Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 39
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41Worksheet # 29: Review for Final
1. Compute the derivative of the given function:
x
2
(a) f (θ) = cos(2θ
+ θ + 2)
2
t +t+1
(c) h(x) =
(t
te
) dt
3599
2
3
(b) g(u) = ln sin
(u)
(d) r(y) = arccos(y
+ 1)
2. Compute the following definite integrals:
π
1
2
6
(a)
sec
(t/4) dt
(c)
x(1 + x)
dx
0
0
1
π/4
x
(b)
xe
dx
(d)
sin(2x) cos(2x) dx
0
0
1
2
8
3. What is the area of the bounded region bounded by f (x) =
, x = e
, x = e
and x-axis? Sketching
x
the region might be helpful.
7
2
4. If F (x) =
cos(t
) dt, find F (x). Justify your work.
3x +1
0.02t
5. Suppose a bacteria colony grows at a rate of r(t) = 100 e
with t given in hours. What is the growth
in population from time t = 1 to t = 3?
5
2
6. Use the left endpoint approximation with 4 equal subintervals to estimate the value of
x
dx. Will
1
this estimate be larger or smaller than the actual value of definite integral? Explain your answer.
3
x
4x
7. Find an antiderivative for the function f (x) =
.
4
1 + x
8. Give the interval(s) for which the function F is increasing. The function F is defined by
x
5t
3
F (x) =
dt
2
t
+ 10
0
x
x
e
e
9. Find a function f (x) such that f (e) = 0 and f (x) =
. (Hint: Consider the Fundamental
ln x
Theorem of Calculus)
x
x
10. Which of the following is an antiderivative for the function f (x) = e
sin(e
). Circle all the correct
answers.
e
x
(a) F (x) =
cos(e
)
(d) F (x) =
cos(t) dt
0
x
t
t
x
(e) F (x) =
e
sin(e
) dt
(b) F (x) = sin(e
)
0
x
e
t
t
(c) F (x) =
e
cos(e
) dt
(f) F (x) =
sin(t) dt
0
0
1
ln(arcsin(x))
11. Which of the following integrals are the same as
dx. Circle all the correct
2
arcsin(x) 1
x
1/2
answers.
(Hint: Use substitution method. You may need to do substitution more than once.)
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