REVIEW FOR MIDTERM I IN MAT 132: FALL 2017 Midterm I for mat 132 is scheduled for October 2 8:45pm-10:15pm. The room assignments are as follows: ESS-001: all students in lecture 03 and in lecture 04 Engineering 143: all students in recitations R04, R05, R06 Engineering 145: all students in recitations R20, R23, R24 Harriman Hall 137: all students in recitations R01, R02, R03 Light Engineering 102: all students in recitations R21, R22 Humanities 1003: all students in recitations R25, R32 Midterm I will cover the following sections from our text book: 5.1-5.7, 5.10. There will be approximately 5 problems on the exam (some of them multiparts). Problems on the exam will be similar to some of the practice problems listed below. So the first way you should prepare for midterm I is to solve all of the practice problems listed below. YOU SHOULD BE ABLE TO GIVE AN EXPLANATION FOR EACH OF YOUR ANSWERS - EITHER BY SHOWING YOUR WORK OR BY GIVING A WRITTEN EXPLANATION. (1) True or False? (a) Let v(t) denote the velocity at time t of a particle traveling along the x-axis during the time interval [t1 , t2 ]. If v(t)>0 is always true then the total distance traveled by the particle during the given time interval is equal to the displacement of the particle during this time interval. (b) If fR(x) is a continuous function defined on the real line then y d dy ( 0 f (x)dx) = f (y) f (0). (c) Define a function f (x) on the real line by f (x) =| x |. Then f (x) has no antiderivative function which is defined on the entire real line. (2) Use any method to compute n X 2i 4 2i 2 limn ( (1 + )3 (1 + )2 ) n n n n i=1 . 1 2 REVIEW FOR MIDTERM I IN MAT 132: FALL 2017 (3) A particle is moving along the x-axis; its acceleration at time t seconds is equal to 2t-6 f eet/sec2 . Its velocity at time t=0 is equal -7 feet/sec. (a) Find the total distance this particle travels in the time interval 0 t 10. (b) Find the total displacement of the particle in the time interval 0 t 10. Rx 2 (4) Define a function f (x) by f (x) = 0 (2 eu )du. (a) At which values of x is f(x) continuous? At which values of x is f(x) differentiable? (b) At which values of x is f(x) a maximum value for this function on the interval [0,10] ? (c) At which values of x is f(x) a maximum value for this function on the interval [-10,0]? (5) Find the area under under the graph of f (x) = xcos(2x) and over the interval [0, 4 ]. (6) Evaluate the following. (Hint: the substitution method, integration by parts, could help.) and the fundamental theorem of calculus to evaluate each of the following intervals. R + ln(x))dx. (a) ( ln(x) Rx2+ex d (b) dx ( sin(3x cos(U 2 )du) R x (c) 1+x 2 dx R (d) R cos(x)sin2 (x) + ex cosxdx (e) cosn (x)dx, for n=3,4. (7) Compute each of the following indefinite integrals. R (a) 2x2x2 dx 4x6 R x3 x (b) (x2 +4)(x1) dx (8) Which of the following improper integrals converge and diverge? If an integral converges then find its value. R 1 dx (a) 0 x2 +4x+3 R x+2 (b) 1 (x+1)(x2 +2) R1 (c) 0 xln(x)dx REVIEW FOR MIDTERM I IN MAT 132: FALL 2017 3 (9) Consider the region in the plane which is below the graph of f (x) = x2 ex , above the x-axis, and two the left of the line y=2. Draw a picture of this region. This is an infinite region in that it is not contained in any disc centered at the origin having finite radius. Question: is the area of this region finite? If it is finite then calculate this area. (10) Define a function f(x) by n 2 X xcos(( xi n) ) ) f (x) = limn ( n i=1 Express the derivative of f(x) as a fairly simple function (e.g. no limits or sums)