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Chapter opener: iceboats

An image of an iceboat in action.
(credit: modification of work by Carter Brown, Flickr)

As we saw at the beginning of the chapter, top iceboat racers ( [link] ) can attain speeds of up to five times the wind speed. Andrew is an intermediate iceboater, though, so he attains speeds equal to only twice the wind speed. Suppose Andrew takes his iceboat out one morning when a light 5-mph breeze has been blowing all morning. As Andrew gets his iceboat set up, though, the wind begins to pick up. During his first half hour of iceboating, the wind speed increases according to the function v ( t ) = 20 t + 5 . For the second half hour of Andrew’s outing, the wind remains steady at 15 mph. In other words, the wind speed is given by

v ( t ) = { 20 t + 5 for 0 t 1 2 15 for 1 2 t 1 .

Recalling that Andrew’s iceboat travels at twice the wind speed, and assuming he moves in a straight line away from his starting point, how far is Andrew from his starting point after 1 hour?

To figure out how far Andrew has traveled, we need to integrate his velocity, which is twice the wind speed. Then

Distance = 0 1 2 v ( t ) d t .

Substituting the expressions we were given for v ( t ) , we get

0 1 2 v ( t ) d t = 0 1 / 2 2 v ( t ) d t + 1 / 2 1 2 v ( t ) d t = 0 1 / 2 2 ( 20 t + 5 ) d t + 1 / 3 1 2 ( 15 ) d t = 0 1 / 2 ( 40 t + 10 ) d t + 1 / 2 1 30 d t = [ 20 t 2 + 10 t ] | 0 1 / 2 + [ 30 t ] | 1 / 2 1 = ( 20 4 + 5 ) 0 + ( 30 15 ) = 25.

Andrew is 25 mi from his starting point after 1 hour.

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Suppose that, instead of remaining steady during the second half hour of Andrew’s outing, the wind starts to die down according to the function v ( t ) = −10 t + 15 . In other words, the wind speed is given by

v ( t ) = { 20 t + 5 for 0 t 1 2 10 t + 15 for 1 2 t 1 .

Under these conditions, how far from his starting point is Andrew after 1 hour?

17.5 mi

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Integrating even and odd functions

We saw in Functions and Graphs that an even function    is a function in which f ( x ) = f ( x ) for all x in the domain—that is, the graph of the curve is unchanged when x is replaced with − x . The graphs of even functions are symmetric about the y -axis. An odd function    is one in which f ( x ) = f ( x ) for all x in the domain, and the graph of the function is symmetric about the origin.

Integrals of even functions, when the limits of integration are from − a to a , involve two equal areas, because they are symmetric about the y -axis. Integrals of odd functions, when the limits of integration are similarly [ a , a ] , evaluate to zero because the areas above and below the x -axis are equal.

Rule: integrals of even and odd functions

For continuous even functions such that f ( x ) = f ( x ) ,

a a f ( x ) d x = 2 0 a f ( x ) d x .

For continuous odd functions such that f ( x ) = f ( x ) ,

a a f ( x ) d x = 0 .

Integrating an even function

Integrate the even function −2 2 ( 3 x 8 2 ) d x and verify that the integration formula for even functions holds.

The symmetry appears in the graphs in [link] . Graph (a) shows the region below the curve and above the x -axis. We have to zoom in to this graph by a huge amount to see the region. Graph (b) shows the region above the curve and below the x -axis. The signed area of this region is negative. Both views illustrate the symmetry about the y -axis of an even function. We have

−2 2 ( 3 x 8 2 ) d x = ( x 9 3 2 x ) | −2 2 = [ ( 2 ) 9 3 2 ( 2 ) ] [ ( −2 ) 9 3 2 ( −2 ) ] = ( 512 3 4 ) ( 512 3 + 4 ) = 1000 3 .

To verify the integration formula for even functions, we can calculate the integral from 0 to 2 and double it, then check to make sure we get the same answer.

0 2 ( 3 x 8 2 ) d x = ( x 9 3 2 x ) | 0 2 = 512 3 4 = 500 3

Since 2 · 500 3 = 1000 3 , we have verified the formula for even functions in this particular example.

Two graphs of the same function f(x) = 3x^8 – 2, side by side. It is symmetric about the y axis, has x-intercepts at (-1,0) and (1,0), and has a y-intercept at (0,-2). The function decreases rapidly as x increases until about -.5, where it levels off at -2. Then, at about .5, it increases rapidly as a mirror image. The first graph is zoomed-out and shows the positive area between the curve and the x axis over [-2,-1] and [1,2]. The second is zoomed-in and shows the negative area between the curve and the x-axis over [-1,1].
Graph (a) shows the positive area between the curve and the x -axis, whereas graph (b) shows the negative area between the curve and the x -axis. Both views show the symmetry about the y -axis.
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Questions & Answers

Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
please how do I learn integration
aliyu Reply
they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function.
Andrew
best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding.
Andrew
example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2
Andrew
great noticeable direction
Isaac
limit x tend to infinite xcos(π/2x)*sin(π/4x)
Abhijeet Reply
can you give me a problem for function. a trigonometric one
geovanni Reply
state and prove L hospital rule
Krishna Reply
I want to know about hospital rule
Faysal
If you tell me how can I Know about engineering math 1( sugh as any lecture or tutorial)
Faysal
I don't know either i am also new,first year college ,taking computer engineer,and.trying to advance learning
Amor
if you want some help on l hospital rule ask me
Jawad
it's spelled hopital
Connor
hi
BERNANDINO
you are correct Connor Angeli, the L'Hospital was the old one but the modern way to say is L 'Hôpital.
Leo
I had no clue this was an online app
Connor
Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 (t=0), total online holiday sales were $42.3 billion, whereas in 2013 they were $48.1 billion. Find a linear function S that estimates the total online holiday sales in the year t . Interpret the slope of the graph of S . Use part a. to predict the year when online shopping during Christmas will reach $60 billion?
Nguyen Reply
Practice Key Terms 1

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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