Math 10b Homework for Section 5.4, 5.3 and 5.5
Show all your work on all assignments.
I. Section 5.4.
1. Do the following problems on page 372–73: # 3, 4a–d, 8, 9, 12, 13, 15, 17, 21 and 22.
2. Find a function F(x) that satisfies both of the following conditions:1,
(a) F(x) is an antiderivative of f(x) = √1 + x3;
(b) F(4) = 0.
3. Find the equation of the line tangent to the graph of g(x) = Zcos x
−2
√4−t2dt at x=π
2.
4. The Fresnel function, named after the French physicist Augustin Fresnel, is defined
as follows:
S(x) = Zx
0
sin(πt2/2) dt.
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.
The graph of f(t) = sin(πt2/2) 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?
y!x2
2
" #
$ %
& '
sin x 0>if
" #
$ %
& '
=
0
0
OVER
1Hint: In #2 for §5.4, use the Fundamental Theorem of Calculus.
