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### Sample_test_1A_Fall_2005

Course: MATH 1501, Fall 2008
School: Georgia Tech
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Word Count: 232

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Test Sample 1A, Math 1501F, Fall 2005 Problem 1. Let f (x) = 5x - 4 and g(x) = |x - 1| (a) Compute (f g)(2); (b) Find the largest possible domain for f g. Problem 2. Answer the following questions on limits. (a) Give a precise definition of the meaning of the statement limxc f (x) = L; (b) Give an - proof that limx-2 (x2 - 1) = 3. Problem 3. Compute the following limits, if they exist. Show your work. x2 - x +...

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Test Sample 1A, Math 1501F, Fall 2005 Problem 1. Let f (x) = 5x - 4 and g(x) = |x - 1| (a) Compute (f g)(2); (b) Find the largest possible domain for f g. Problem 2. Answer the following questions on limits. (a) Give a precise definition of the meaning of the statement limxc f (x) = L; (b) Give an - proof that limx-2 (x2 - 1) = 3. Problem 3. Compute the following limits, if they exist. Show your work. x2 - x + 2 ; x-2 x-1 x2 + x - 12 (b) lim ; x3- x-3 x2 + x - 2 (c) lim 2 ; x1 x - 2x + 1 tan(3x) (d) lim 2 . x0 2x + 5x (a) lim Problem 4. Let 1, 2 x , f (x) = 1, x, x0 0<x<1 1x<2 2 x. Sketch the graph of f (x) and the classify discontinuities (if any) as being a removable discontinuity, a jump discontinuity, or an infinite discontinuity. Problem 5. Show that the equation x3 - 4x + 2 = 0 has three distinct roots in [-3, 3] and locate the roots between consecutive integers. Problem 6. Find the points where the tangent to the graph of f (x) = -x2 - 6 is parallel to the line y = 4x - 1. Problem 7. A stone is thrown upward from ground level. The initial speed is 32 feet per second. (a) In how many seconds will the stone hit the ground? (b) How high will it go? (c) With what minimum speed should the stone be thrown so as to reach a height of 36 feet?
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Georgia Tech - MATH - 1501
Answers to Sample Test 2A, Math 1501F, Fall 2005Problem 1. 20x19 f (x) = (1 - x)21 and 3 cos( x3 + 7) x2 g (x) = . 2 x3 + 7Problem 2. See exercise 11, Section 4.10. Problem 3. (a) an+1 - an = q n (q - 1) &lt; 0, i.e. an is decreasing. It is clear t
Georgia Tech - MATH - 1501
Sample Test 3A, Math 1501F, Fall 2005Problem 1. Let f (x) = sin(x) cos(x) - 3 sin(x) + 2x for 0 &lt; x &lt; 2. Find the critical numbers of f and the local extreme values. Problem 2. A pentagon with a perimeter of 30 inches is to be constructed by adjoini
Georgia Tech - MATH - 1501
Sample Test 3B, Math 1501F, Fall 2005|x + 1|, -3 x &lt; 0 - 4x + 2, 0 x &lt; 3 Problem 1. Let f (x) = 2x - 7, 3 x &lt; 4. Find the critical numbers of f and classify the extreme values. x2 Problem 2. Let f (x) = 2 sin3 (x) + 3 sin(x), x [0, ]. Sketch th
Georgia Tech - MATH - 1501
Answers to Sample Test 4B, Math 1501F, Fall 2005Problem 1. See exercise 3, Section 6.6. Problem 2. (a) dom(f ) = (, 2) (2, 1) (1, +), f (x) = (b) See exercise 23, Section 7.7. (c) See exercise 31, Section 7.7. (d) See exercise 27, Section 7.5. Pro
Georgia Tech - MATH - 1501
Quiz #3, Math 1501F, Fall 2006Name:GTID#:Problem. (20 points) Study the boundedness and monotonicity of the sequences an = 4 -1 nand bn =1 n + n.
Georgia Tech - MATH - 1501
Quiz #6, Math 1501F, Fall 2006Name:GTID#:Problem. (20 points) Find the area between the graph of f (x) = x3 and the x-axis for x [0, 1].
Georgia Tech - MATH - 1501
Answers to sample final exam A, Math 1501Problem 1. (a) See problem 13, Section 2.3. (b) See problem 21, Section 2.3. (c) See problem 31, Section 2.5. Problem 2. (a) We say that limxc f (x) = L if for each 0 &lt; |x - c| &lt; , (see Definition 2.2.1 on pa
Georgia Tech - MATH - 1501
Quiz #5, Math 1501N, Fall 2007Name:GTID#:Problem. (20 points) Find the critical points and classify x + 9, x2 + x, ues (local and absolute) of the function f (x) = 5x - 4,the extreme val-8 x -3 -3 &lt; x &lt; 2 2x5
Georgia Tech - MATH - 1501
Quiz #6, Math 1501N, Fall 2007Name:GTID#:Problem. (7+7+6 points) Let f (x) = sin(x), x [0, ], P = {0, , , }. 6 2 Calculate: (a) Lf (P )(b) Uf (P )(c) The norm of P .
Georgia Tech - MATH - 1501
Quiz #7, Math 1501N, Fall 2007Name:GTID#:Problem. (10+10 points) Evaluate the integrals: 1 (a) 0 x 1 x2 dx(b)1 1x2 tan(x) + x4 sin(x) dx
Georgia Tech - MATH - 1501
Quiz #2, Math 1501N, Fall 2008Name: Problem. Let f (x) = x = -5, 0, 5 x = -5, 0, 5. (a) Evaluate the limits that exist (3+3+3 points)x0 x2 -6x+5 , x2 -25 2 5,GTID#:lim f (x)x5lim f (x)x-5lim f (x)(b) At which points is f (x) disconti
Georgia Tech - MATH - 1501
Quiz #3, Math 1501N, Fall 2008Name:GTID#:Problem 1. (9 points) Find the slope of the tangent line of the curve sin(xy) = x 2 at the point ( 1 , ). (Hint: Use implicit differentiation) 2 2Problem 2. (6 points) Find the least upper bound (if
Georgia Tech - MATH - 1501
Quiz #4, Math 1501N, Fall 2008Name:GTID#:Problem. (5+5+5 points) Evaluate the limits that exist. (a) lim 2n3 + n - 1 n n3 + 2n2 - 1(b) limn1 1+ n3n+2(c) lim tannn + 2 4n +
Georgia Tech - MATH - 1501
Quiz #6, Math 1501N, Fall 2008Name:GTID#:Problem 1. (6 points) Let F (x) =x t-1 0 t2 +1 dt.Calculate F (2).Problem 2. (9 points) Sketch the region bounded by y = and find its area.x and y = x2
Georgia Tech - MATH - 1501
Quiz #7, Math 1501N, Fall 2008Name:GTID#:Problem 1. (8 points) Calculate x4 1 + 2x5 dx.Problem 2. (7 points) A rod of length 1 is placed on the x-axis from x = 0 to x = 1. Find the mass of the rod and the center of mass given that the densi
Georgia Tech - MATH - 1501
Test #1, Math 1501N, Fall 2008Name: Problem 1. (8 points) Show that positive integers n.1 2GTID#: +2 22+3 23+ +n 2n=2n+2 2nfor allName:GTID#:Problem 2. (3+3+3 points) Evaluate the limits that exist. (a) lim x2 5x + 6 x2 x2
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