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Final

MATH 157 Study Guide - Final Guide: Wellington Koo, Periodic Function, Inflection Point

6 pages105 viewsSpring 2014

Department
Mathematics
Course Code
MATH 157
Professor
Shiva Gol Tabaghi
Study Guide
Final

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Review Notes
1. Limit of a function: we write
ax
Lxf
)(lim
and say “the limit of f(x), as x approaches a, equals L” , if we can make the value of f(x)
arbitrary close to L (as close to L as we like) by taking x to be sufficiently close to a (on
either side of a) but not equal to a.
Fact:
axaxax
LxfandLxfLxf ))(lim)(lim()(lim
Special Limits
0
1
sin
lim71828.2.)
1
1(lim
xn x
x
ee
n
n
2. Limit definition of Continuity of a function. A function is continuous at a number a if
ax
afxf
)()(lim
.
Note. a belongs to the domain of f . b)
ax
xf
)(lim
exist . c)
ax
afxf
)()(lim
Discontinuity of a function at a. If
1) f is defined on an open interval containing a, except perhaps at a, and
2) f is not continuous at a
we say that f is discontinuous at a.
Theorem: Intermediate value theorem. Suppose that f is continuous on the closed interval
[a,b] and let N be any number between f(a) and f(b), where f(a)
f(b). Then there exists a
number c in (a,b) such that f(c)=N.
Theorem: Existence of Zeros of a continuous function. If f is a continuous function on a
closed interval [a,b], and if f(a) and f(b) have opposite signs, then there is at least on solution
of the equation f(x)=0 in the interval (a,b).
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(
a
a
b
b
)
3. Limit Definition of Derivative at the point (a, f(a)). The derivative of a function f at a
defined as
0
)()(
lim
hh
afhaf
.
It is also denoted by
)(xf
and also called instantaneous rate of change. The derivative of a
function at a point x=a is the slope of the tangent line to the graph of f at (a,f(a)).
4. Differentiation Rules:
(a) General formulas:
0)( c
dx
d
)(.))(( xf
dx
d
cxcf
dx
d
1
)(
nn nxx
dx
d
)()())()(( xg
dx
d
xf
dx
d
xgxf
dx
d
)()()()()())()(( ruleproductxfxgxgxfxgxf
dx
d
2
)]([
]
)(
)(
[xg
gfgf
xg
xf
dx
d
)()()).(())](([ rulechainxgxgfxgf
dx
d
x
x
x
dx
d
(b) Exponential and Logarithmic Functions:
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