Math 10b
Homework Assignments for Inverse Trig (Section 3.6), Sigma Notation
and Sections 5.1, 5.2 and 5.4
SHOW ALL YOUR WORK ON ALL HOMEWORK ASSIGNMENTS.
I. Homework Assignment for Inverse Trig (Section 3.6).
1. Find the following:
(a) sin−1 (
√
3
)
2
(d) arcsin(−1)
(b) tan−1 (− √13 )
(c) arctan(1)
(e) sec(arcsin( 12 )
(f) sin arctan (− 23 )
Hint: In part (f) draw a triangle.
2. Do problem 4a on page 220. Remember that the range of tan−1 x is (− π2 , π2 ).
3. In each of the following, find f 0 (x). Don’t simplify your answers.
(b) f (x) = tan(sin−1 x)
(a) f (x) = x2 arctan(ex )
√
(c) f (x) = e
tan−1 x
4. Find the equation of the line tangent to f (x) = tan−1 (ln x) at x = 1e .
5. Find the following limits:
(a) lim
x→0
tan−1 x
x
(b) lim tan−1 x
x→−∞
tan−1 (3 + h) − tan−1 (3)
6. Find the following limit by using the definition of the derivative: lim
.
h→0
h
7. Find the general antiderivative F (x) of the function f (x) = 2x + sec2 x −
8. Suppose that f 0 (x) = √
1
.
4(1 + x2 )
6
and f ( 12 ) = 1. Find f (x).
1 − x2
II. Homework Assignment for Sigma Notation. You can find the sum formulae in
Theorem 3 on page A41.
1. Do the following problems on page A42 in Appendix F: #1, 5, 13, 18–20, 28, 36.
2. Use the properties of sums and the sum formulae to evaluate the following:
(a)
12
X
2
(3i + 4i)
(b)
i=1
100
X
4
(c)
i=1
3. Evaluate the following limit: lim
n→∞
i=1
" n
X 5i
i=1
6
X
#
1
−3 · .
n
n
(i3 − 1)
III. Section 5.1.
1. Read about the Distance Problem (middle of page 339 to the end of section). Notice
that a similar argument will hold if v(t) is replaced by any rate function (not just a
velocity function).
2. Do the following problems on page 341–42: # 2a–c, 5, 13, and 14. Hint: In problem
#2, no formula for the function is given, so use the graph to approximate the heights
of the different rectangles.
3. Speedometer readings for a motorcycle at 12-second intervals are shown in the following
table:
Time t (in sec)
Velocity v (in ft/sec)
0
30
12 24 36 48 60 72
28 25 22 24 27 25
(a) Estimate the total distance traveled by the motorcycle using during this time
period using the velocities at the beginning of six time intervals.
(b) Estimate the total distance traveled by the motorcycle using during this time
period using the velocities at the end of six time intervals.
(c) Is it possible to determine if your estimates in parts (a) and (b) are overestimates
or underestimates? If it’s possible, which is an underestimate and which is an
underestimate? If it’s not possible, explain why not. Be careful!
4. Estimate the area under the graph of f (x) = arcsin x over the interval [0, 1] using 2
rectangles. Take your sample points to be the righthand endpoints of the subintervals.
Sketch the graph of f (x) and the rectangles.
IV. Section 5.2.
1. Do the following problems on page 353–55. In problems #23 and 25, use the appropriate formulas from 5 – 7 on page 346. Note: No credit for shortcuts on problems
#23 and 25.
# 1, 5, 7, 23, 25, 31, 34, 35, 38, 41–43, 48.
2. Let f (x) and g(x) be continuous functions such that
Z −1
Z 3
f (x) dx = 1.5,
−3
f (x) dx = 3.5,
−3
Z 5
g(x)dx = 2.5,
−1
Z 5
g(x)dx = −2.
3
Some of the integrals listed below can be evaluated using properties of integrals and
some can not. Determine which ones can be evaluated and evaluate them. Note: If an
integral can’t be evaluated, you don’t need to explain why; just write “Can’t evaluate”.
(a)
Z 3
f (x) dx
(b)
−1
(d)
Z 3 −1
Z −1
g(x) dx
3
2f (x) − 4g(x) dx
(e)
(c)
Z 3
−1
f (x)
dx
g(x)
Z 5
g(x) dx
3
OVER FOR REST OF ASSIGNMENT →
3. Express the limit lim
n→∞
n
X
(5 +
2i 5
)
n
·
2
n
as a definite integral
Z b
f (x) dx.
a
i=1
Note: For this problem, you may choose any value of a. Once you find the integral,
you do not have to evaluate it.
V. Section 5.4.
1. Do the following problems on page 372–73: # 3, 4a–d, 8, 9, 13, 14, 17 and 22.
2. Find a function F (x) such that satisfies both the following conditions:
√
(a) F (x) is an antiderivative of f (x) = 1 + x5 ;
&
& !x2 '
'
y = $ sin $
% if x > 0%
"
" 2 #
#
(b) F (2) = 0.
Hint: Use the Fundamental Theorem of Calculus.
3. Find the equation of the line tangent to the graph of g(x) =
Z cos x √
−2
4 − t2 dt at x = π2 .
4. The Fresnel function, named after the French physicist Augustin Fresnel, is defined
as follows:
Z x
2
sin( πt2 ) dt.
S(x) =
0
0
It is one of many functions in physics and engineering that cannot be written in a
simpler form. The function first appeared in Fresnel’s theory of the diffraction of light
waves, but more recently it has been applied to the design of highways.
0
2
The graph of f (t) = sin( πt2 ) is shown below. Use it to answer the following questions:
(a) Give a rough estimate for S(1).
(b) At what value of x (for x > 0) does S(x) attain its first local maximum? Note:
your answer should be an exact number, not an estimate.
(c) Is S(x) concave up or concave down on the interval (0, 1)? Why?