4, Exam MATH 113 - Fall 2007 Name:
09/12/2007 Instructor: Bong-Sik Kim
Your writing should be clear and contain the detailed algebraic procedure. If you fail to do so, you will get no credits on the problem. 1. [5 points each] Evaluate the integral if it exists: (a)
t4 +2+ t t
dx
(b)
9t 8 2 dt
(c)
sin3 x cos xdx
(d)
cos x dx x
2. [5 points] Compute the limit:
x0
lim
x2 x
2x 25 6t2 dt
3. [5 points] Sketch the region enclosed by the given curves, 4x + y 2 = 12, , and nd the area of the region. x=y
4. [6 points] Let g (x) =
x 0 f (t)dt,
0 x 6 where f is the function pictured below:
(a) At what value of x does g achieve its absolute minimum value on [0, 6]? (b) At what value of x does g achieve its absolute maximum value on [0, 6]? (c) Find the values (if any) where g has local extrema, and classify the extrema on [0, 6].
5. points [5 each]Consider the function f (x) = 1 + 2x3 . In this problem you will calculate 5 3 dx by using the denition 0 1 + 2x
b n
f (x) dx = lim
a
n
f (x ) x i
i=1
The summation inside the brackets is Rn which is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval. (a) Calculate Rn for f (x) = 1 + 2x3 on the interval [0, 5]. ( Note:
n 3 i=1 i
=
n(n+1) 2
2
)
(b) Calculate limn Rn .
6. [Bonus 5 points] Breathing is cyclic and a full respiratory cy...