Unformatted text preview: Math 1110 Name: HW 1 Solutions Due 9/1 or 9/2 in class Please print out these pages. Answer the “presentation problems”
in the spaces provided. Include full explanations and write your answers
in complete, mathematically and grammatically correct sentences. Your
answers will be assessed for style and accuracy; you will be given written
feedback on these problems.
Please write your answers to the “text exercises” on separate paper
and staple it to these pages. You should include computational details.
These problems will be assessed for completeness. Always write neatly
and legibly. Tara Holm GRADES
Text exercises 20 / 20 Pres probs 20 / 20 Staple 1 /1 Text exercises. Please do the following problems from the book.
§1.1 # 4, 8, 10, 11, 24, 28, 32, 34, 40, 54, 66, 72
§1.2 #4, 6, 8, 11abc, 24, 26, 42, 47, 48, 59
§1.3 #6, 7, 18, 43, 50, 55 The following problems are recommended if you would like to practice with your graphing calculator, or play with one online (here, for example). You should not turn these in.
§1.4 #2. 3. 20, 26, 33
Presentation problem 1. (6 points) (Thomas, §1.3 #52) For 0 ≤ θ ≤ 2π, ﬁnd the values of θ that
sin2 θ = cos2 θ.
We know that sin2 θ + cos2 θ = 1, so if sin2 θ = cos2 θ, we must have sin2 θ = cos2 θ = 1 . That
means we are looking for values of θ where sin θ = ± √2 . For those θ, it will also be true that
cos θ = ± √2 . We can ﬁnd such a θ using the right triangle drawn in Figure 1. This gives us an answer θ = π . We also get the same triangle, reﬂected
around R2 , when θ = 34 , θ = 54 , or θ = 74 . Thus, our
ﬁnal answer is
θ= π 3π 5π
4, 4 , 4 or 7π
! Figure 1
Alternatively, we could look at the unit circle and see where it intersects the lines Figure 2 y = x and y = −x.
Because points on the unit circle have (x, y)-coordinates equal to (sin θ, cos θ), the intersection
points shown Figure 2 tell us the angles where sin2 θ = cos2 θ. These are the same four values of θ
given above. Math 1110 (Fall 2011) HW1 2 Presentation problem 2. (6 points) (Thomas, §1.1 #64) In the ﬁgure shown below, there is a rectangle inscribed in an isosceles right triangle whose hypotenuse is on the x-axis and is 2 units long.
(a) Express the y-coordinate of P in terms of x. (You might start
by writing an equation of the line that joins A and B.)
Let’s call the horizontal axis the t–axis. The line joining A
and B is described by the equation
$ y = −t + 1.
%&'&()&*&++, The point P is on that line, so it has coordinates (x, −x + 1).
In particular, the y–coordinate of the point P is −x + 1 . #
!" - ) " (b) Express the area of the rectangle in terms of x.
The shaded rectangle has width 2x, because it goes along
the t–axis from −x to x.
As we ﬁgured out above, it has height −x + 1.
Thus, the area of the rectangle is
A = (width) · (height) = 2x(−x + 1) = 2 − 2x2 .
Presentation problem 3. (8 points) The graph of f(x) is shown in the ﬁgure below. It has
[−4, 2] and range [0, 4]. Please sketch the graph of the function g(x) = − 1 f(x − 1) + 3 on the
same set of axes, and indicate its domain and range in the space provided.
!"#$%& $'( The graph of g(x) is shown on the
graph to the left in red. It has
Domain(g(x)) = [−3, 3]
$"#$%& !"'$%& Range(g(x)) = [−5, −3] Math 1110 Fall 2011 Homework 1 Book Problem Answers:
Section 1.1 Section 1.2: Section 1.3: Section 1.4 Optional Problems: ...
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This note was uploaded on 03/01/2012 for the course MATH 1110 at Cornell.