Solutions to Math 1411- Test 4
1.
Find an
antiderivative 
with 
and 
when 
Answer: 
Solution:
Since , 
, and 
. Therefore,
.
2.
Use the Fundamental Theorem of Calculus to evaluate 
.
Answer: 50
Solution:
3.
Find the exact area of the region between 
and 
.
Answer:
Solution: The two
graphs intersect at 
and 
. Therefore, the limits of integration are 
and 
. We now need to
find the area of a representative rectangle within the region. The height
of the rectangle is 
and the width of the rectangle is dx.
4.
Find the area between the x-axis and the graph of the function 
for one cycle of
the graph.
Answer: 10
Solution: The period (one
cycle of the graph) is 
. Since the area above and below the graph are the same, we
can calculate the area as 
5.
Find the general solution of the differential equation 
.
Answer:
6. An object falls from the top of a 400-foot building. When does it hit the ground and how fast is it going at the time of impact?
Answer: It hits the ground in 5 seconds at a rate of 160 feet per second
Solution:
, 
, 
The object hits the
ground when 
The object hits
the ground in 5 seconds. At 5 seconds, the velocity of the object is
7. Find antiderivatives for the following functions. Check by differentiating.
a) 
Answer:
b) 
where a is a constant
Answer
8. Find antiderivatives for the following functions. Check by differentiating.
a) 
, a is constant.
Answer:
b)
Answer:
9. Is
the statement 
a true statement or a false statement?
Explain.
Answer:
is the area
of the top half of the unit circle. The area of the unit circle is
and the
area of half the unit circle is 
. The statement is true.
10. Is the statement “If a function is concave up, then the left-hand Riemann sums are always less than the right-hand Riemann sums with the same subdivisions, over the same interval.” a true statement or a false statement? Explain.
Answer: This
is a false statement. For example, take the function 
. We want to determine the
area between the graph of the function (concave up everywhere), the x-axis, and
the lines 
and . Let’s divide the interval into subintervals of length 1
units.
Left Hand
Sum: 
Right Hand
Sum: 
In this case, the left hand sum is the same as the right hand sum. This one exception, and there are many, means the original statement is false.