calculus function

7/25/2019 Calculus Function

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Calculus

The Computational Method

(mathematics)The Mineral growth in a hollow organ of the body, e.g. kidney stone(medical term)


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Function

A function is a rule that assigns to each

elementxin a set A exactly one element,

called f(x), in a set B.


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Linear Function

y =f (x) = mx +b

here m is the slope o! the line and b is

the y"intercept.


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ENGINEERING EXAMPLE

(a)As dry air mo#es upard, it expands and cools. $! the ground temperature is %& 'C and the

temperature at a height o! m is & 'C , express the temperature T (in 'C) as a !unction o! the

height h (in ilometers), assuming that a linear model is appropriate.

(b)*ra the graph o! the !unction in part (a). +hat does the slope represent(c)+hat is the temperature at a height o! %.- m

/L0T$/1

(a) Because e are assuming that T is a linear !unction o! h, e can rite

T = mh + b

+e are gi#en that T2 %& 'Chen h2 &, so

%& 2 m .& 3 b 2 b

$n other ords, the y"intercept is b2 %&.

+e are also gi#en that T 2 & 'C hen h2 , so

& 2 m . 3 %&The slope o! the line is there!ore m = "& and the re4uired linear !unction is

T = "&h + %&


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(5) The graph is setched in Figure 6. The slope is m ="& 'C7m, and this represents the rate o!

change o! temperature ith respect to height.

(c) At a height o! %.- m, the temperature is

T 2 "&(%.-) 3 %& 2 " - 'C


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tudent Assignment

9. The relationship 5eteen the Fahrenheit and Celsius temperature scales isgi#en 5y the linear !unction 8

(a) etch a graph o! this !unction.(5) +hat is the slope o! the graph and hat does it represent

+hat is the F"intercept and hat does it represent


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9olynomials Function

Quadratic function2 9olynomial degree %

P(x) = ax% 3 bx +c

Cubic function2 9olynomial degree 6

P(x)2 ax63 bx%3 cx +d


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9olynomial n"degree

P(x)2 anxn+an-xn-3:3 a%x%3 ax +a&


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EXAMPLE 4A 5all is dropped !rom the upper o5ser#ation dec o! the C1 Toer, ;-& m

a5o#e the ground, and its height h a5o#e the ground is recorded at "second inter#als in

Ta5le %. Find a model to !it the data and use the model to predict the time at hich the

5all hits the ground.


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/L0T$/1

+e dra a scatter plot o! the data in Figure < and o5ser#e that a linear model is inappropriate. But itloos as i! the data points might lie on a para5ola, so e try a 4uadratic model instead. 0sing a graphingcalculator or computer alge5ra system (hich uses the least s4uares method), e o5tain the !olloing4uadratic model8

h =449.36 !.96t +4.9!t "

$n Figure & e plot the graph o! The =4uation together ith the data points and see that the 4uadraticmodel gi#es a #ery good !it. The 5all hits the ground hen h2&, so e sol#e the 4uadratic e4uation8

ax% 3 bx 3 c 2 &,

The 4uadratic !ormula gi#es

The positi#e root is t ?, so e predict that the 5all ill hit the ground a!ter a5out


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tudent Assignment

#.6 Let f (x) 2x%3 %x @ !or allx. =#aluate8

(a) f (%),

(b) f (@%),

(c) f (@x),

(d) f (x 3 )

(e) f (x @ )

(f) f (x 3 h)

(g) f (x 3 h) @ f (x)

(h) f (x 3 h) @ f (x)

h


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9oer Function

A !unction o! the !orm f (x)2xa, here is aconstant, is called a $o%&r function.

(i) a =n' %&r& n i a $oiti*& int&+&r


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(ii) a =7n' %&r& n i a $oiti*& int&+&r. The !unction is a root function.

(ii) a = ". The !unction is a r&ci$roca, function.


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Trigonometric Function


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=xponential Function

The &-$on&ntia, function are the !unctions o! the !orm here the 5ase ais a

positi#e constant.


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Logarithmic Function

calculus function

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