Evaluating a Limit Consider the limit lim x → 0 + ( − x ln x ) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. Use a graphing utility to verify the result. M FOR FURTHER INFORMATION for a geometric approach to this exercise, see the article "A Geometric Proof of lim d → 0 + ( − d ln d ) = 0 " by John H. Mathews in The College Mathematics Journal, To view this article, go to MathArticles.com.
Evaluating a Limit Consider the limit lim x → 0 + ( − x ln x ) (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. Use a graphing utility to verify the result. M FOR FURTHER INFORMATION for a geometric approach to this exercise, see the article "A Geometric Proof of lim d → 0 + ( − d ln d ) = 0 " by John H. Mathews in The College Mathematics Journal, To view this article, go to MathArticles.com.
Evaluating a Limit Consider the limit
lim
x
→
0
+
(
−
x
ln
x
)
(a) Describe the type of indeterminate form that is obtained by direct substitution.
(b) Evaluate the limit. Use a graphing utility to verify the result.
M FOR FURTHER INFORMATION for a geometric approach to this exercise, see the article "A Geometric Proof of
lim
d
→
0
+
(
−
d
ln
d
)
=
0
"
by John H. Mathews in The CollegeMathematics Journal, To view this article, go to MathArticles.com.
The graph of the function f in the figure below consists of line segments and a quarter of a circle. Let g be the function given by
x
g(x) = __ f (t)dt. Determine all values of a, if any, where g has a point of inflection on the open interval (-9, 9).
8
y
7
76
LO
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
1
2 3
♡.
-1
-2
3
-4
56
-5
-6
-7
-8
Graph of f
4 5
16
7
8
9 10
The areas of the regions bounded by the graph of the function f and the x-axis are labeled in the figure below. Let the function g be
C
defined by the equation g(x) = [* f(t)dt. What is the maximum value of the function g on the closed interval [-7, 8]?
17
y
Graph of f
00
8
76
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3-2-1
-2
702
4
1
21
3 4
568
-4
-5
--6
-7
-8
x
5
6
7
8
9 10
17
A tank holds a 135 gal solution of water and salt. Initially, the solution contains 21 lb of salt. A salt solution with a concentration of 3 lb of salt per gal begins flowing into the tank at the rate of 3 gal per
minute. The solution in the tank also begins flowing out at a rate of 3 gal per minute. Let y be the amount of salt present in the tank at time t.
(a) Find an expression for the amount of salt in the tank at any time.
(b) How much salt is present after 51 minutes?
(c) As time increases, what happens to the salt concentration?
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