at empty?

12. A rope, attached to a weight, goes up through a pulley at the ceiling and back down to a worker. The man holds the rope at the same height as the connection point between rope and weight.
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Suppose the man stands directly next to the weight (i.e., a total rope length of 60ft) and begins to walk away at a rate of 2ft/s. How fast is the weight rising when the man has walked:
(a) 10 feet?
(b) 40 feet?

How far must the man walk to raise the weight all the way to the pulley?

13. Consider the situation described in Exercise 12. Suppose the man starts 40ft from the weight and begins to walk away at a rate of 2ft/s.
(a) How long is the rope?
(b) How fast is the weight rising after the man has walked 10 feet?
(c) How fast is the weight rising after the man has walked 40 feet?
(d) How far must the man walk to raise the weight all the way to the pulley?

14. A hot air balloon lifts off from ground rising vertically. From 100 feet away, a 5' woman tracks the path of the balloon. When her sightline with the balloon makes a 45 degree angle with the horizontal, she notes the angle is increasing at about \(5^\circ\) /min.

15. A company that produces landscaping materials is dumping sand into a conical pile. The sand is being poured at a rate of 5ft\(^3\)/sec; the physical properties of the sand, in conjunction with gravity, ensure that the cone’s height is roughly 2/3 the length of the diameter of the circular base.
How fast is the cone rising when it has a height of 30 feet?

4.3: Optimization

Terms and Concepts

1. T/F: An "optimization problem" is essentially an "extreme values" problem in a "story problem" setting.

2. T/F: This section teaches one to find the extreme values of function that have more than one variable.

Problems

3. Find the maximum product of two numbers (not necessarily integers) that have a sum of 100.

4. Find the minimum sum of two numbers whose product is 500.

5. Find the maximum sum of two numbers whose product is 500.

6. Find the maximum sum of two numbers, each of which is in [0,300] whose product is 500.

7. Find the maximal area of a right triangle with hypotenuse of length 1.

8. A rancher has 1000 feet of fencing in which to construct adjacent, equally sized rectangular pens. What dimensions should these pens have to maximize the enclosed area?

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9. A standard soda can is roughly cylindrical and holds 355cm\(^3\) of liquid. What dimensions should the cylinder be to minimize the material needed to produce the can? Based on your dimensions, determine whether or not the standard can is produced to minimize the material costs.

10. Find the dimensions of a cylindrical can with a volume of 206in\(^3\) that minimizes the surface area.

The “#10 can”is a standard sized can used by the restaurant industry that holds about 206in\(^3\) with a diameter of 6 2/16in and height of 7in. Does it seem these dimensions where chosen with minimization in mind?

11. The United States Postal Service charges more for boxes whose combined length and girth exceeds 108” (the “length” of a package is the length of its longest side; the girth is the perimeter of the cross section, i.e., 2w + 2h)

What is the maximum volume of a package with a square cross section (w = h) that does not exceed the 108” standard?

12. The strength \(S\) of a wooden beam is directly proportional to its cross sectional width w and the square of its height h; that is, \(S = kwh^2\) for some constant \(k\).

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Given a circular log with diameter of 12 inches, what sized beam can be cut from the log with maximum strength?

13. A power line is to be run to an offshore facility in the manner described in Example 104. The offshore facility is 2 miles at sea and 5 miles along the shoreline from the power plant. It costs $50,000 per mile to lay a power line underground and $80,000 to run the line underwater.

How much of the power line should be run underground to minimize the overall costs?

14. A power line is to be run to an offshore facility in the manner described in Example 104. The offshore facility is 5 miles at sea and 2 miles along the shoreline from the power plant. It costs $50,000 per mile to lay a power line underground and $80,000 to run the line underwater.

How much of the power line should be run underground to minimize the overall costs?

15. A woman throws a stick into a lake for her dog to fetch; the stick is 20 feet down the shore line and 15 feet into the water from there. The dog may jump directly into the water and swim, or run along the shore line to get closer to the stick before swimming. The dog runs about 22ft/s and swims about 1.5ft/s.

How far along the shore should the dog run to minimize the time it takes to get to the stick? (Hint: the figure from Example 104 can be useful.)

16. A woman throws a stick into a lake for her dog to fetch; the stick is 15 feet down the shore line and 30 feet into the water from there. The dog may jump directly into the water and swim, or run along the shore line to get closer to the stick before swimming. The dog runs about 22ft/s and swims about 1.5ft/s.

How far along the shore should the dog run to minimize the time it takes to get to the stick? (Google "calculus dog" to learn more about a dog's ability to minimize times.)

17. What are the dimensions of the rectangle with largest area that can be drawn inside the unit circle?

4.4: Differentials

Terms and Concepts

1. T/F: Given a differentiable function \(y=f(x)\), we are generally free to choose a value for \(dx\), which then determines the value of \(dy\).

2. T/F: the symbols \("dx"\) and \("\Delta x"\) represent the same concept.

3. T/F: the symbols \("dy"\) and \("\Delta y"\) represent the same concept.

4. T/F: Differentials are important in the study of integration.

5. How are the differentials and tangent lines related?

Problems

In Exercises 6-17, use differentials to approximate the given value by hand.

6. \(2.05^2\)

7. \(5.93^2\)

8. \(5.1^3\)

9. \(6.8^3\)

10. \(\sqrt{16.5}\)

11. \(\sqrt{24}\)

12. \(\sqrt[3]{63}\)

13. \(\sqrt[3]{8.5}\)

14. \(\sin 3\)

15. \(\cos 1.5\)

16. \(e^{0.1}\)

In Exercises 17-29, compute the differential \(dy\).

17. \(y=x^2+3x-5\)

18. \(y=x^7-x^5\)

19. \(y=\frac{1}{4x^2}\)

20. \(y=(2x+\sin x)^2\)

21. \(y=x^2e^{3x}\)

22. \(y=\frac{4}{x^4}\)

23. \(y=\frac{2x}{\tan x+1}\)

24. \(y=\ln (5x)\)

25. \(y=e^x\sin x\)

26. \(y=\cos (\sin x)\)

27. \(y = \frac{x+1}{x+2}\)

28. \(y=3^x\ln x\)

29. \(y=x\ln x -x\)

30. A set of plastic spheres are to be made with a diameter of 1cm. If the manufacturing process is accurate to 1mm, what is the propagated error in volume of the spheres?

31. The distance, in feet, a stone drops in \(t\) seconds is given by \(d(t) = 16t^2\). The depth of a hole is to be approximated by dropping a rock and listening for it to hit the bottom. What is the propagated error if the time measurement is accurate to 2/10ths of a second and the measured time is:
(a) 2 seconds?
(b) 5 seconds?

32. What is the propagated error in the measurement of the cross sectional area of a circular log if the diameter is measured at 15′′, accurate to 1/4 ′′?

33. A wall is to be painted that is 8′ high and is measured to be 10′ , 7′′ long. Find the propagated error in the measurement of the wall’s surface area if the measurement is accurate to 1/2 ′′ .

Exercises 34-38 explore some issues related to surveying in which distances are approximated using other measured distance and measured angles. (Hint: Convert all angle to radians before computing.)

34. The length \(l\) of a long wall is to be approximated. The angle \(\theta\), as shown in the diagram (not to scale), is measured to be \(85.2^\circ\) , accurate to \(1^\circ\) . Assume that the triangle formed is a right triangle.
4434.PNG
(a) What is the measured length \(l\) of the wall?
(b) What is the propagated error?
(c) What is the percent error?

35. Answer the questions to Exercise 34, but with a measured angle of \(71.5^\circ\) , accurate to \(1^\circ\), measured with a point 100' from the wall.

36. The length \(l\) of a long wall is to be calculated by measuring the angle \(\theta\) shown in the diagram (not to scale). Assume the formed triangle is an isosceles triangle. The measured angle is \(143^\circ\), accurate to \(1^\circ\).
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(a) What is the measured length of the wall?
(b) what is the propagated error?
(c) What is the percent error?

37. The length of the walls in Exercise 34-36 are essentially the same. Which setup gives the most accurate result?

38. consider the setup in Exercises 36. This time, assume the angle measurement of \(143^\circ\) is exact but the measured \(50^\circ\) from the wall is accurate to \(6"\). What is the approximate percent error?