Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.
-
Upload
helen-blair -
Category
Documents
-
view
257 -
download
0
Transcript of Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.
![Page 1: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/1.jpg)
Calculus I Hughes-Hallett
Math 131Br. Joel Baumeyer
Christian Brothers University
![Page 2: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/2.jpg)
Function: (Data Point of View) One quantity H, is a function of
another, t, if each value of t has a unique value of H associated with it. In symbols: H = f(t).
We say H is the value of the function or the dependent variable or output; and
t is the argument or independent variable or input.
![Page 3: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/3.jpg)
Working Definition of Function: H = f(t)
A function is a rule (equation) which assigns to each element of the domain (independent variable) one and only one element of the range (dependent variable).
![Page 4: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/4.jpg)
Working definition of function continued:
Domain is the set of all possible values of the independent variable (t).
Range is the corresponding set of values of the dependent variable (H).
![Page 5: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/5.jpg)
Questions?
![Page 6: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/6.jpg)
General Types of Functions (Examples):
Linear: y = m(x) + b; proportion: y = kx Polynomial: Quadratic: y =x2 ; Cubic: y= x3 ;
etc Power Functions: y = kxp
Trigonometric: y = sin x, y = Arctan x Exponential: y = aebx ; Logarithmic: y = ln x
![Page 7: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/7.jpg)
Graph of a Function:
The graph of a function is all the points in the Cartesian plane whose coordinates make the rule (equation) of the function a true statement.
![Page 8: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/8.jpg)
Slope
• m - slope :
b: y-intercept• a: x-intercept
• .
run
rise
x
y
xx
yym
12
12
sintpoareyxandyx 2211 ,,
![Page 9: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/9.jpg)
5 Forms of the Linear Equation
• Slope-intercept: y = f(x) = b + mx• Slope-point:• Two point:
• Two intercept:
• General Form: Ax + By = C
)( 11 xxmyy
)( 112
121 xx
xx
yyyy
1b
y
a
x
![Page 10: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/10.jpg)
Exponential Functions: If a > 1, growth; a<1, decay
• If r is the growth rate then a = 1 + r, and
• If r is the decay rate then a = 1 - r, and
taPP 0
tt rPaPP )1(00 0P
tt rPaPP )1(00
![Page 11: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/11.jpg)
Definitions and Rules of Exponentiation:• D1: • D2:• R1:
• R2:
• R3:
0,,,1 1110 aaandaa xa
xa
evennforaaaandaa nn 0;1
21
txtx aaa
txt
x
aa
a
xttx aa
![Page 12: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/12.jpg)
Inverse Functions:
• Two functions z = f(x) and z = g(x) are inverse functions if the following four statements are true:
• Domain of f equals the range of g.• Range of f equals the domain of g.• f(g(x)) = x for all x in the domain of g.• g(f(y)) = y for all y in the domain of f.
)()( 1 xfxg
![Page 13: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/13.jpg)
A logarithm is an exponent.
.
bameanscb
generalinand
xemeanscxx
xmeanscx
ca
ce
c
log
:
,logln
10log10
![Page 14: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/14.jpg)
General Rules of Logarithms:
log(a•b) = log(a) + log(b) log(a/b) = log(a) - log(b)
b
aaalso
ccbecause
xcandxc
apa
c
cb
c
xxc
p
c
log
loglog
0,101log
log
)log()log(
0
log
![Page 15: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/15.jpg)
e = 2.718281828459045...
• Any exponential function
can be written in terms of e by using the fact that
So that
kxaby beb ln
kxbkx ea yaby )(becomes ln
![Page 16: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/16.jpg)
Making New Functions from Old
Given y = f(x):
(y - b) =k f(x - a) stretches f(x) if |k| > 1
shrinks f(x) if |k| < 1
reverses y values if k is negative
a moves graph right or left, a + or a -
b moves graph up or down, b + or b -
If f(-x) = f(x) then f is an “even” function.
If f(-x) = -f(x) then f is an “odd” function.
![Page 17: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/17.jpg)
Polynomials:
• A polynomial of the nth degree has n roots if complex numbers a allowed.
• Zeros of the function are roots of the equation.
• The graph can have at most n - 1 bends.• The leading coefficient determines the
position of the graph for |x| very large.
nn
kn
kk xaxaxaxay
10
00
na
![Page 18: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/18.jpg)
Rational Function: y = f(x) = p(x)/q(x)where p(x) and q(x) are polynomials.• Any value of x that makes q(x) = 0 is called a
vertical asymptote of f(x).• If f(x) approaches a finite value a as x gets larger
and larger in absolute value without stopping, then a is horizontal asymptote of f(x) and we write:
• An asymptote is a “line” that a curve approaches but never reaches.
axfx
)(lim
![Page 19: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/19.jpg)
Asymptote Tests y = h(x) =f(x)/g(x)
• Vertical Asymptotes: Solve: g(x) = 0If y as x K, where g(K) = 0,
then x = K is a vertical asymptote.• Horizontal Asymptotes:
If f(x) L as x then y = L is a vertical asymptote. Write h(x) as:
, where n is the highest power of x in f(x) or g(x).
n
n
x1
x1
)x(g
)x(f)x(h
![Page 20: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/20.jpg)
Basic Trig
• radian measure: = s/r and thus s = r , • Know triangle and circle definitions of the
trig functions.• y = A sin (Bx + C) + k
• A amplitude; • B - period factor; period, p = 2/B - phase shift; -C/B• k (raise or lower graph factor)
![Page 21: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/21.jpg)
Continuity of y = f(x)
• A function is said to be continuous if there are no “breaks” in its graph.
• A function is continuous at a point x = a if the value of f(x) L, a number, as x a for values of x either greater or less than a.
![Page 22: Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University.](https://reader033.fdocuments.in/reader033/viewer/2022061514/56649edb5503460f94bea54d/html5/thumbnails/22.jpg)
Intermediate Value Theorem
• Suppose f is continuous on a closed interval [a,b]. If k is any number between f(a) and f(b) then there is at least one number c in [a,b] such that f(x) = k.