Dr. Sophie Marques

MAM1020S Tutorial 11 August 2017

1. A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 metersof framing materials what must the dimensions of the window be to let in the most light?

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2. Determine the area of the largest rectangle that can be inscribed in a circle of radius 4.

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3. Determine the point(s) on y = x2 + 1 that are closest to (0,2).

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4. A 2 feet piece of wire is cut into two pieces and one piece is bent into a square and the other is bent intoan equilateral triangle. Where should the wire cut so that the total area enclosed by both is minimumand maximum?

Solution:

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5. A piece of pipe is being carried down a hallway that is 10 feet wide. At the end of the hallway the thereis a rightangled turn and the hallway narrows down to 8 feet wide. What is the longest pipe that can becarried (always keeping it horizontal) around the turn in the hallway?

Solution:

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6. Two poles, one 6 meters tall and one 15 meters tall, are 20 meters apart. A length of wire is attached tothe top of each pole and it is also staked to the ground somewhere between the two poles. Where shouldthe wire be staked so that the minimum amount of wire is used?

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7. Two poles, one 6 meters tall and one 15 meters tall, are 20 meters apart. A length of wire is attached tothe top of each pole and it is also staked to the ground somewhere between the two poles. Where shouldthe wire be staked so that the angle formed by the two pieces of wire at the stake is a maximum?

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8. A trough for holding water is be formed by taking a piece of sheet metal 60 cm wide and folding the 20cm on either end up as shown below. Determine the angle that will maximize the amount of water thatthe trough can hold.

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9. We need to enclose a rectangular field with a fence. We have 500 feet of fencing material and a buildingis on one side of the field and so won’t need any fencing. Determine the dimensions of the field that willenclose the largest area.

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10. We want to construct a box whose base length is 3 times the base width. The material used to build thetop and bottom cost 10/ft2 and the material used to build the sides cost 6/ft2. If the box must have avolume of 50ft3 determine the dimensions that will minimize the cost to build the box.

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11. We want to construct a box with a square base and we only have 10m2 of material to use in constructionof the box. Assuming that all the material is used in the construction process determine the maximumvolume that the box can have.

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12. A manufacturer needs to make a cylindrical can that will hold 1.5 liters of liquid. Determine the dimensionsof the can that will minimize the amount of material used in its construction.

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13. We have a piece of cardboard that is 14 inches by 10 inches and we’re going to cut out the corners asshown below and fold up the sides to form a box, also shown below. Determine the height of the box thatwill give a maximum volume.

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14. A printer need to make a poster that will have a total area of 200in2 and will have 1 inch margins on thesides, a 2 inch margin on the top and a 1.5 inch margin on the bottom. What dimensions will give thelargest printed area?

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15. Use Newton’s Method 2− x2 = sin(x) to find all the roots of accurate to six decimal places.

Solution:

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16. Use Newton’s Method to find all the roots of x3 − x2 − 15x+ 1 = 0 accurate to six decimal places.

Solution:

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17. Use Newton’s Method to find the root 2x2 + 5 = ex of accurate to six decimal places in the interval [3.4].

Solution:

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18. Use Newton’s Method to find the root x4 − 5x3 + 9x + 3 = 0 of accurate to six decimal places in theinterval [4, 6].

Solution:

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19. Use Newton’s Method to determine x2 for f(x) = xcos(x)− x2 if x0 = 1.

Solution:

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20. Use Newton’s Method to determine x2 for f(x) = x3 − 7x2 + 8x− 3 if x0 = 5.

Solution:

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21. Use Newton’s Method to determine an approximation to the solution cos(x) = x to that lies in the interval[0, 2]. Find the approximation to six decimal places.

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22. Use x0 = 1 to find the approximation to the solution to 3√x = 0.

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23. A ball is thrown straight up 6 feet from the ground (it is released 6 feet above the ground). When it isreleased it is traveling at the rate of 100 feet per second.

(a) Find the velocity function.

(b) Find the position function.

(c) How high does the ball go?

(d) How long does it take for the ball to reach the ground?

Solution:

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24. An object starts from rest and has an acceleration of a(t) = t2.

(a) What is its velocity after 3 seconds?

(b) What is its position after 3 seconds?

Solution:

25. Find all the antiderivatives of each function bellow

(a) f(t) = 3/t− 2/t2

(b) h(Ψ) = 3cos(Ψ) + 3√

Ψ

(c) f(x) = x2+x+1x

(d) f(x) = 3cos(x)− 7sin(x)

(e) g(x) = 1cos2(x)

(f) f(x) = cos(x)esin(x)

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(g) f(θ) = sin(θ)(cos(θ) + 5)7

(h) h(x) = 1√4−x

(i) f(x) = xe−x2

(j) f(x) = xcos(x2)√sin(x2)

(k) f(x) = ex−e−x

ex+e−x

(l) f(z) = [ln(z)]2]z

(m) f(x) = 18x2

(n) f(x) = x−3/5

(o) f(x) = 2x4 − 24x2 + 12x−1

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(p) f(x) = 9x+ 15x−2

(q) f(x) = 2cos(x)− 9sin(x)

(r) f(x) = 4x7 − 3cos(x)

(s) f(x) = 12ex − 5x−2

(t) f(x) = ex − 4sin(x)

26. Match the function f with they antiderivatives F

(a) f(x) = sin(x)

(b) f(x) = xsin(x2)

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(c) f(x) = sin(1− x)

(d) f(x) = xsin(x)

(i) F (x) = cos(1− x)

(ii) f(x) = −cos(x)

(iii) f(x) = −1/2cos(x2)

(iv) f(x) = sin(x)− xcos(x)

27. Solve the initial value problems Find y when

(a) y′(x) = x3 and y(0) = 4

(b) y′(t) = 3− 2t and y(0) = −5

(c) y′(t) = 2t+ 9t2 and y(1) = 2

(d) y′(t) =√t and y(1) = 1

(e) y′(x) = (3x+ 2)3 and y(0) = 1

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(f) y′(t) = (4t+ 3)−2 and y(1) = 0

(g) y′(x) = sin(x) and y(π/2) = 1

(h) y′(z) = sin(2z) and y(π/4) = 4

(i) y′(x) = cos(5x) and y(π) = 3

(j) y′(x) = ex and y(2) = 0

(k) y′(t) = e−t and y(0) = 0

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(l) y′(t) = 9e12−3t and y(4) = 7

(m) y′(t) = t+ 2et−9, y(9) = 4.

28. First find f ′ then find f

(a) f ′′(x) = 12x, f ′(0) = 1, and f(0) = 2.

(b) f ′′(x) = x3 − 2x, f ′(1) = 0 and f(1) = 2.

(c) f ′′(x) = x3 − 2x+ 1, f ′(0) = 1 and f(0) = 0.

(d) f ′′(x) = x3 − 2x+ 1, f ′(1) = 0 and f(0) = 0.

(e) f ′′(x) = x3 − 2x+ 1, f ′(1) = 0 and f(1) = 4.

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(f) f ′′(t) = t−3/2, f ′(4) = 1 and f(4) = 4.

(g) f ′′(θ) = cos(θ), f ′(π/2) = 1 and f(π/2) = 6.

(h) f ′′(t) = t− cos(t), f ′(0) = 2 and f(0) = −2.

29. Suppose that F ′(x) = f(x) and G′(x) = g(x). Which of the following statements are true? Explain.

(a) If f = g, then F = G.

(b) If F and G differ by a constant, then f = g.

(c) If f and g differ by a constant, then F = G.

30. Is y = x a solution of the following initial Value problem

dy

dx= 1, y(0) = 1

31. Show that F (x) = tan2(x) and G(x) = sec2(x) have same derivative. What can you conclude about Fand G? Verify this conclusion directly.

32. A particle located at the origin at t = 1 s moves along the x-axis with velocity v(t) = (6t2− t) m/s. Findthe position function.

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33. A mass ocillates at the end of a spring. Let s(t) be the displacement of the mass from the equilibriumposition at time t. Assuming that the mass is located at the origin t = 0 and has velocity v(t) =sin(πt/2) m/s. Find s(t).

34. Beginning at t = 0 with initial velocity 4 m/s, a particle moves in a straight line with accelerationa(y) = 3t1/2 m/s2. Find the distance traveled after 25 seconds.

35. A car traveling 25 m/s begins to decelerate at a constant rate of 4 m/s2. After how many seconds doesthe car come to a stop and how far will the car have traveled before stopping?

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36. At time t= 1 s, a particle is traveling at 72 m/s and begins to decelerate at the rate a(t) = −t−1/2 untilit stops. How far does the particle travel before stopping?

37. A 900-kg rocket is released from a space station. As it burns fuel, the rocket?s mass decreases and itsvelocity increases. Let v(m) be the velocity (in meters per second) as a function of mass m. Find thevelocity when m = 729 if dv/dm = −50m−1/2. Assume that v(900) = 0.

38. A water flows through a tube of radius R = 10cm, the velocity v of an individual water particle dependsonly on its distance r from the center of the tube. The particles at the walls of the tube have zero velocityand dv/dr = −0.06r. Determine v(r).

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39. Find constants c1 and c2 such that F (x) = c1xsin(x) + c2cos(x) is an antiderivative of f(x) = xcosx.

40. Use the Riemmann sums to evaluate the area delimited by the graph of the function defined by y = 2x2+3,the x-axis and the line of equation x = 0 and x = 4.

Solution:

41. Use the Riemmann sums to evaluate the area delimited by the graph of the function defined by y = 8x−x2,the x-axis and the line of equation x = 2 and x = 5.

Solution:

42. Use the Riemmann sums to evaluate the area delimited by the graph of the function defined by y = 4x,the x-axis and the line of equation x = 0 and x = 5.

Solution:

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43. Use the Riemmann sums to evaluate the area delimited by the graph of the function defined by y = x2+10,the x-axis and the line of equation x = 0 and x = 2.

Solution:

44. Use the Riemmann sums to evaluate the area delimited by the graph of the function defined by y =5x2 − 2x, the x-axis and the line of equation x = 0 and x = 3.

Solution:

45. Use the Riemmann sums to evaluate the area delimited by the graph of the function defined by y =−3x2 + 5x− 1, the x-axis and the line of equation x = 0 and x = 4.

Solution:

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46. Use the Riemmann sums to evaluate the area delimited by the graph of the function defined by y = 2x2,the x-axis and the line of equation x = 4 and x = 6.

Solution:

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