Math 1710 Final Review 1 1

Transcription

1 Math 7 Final Review. Use the ɛ δ definition of it to prove 3 (2 2 +)=4. 2. Use the ɛ δ definition of it to prove =3. 3. Use the ɛ-δ definition of it to prove (32 +5 ) = Prove that if f() = L and a g() =K, then (f() +g()) = a a L + K. 5. Use the definition to derive the formula for the derivative of f() = Use the definition of derivative to derive f () wheref() = Use the definition of derivative to derive g 2 + () whereg() =. 8. Derive the formula for the derivative of sin. 9. Use the derivative formulas for sin and cos to derive the formula for the derivative of tan.. Let f() = for 4. Find the maimum and minimum values for f() on this interval. (Be sure to say how you know your answer is correct.). Let f() = Findthe minimum and maimum values of f() for 2 2. Be sure to justify all your conclusions. 2. An object moves along a straight line with position given by s(t) =t 2 2t +3. Find the velocity and acceleration of the object at time t =2. 3. An object moves along a straight line with position given by s(t) =5cos( 2πt )+ 3 2 sin( 2πt ). Find the velocity and acceleration of the object at time t 3 =3. 4. Find an equation of the tangent line to y =sec 2 at the point where = π Let f() = 3 +. Find the tangent 2 line to f() atthepointwhere =4. 6. Find equations for all the lines that pass through the point (, 8) and are tangent to the graph of y = (Note that the point is not on the graph.) 7. Find all the horizontal and vertical asymptotes of the graph of f() = Compute 2sin Compute sec 2 3 sin. 2. Compute θ cos 2 θ θ Compute the derivative implicitly given that 42 y+2y y y 3 =. 22. Find an equation of a line tangent to the curve defined by the equation 3 y +y + 2 y =2y 3 at the point (, ). 23. A cone made of very fleible material keeps its volume at 5π cubic feet while

2 Math 7 Final Review 2 its radius and height vary. When the radius is 3 feet the radius is increasing at the rate of. feet per second. How fast is the height increasing? 24. Suppose that an object moves along a straight horizontal line with its displacement given by s(t) =t 3 3t 2 + t. Find the velocity and acceleration of the object at all times t. Find all values of t where the object is moving to the left. 25. On a certain planet the acceleration due to gravity is given by -5.7 m/sec 2. If a rock is thrown straight up at a velocity of 8 m/sec, how long will it take to hit the ground? 26. Let f() = sin 2 : < : < : = tan 5 Is f continuous at =? Ifso,show why. If not, determine if is a removable discontinuity. 27. Let 2 +3 : < f() = 3 7 : 2 3 : > Find a) f() + b) f() c) f() + d) f() e) f() Use all the information you can concerning f() = Use all the information you can concerning g() = Use all the information you can concerning f() = Eplain how the mean value theorem implies that if you average speed over some time interval is 7 miles per hour, then at some time in that interval you were traveling with a speed of 7 miles per hour. 32. Given that =, find 2 when using Newton s method to solve =. 33. Use the linearization of the appropriate function to estimate sin(π/3 +.). 34. A beacon of light 2 m offshore rotates twice per minute, forming a spot of light

3 Math 7 Final Review 3 that moves along a straight wall along the shore. How fast is the spot of light movingwhenthespotisatthepointp, directly opposite the beacon? When it is 2 meters from point P? 35. A man walks directly under a street light. The man is 6 feet tall, the light is 5 feet above the ground, and the man walks at 3 feet per second. How fast is the shadow of the man growing when he is 5 feet from directly under the light? 36. A conical cup is to hold cm 3.What dimensions will give it the least surface area? 37. A woman is in a boat meters from shore. If she can row the boat at the rate of meter per second and she can walk on shore at the rate of 2 meters per second, find her fastest path to a point 2 meters down the (straight) shore. 38. A rectangle is formed by making the lower left corner at the origin, one side along the -ais, another side along the y-ais and the upper right corner on the graph of y = 9 2. Find the point 4 on the graph that maimizes the area of the rectangle. 39. Compute sin 2d in two different ways. 4. Compute cos d Compute d 42. Compute 43. Compute 44. Compute 45. Compute 46. π/2 4 4 sin 2 d 47. Compute sin 2 2 cos 2 2 d 2 +9d 2 +d d +) d 48. Compute csc 2 θ cot 2 θdθ 49. Use a its of Riemann sums to compute ( 2 4 +)d Compute ( 2 + ) d using a it of Riemann sums. No credit will be given unless you actually set up Riemann sums and compute the it, so be sure to write you answer so that the grader can follow your calculations. 5. Compute n Σ n (n+3k) 3 k=. n Compute n Σ n k= sin( kπ n ) n. 53. Prove the Mean Value Theorem using Rolles Theorem. 54. Prove Rolles Theorem.

4 Math 7 Final Review Prove that if f () =g () for every real number, then there is a constant c such that f() =g()+c for every real number. 56. Find the area bounded by the functions f() =sin and g() =( π). 57. Find the area bounded between the graphs of y 2 + = and y =. 58. Find the area bounded by the graphs of y =4 +6andy = Find the volume of a room whose floor is an ellipse with equation 2 + y2 =and 4 9 whose cross section is a semicircle when sliced perpendicular to the -ais. 6. Find a value of n that will give an error of at most.6 when estimating 2 cos3 dusing the Trapezoid rule. Be sure to clearly show your calculuations to justify your answer. 6. Find the value of n needed to estimate π sin 3 d using the trapezoid rule if you want your estimate to be within. of the correct answer. Use the n you found to estimate the integral. 62. Find the value of n needed to estimate π sin 3 d using the Simpson s rule if you want your estimate to be within. of the correct answer. Use the n you found to estimate the integral. 63. Derive the formula for the volume of a cone with base radius r and height h. 64. The region R is bounded between the functions f() = and g() =.Find the volume obtained when R is rotated about the ais. 65. Find the volume generated if the region in the plane between =and = π, below y = sin, and above the -ais is rotated about the -ais. 66. A spherical tank with radius 2 is filled with water. The water is drained at the rate of.5 cubic meters per second. How fast is the water level in the tank changing at the instant that the water has a depth of 3 meters. (Depth of three meters means that the water level is one meter above the center.) I suggest that you first find the volume obtained from the region in the plane inside the circle 2 +y 2 = r 2 and below the line y = h for afiedh with h r rotated about the y-ais. 67. The region between the graphs of y = 2 and y = is revolved about the y- ais to form a lens. Compute the volume of the lens. 68. Find the volume of the solid formed by revolving the region enclosed by the graph of about the -ais. 2/3 + y 2/3 = a 2/3

5 Math 7 Final Review Find the volume of the solid obtained by rotating the area in the plane bounded by y =sin( 2 ), the ais, =and = π/2 aboutthey ais. 7. Find the center of mass of a wire that etend along the number line from to 4 and whose cross section area when sliced perpendicular to the number line is a circle with radius Find the center of mass of the triangle with vertices (, ), (3, ), and (, 2). 72. Find the center of mass of the region in the first quadrant bounded by y = 2 and y = Find the area of the surface generated by revolving the graph of = y for 2 y 6aboutthey ais. 79. Find the arc length of the graph of y = 5 2 6/ /5 for The center of mass of a region in the upper half-plane is (, 5). Find the volume of the solid created by revolving the area about the ais. 8. A bug rides on the perimeter of a wheel having radius. What is the length of the path the bug travels as the wheel rotates once. 73. Find the center of mass of the region in the plane bounded by y = 3and =(y +3)(y ). 74. Find the center of mass of the planar region bounded by y = 2 and y = 3. Nowdothesamethingwithy = n and y = n+. Is the center of mass of a connected planar object always inside the object? 75. State the first form of the Fundamental Theorem of Calculus. Prove it. 76. State the second form of the Fundamental Theorem of Calculus. Prove it. (You may use the first form in your proof.) 77. Find the area of the surface generated by revolving the graph of y = 3 /3for aboutthe ais.