Math 1710

Topics for first exam

Chapter 1: Limits and Continuity

§ 1:
Rates of change and limits
Calculus = Precalculus + (limits)
Limit of a function f at a point x0 = the value the function `should' take at the point
= the value that the points `near' x0 tell you f should have at x0
limx®x0f(x) = L means f(x) is close to L when x is close to (but not equal to) x0
Idea: slopes of tangent lines

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limx®x0f(x) = L does not care what f(x0) is; it ignores it
limx®x0f(x) need not exist! (function can't make up it's mind?)

§ 2:
Rules for finding limits
If two functions f(x) and g(x) agree (are equal) for every x near a
(but maybe not at a), then limx® af(x)limx® ag(x)
Ex.: limx®2 [(x2-3x+2)/(x2-4)] = limx®2 [(x-1)/(x+2)]
If f(x)®L and g(x)®M as x®x0 (and c is a constant), then
f(x)+g(x)®L+M ; f(x)-g(x)®L-M ; cf(x)®cL ;
f(x)g(x)®LM ; and f(x)/g(x)®L/M provided M ¹ 0
If f(x) is a polynomial, then limx® x0f(x)= f(x0)
Basic principle: to solve limx® x0 , plug in x = x0 !
If (and when) you get 0/0 , try something else! (Factor (x-x0) out of top and bottom...)
If a function has something like Öx - Öa in it, try multiplying (top and bottom)
with Öx + Öa
Sandwich Theorem: If f(x) £ g(x) £ h(x) , for all x near a (but not at a), and
limx® af(x)limx® ah(x)L , then limx® ag(x)L .

§ 4:
Extensions of the limit concept
Motivation: the Heaviside function

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Limit from the right: limx® a+f(x)L means f(x) is close to L
when x is close to, and bigger than, a
Limit from the left: limx® a-f(x)M means f(x) is close to M
when x is close to, and smaller than, a
limx® af(x)L then means limx® a+f(x)limx® a-f(x)L

Infinite limits: ¥ represents something bigger than any number we can think of
limx® af(x)¥ means f(x) gets really large as x gets close to a
Also have limx® af(x)-¥ ; limx® a+f(x)¥ ;
limx® a-f(x)¥ ; etc....
Typically, an infinite limit occurs where the denomenator of f(x) is zero
(although not always)

§ 5:
Continuity
A function f is continuous (cts) at a if limx® af(x)f(a)
This means: (1) limx® af(x) exists ; (2) f(a) exists ; and
(3) they're equal.
Limit theorems say (sum, difference, product, quotient) of cts functions are cts.
Polynomials are continuous at every point;
rational functions are continuous except where denom=0.
Points where a function is not continuous are called discontinuities
Four flavors:
removable: both one-sided limits are the same
jump: one-sided limts exist, not the same
infinite: one or both one-sided limits is ¥ or -¥
oscillating: one or both one-sided limits DNE

Intermediate Value Theorem:
If f(x) is cts at every point in an interval [a,b], and M is between f(a) and f(b),
then there is (at least one) c between a and b so that f(c) = M.
Application: finding roots of polynomials

§ 6:
Tangent lines
Slope of tangent line = limit of slopes of secant lines; at (x0,f(x0) :
limx® x0[(f(x)-f(x0))/(x-x0)] Notation: call this limit f¢(x0)derivative of f at x0
Different formulation: h = x-x0, x = x0+h
f¢(x0)limh® 0[(f(x0+h)-f(x0))/h]

Chapter 2: Derivatives

§ 1:
The derivative of a function
derivative = limit of difference quotient (two flavors)
f¢(x0) exists, say f is differentiable at x0
Fact: f differentiable (diff'ble) at x0, then f cts at x0
h® 0 notation: replace x0 with x (= variable), get f¢(x) new function
f¢(x)the derivative of f = function whose vaules are the slopes of the tangent
lines to the graph of y=f(x) . Domain = every point where the limit exists
Notation:
f¢(x)[dy/dx][d/dx](f(x)) = [df/dx] = y¢ = Dx f = Df = (f(x))¢

§ 2:
Differentiation rules
[d/dx](constant) = 0
[d/dx](x) = 1

(f(x)+g(x))¢ = (f(x))¢+ (g(x))¢
(f(x)-g(x))¢= (f(x))¢- (g(x))¢
(cf(x))¢= c(f(x))¢

(f(x)g(x))¢= (f(x))¢g(x)+ f(x)(g(x))¢
([f(x)/g(x)])¢= [(f¢(x)g(x)-f(x)g¢(x))/(g2(x))]

(xn)¢= nxn-1 , for n=0,1,-1,2,-2,3,.......
(( (1/g(x))¢= -ggp/(g(x))2 ))

f¢(x) is `just' a function, so we can take its derivative!
(f¢(x))¢= f¢¢(x)(= y¢¢ = [(d2y)/(dx2)] = [(d2f)/(dx2)])
= second derivative of f (=rate of change of rate of change of f !)
Keep going! f¢¢¢(x) = 3rd derivative, f(n)(x) = nth derivative

§ 3:
Rates of change
Physical interpretation:
f(t)= position at time t
f¢(t)= rate of change of position = velocity
f¢¢(t)= rate of change of velocity = acceleration
|f¢(t)| = speed
Basic principle: for object to change direction (velocity changes sign),
f¢(t)= 0 somewhere (IVT!)
Examples:
Free-fall: object falling near earth; s(t) = s0+v0 t-[g/2] t2
s0 = s(0) = initial position; v0 = initial velocity; g= acceleration due to gravity
Economics:
C(x) = cost of making x objects; R(x) = revenue from selling x objects;
P = R-C = profit
C¢(x) = marginal cost = cost of making `one more' object
R¢(x) = marginal revenue ; profit is maximized when P¢(x) = 0 ;
i.e., R¢(x) = C¢(x)

§ 4:
Derivatives of trigonometric functions
Basic limit: limx®0[sinx/x] = 1 ; everything else comes from this!
Note: this uses radian measure! limx®0[sin(bx)/x] = b
Then we get:
(sinx)¢= cosx (cosx)¢= -sinx
(tanx)¢= sec2 x (cotx)¢= = csc2 x
(secx)¢= secx tanx (cscx)¢= = cscxcotx

§ 5:
The Chain Rule
Composition (g°f)(x0) = g(f(x0)) ; (note: we don't know what g(x0) is.)
(g°f)¢ ought to have something to do with g¢(x) and f¢(x)
in particular, (g°f)¢(x0) should depend on f¢(x0) and g¢(f(x0))

Chain Rule: (g°f)¢(x0) = g¢(f(x0))f¢(x0)
= (d(outside) eval'd at inside fcn)·(d(inside))
Ex: ((x3+x-1)4)¢= (4(x3+1-1)3)(3x2+1)

Different notiation:
y = g(f(x)) = g(u), where u = f(x), then [dy/dx] = [dy/du][du/dx]

§ 6:
Implicit differentiation
We can differentiate functions; what about equations? (e.g., x2+y2 = 1)
graph looks like it has tangent lines

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Idea: Pretend equation defines y as a function of x : x2+(f(x))2 = 1 and differentiate!
2x+2f(x) f¢(x) = 0 ; so f¢(x) = [(-x)/f(x)] = [(-x)/y]
Different notation:
x2+xy2-y3 = 6 ; then 2x+(y2+x(2y[dy/dx])-3y2[dy/dx] = 0
[dy/dx] = [(-2x-y2)/(2xy-3y2)]
Application: extend the power rule
[d/dx](xr) = rxr-1 works for any rational number r

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