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Unformatted text preview: Math 1110 Name: Homework 12 Due 12/1 or 12/2 in class Please print out these pages. 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.
Please 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, and you will be given written feedback on these
problems. GRADES
Text exercises / 20 Pres probs / 20 Staple /1 Text exercises. Please do the following problems from the book.
§5.5 #2, 10, 11, 17, 18, 34, 35, 38, 56, 63, 68, 72, 75, 79
§5.6 #2, 3, 8, 13, 20, 31, 50, 53, 58, 68, 73, 84, 89, 93, 99, 104, 107, 111, 112
Question 1. Please solve the following two problems.
x
x
(a) tan3
sec2
dx.
4
4
x
sec2 x dx
4
Let u = tan
, so that du =
.
4
4
x
x
x
Then tan3
sec2
dx = 4u3 du = u4 + C = tan4
+ C, where C is an arbitrary
4
4
4
constant.
(b) Find the area of the region enclosed between the line y = −3 and the curve y = 2x − x2.
First we ﬁnd the points where the line and the curve intersect.
−3 = 2x − x2
x2 − 2x − 3 = 0
(x − 3)(x + 1) = 0 So the points of intersection occur at x = 3 and x = −1.
Now we need to determine whether the line or the curve has greater values on the entire interval (−1, 3). To do this, we evaluate both expressions at point in the interval; for simplicity
we pick x = 0. For the line we have a yvalue of −3, and for the curve we have a yvalue of
0. We conclude that the curve takes on greater values on the entire interval (−1, 3).
3
3
3
2
13
2
2
2
To ﬁnd the area, we compute
2x − x − (−3) dx =
2x − x + 3 dx = x − x + 3x = 10
3
3
−1
−1
−1 Math 1110 (Fall 2011) HW5 Presentation Problems 2 Question 2. Let f be a function that is differentiable on [a, b]. In Chapter 2 of our text, we deﬁned
the average rate of change of f over [a, b] to be
f(b) − f(a)
b−a
and the instantaneous rate of change of f at x to be f (x). In Chapter 5, we deﬁned the average
value of a function. For the new deﬁnition of average value to be consistent with the old one, we
should have
f(b) − f(a)
= average value of f on [a, b].
b−a
Is this the case? Give reasons for your answer.
b
1
Using the deﬁnition in Chapter 5, the average value of on [a, b] is given by
f (x) dx.
b−a a
f(b) − f(a)
Using the Fundamental Theorem of Calculus we see that this is equal to
, since f is an
b−a
on [a, b].
antiderivative of f
f Homework 12 Book Problem Answers:
Section 5.5: Section 5.6: ...
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This note was uploaded on 03/01/2012 for the course MATH 1110 at Cornell University (Engineering School).
 '06
 MARTIN,C.

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