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Material Type: Exam; Class: CALCULUS II; Subject: Mathematics; University: Clark University; Term: Spring 2004;
Typology: Exams
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Test #2 Name: Section 1 2 3 Instructor:
This exam is CLOSED NOTES and CLOSED BOOK. There are NO CALCULATORS allowed. To get full credit you must show all work neatly in the space provided on the test paper.
a.
∫ sin(x) cos(x) dx
b.
∫ (^) x √ 1 + 4x^2
dx.
c.
∫ sec x[sec x + tan x] dx
a.
d dx
ln
2 x + 1
b.
d dx
x^2 ln(3x − 1)
c.
d dx
[∫ (^) x
1
t
dt
]
a. Find the intervals of x over which y is increasing, and those for which y is decreasing.
b. Find the intervals of x over which y is concave up, and those for which y is concave down.
c. Make a neat sketch of the graph of y = ln x and label two specific points through which the graph passes.
6
y
x
x − 4, is one to one.
b. Compute the inverse function of f (x).
c. What is the domain of the inverse function.
d. On the axes to the left below, draw the graph of any concave down function which has no inverse function.
e. On the axes to the right below draw the graph of any concave down function which does have an inverse function.
6
y
x
6
y
x
8.(16 pts.) Consider the region sketched below, which is bounded by y = 4 − x^2 and y = 2 + x. For each of the axes below, set up the integral which gives the volume of rotation about that axis with respect to x. In each case determine the shape of the volume element (disk, washer or shell) and indicate its dimensions.
[Both integrals with respect to x. Do NOT evaluate these integrals.]
a. The x-axis:
-2 -1 1
y 6
3
°
°
°
°
°
°
°
°°
x
b. The axis x = −2.
-2 -1 1
y 6
3
°
°
°
°
°
°
°
°°
x
