MATH 141 Midterm: MATH141 South Carolina 141 96 4 nospace
Exam 4, Math 141, 1996
PRINT Your Name: Section:
There are 10 problems on 5 pages. Each problems is worth 10 point. SHOW
your work. CIRCLE your answer. NO CALCULATORS! You might find the
following formulas to be useful:
n
X
k=1
k2=n(n+ 1)(2n+1)
6and
n
X
k=1
k3=n2(n+1)
2
4.
1. State the Mean Value Theorem.
2. Define the definite integral
b
Z
a
f(x)dx .
3. Find Zx2x2+1
xdx . (Check your answer.)
4. Find Z(cos4x3)(x2sin x3)dx . (Check your answer.)
5. Find Zx√x+1dx . (Check your answer.)
6. Solve the Initial Value Problem dy
dt =t3y2,y(2) = 1 . (Check your answer.)
7. Consider the region A, which is bounded by the x−axis, y=(x−1)2,x=1,
and x= 2 . Consider 50 rectangles, all with base 1/50 , which UNDER
estimate the area of A. How much area is inside the 50 rectangles? (You must
answer the question I asked. I expect an exact answer in closed form.)
8. Let f(x)=x
5/3
−x
2/3. Where is f(x) increasing, decreasing, concave up,
and concave down? What are the local extreme points and points of inflection
of y=f(x) . Find all vertical and horizontal asymptotes. Graph y=f(x).
9. Findthepointsonthecurve y=10−x
2which are closest to the point (0,0) .
10. A 30−foot ladder is leaning against a wall. If the bottom of the ladder is pulled
along the level pavement directly away from the wall at 3 feet per second, how
fast is the top of the ladder moving down the wall when the foot of the ladder
is 5 feet from the wall?
Document Summary
You might nd the following formulas to be useful: Xk=1 k2 = n(n + 1)(2n + 1) 4: state the mean value theorem, de ne the de nite integral b. 1 x(cid:19) dx . (check your answer. : find z x(cid:18)2x2 , find z (cos4 x3)(x2 sin x3)dx . (check your answer. , find z x x + 1dx . (check your answer. , solve the initial value problem dy dt. = t3y2 , y(2) = 1 . (check your answer. : consider the region a , which is bounded by the x axis, y = (x 1)2 , x = 1 , and x = 2 . Consider 50 rectangles, all with base 1/50 , which under estimate the area of a . How much area is inside the 50 rectangles? (you must answer the question i asked. I expect an exact answer in closed form. : let f (x) = x5/3.
