Calc 1
Polynomials Continuous?
All real numbers = (-∞,∞)
Removable Discontinuity
Discontinuity can be removed by redefining ƒ at a single number A hole in a graph.
Easy way to FOIL (a+b)² (a+b)³ (a+b)₄
Fibonacci triangle pattern Start your a in descending powers Start your b in ascending powers Apply numbers in triangle (a+b)² = a² + 2ab + b² (a+b)³ = a³ + 3a²b + 3ab² +b³ (a+b)₄ = a⁴ +4a³b +6 a²b² + 4ab³ +b⁴ Caution: make sure each term's exponents add to the original amount ex. 4a¹b³ is a 4th degree same as (a+b)⁴
The Precise Definition of a Limit
Given that ε>0 and δ>0 such that if 0<|x-a|<δ find |f(x) - L|<ε
Direct Substitution Property
If ƒ is a polynomial or rational function and "a" is in the domain of ƒ, then lim x→a f(x) = f(a)
The definition of Continuity
Let a be a point in the domain of the function f(x). Then f is continuous at x=a if and only if lim f(x) = f(a) x --> a A function f(x) is continuous on a set if it is continuous at every point of the set. Finally, f(x) is continuous (without further modification) if it is continuous at every point of its domain.
LL6 Power Law
Lim x→a [f(x)]ⁿ = [lim x→a f(x)]ⁿ where "n" is a positive integer
LL11 Square Root Law
Lim x→a ⁿ√f(x) = ⁿ√lim x→a f(x) where "n" is a positive integer If "n" is even, then lim x→a f(x) >0
Composition Theorem if ƒ and g are continuous at a and c is a constants, then the following functions are also continuous
f + g f - g cf fg f/g if g(a)≠0
Defn. # 2 Derivatives
f'(a) = lim f(x) - f(a) x→a x-a
The Squeeze Theorem
f(x) ≤ g(x) ≤ h(x) if lim x→a f(x) =lim x→a h(x) = L the so does lim x→a g(x)
A tangent line...
is a straight line that touches a function at only one point.
The slope of the tangent line at a point on the function...
is equal to the derivative of the function at the same point
LL3 Constant Multiple Law
lim x→a [cf(x)] = c lim x→a f(x)
LL1 Sum Law
lim x→a [f(x)+g(x)] = lim x→a f(x) + lim g(x)
LL2 Difference Law
lim x→a [f(x)-g(x)] = lim x→a f(x) - lim g(x)
LL4 Product Law
lim x→a [f(x)g(x)] = lim x→a f(x) * lim g(x)
LL7 Special Limits Power Law
lim x→a c = c
Theorem 8 If ƒ is continuous at b and lim x→a g(x) = b then lim x→a f(g(x)) = f(b)
lim x→a f(g(x)) = f (lim x→a g(x))
A function ƒ is continuous at a number "a" if
lim x→a f(x) = f(a)
LL5 Quotient Law
lim x→a f(x) = lim x→a f(x) g(x) lim x→a g(x) Where g(x) ≠0
LL8 Special Limits Power Law
lim x→a x = a
LL9 Special Limits Power Law
lim x→a xⁿ = aⁿ where "n" is a positive integer
LL10 Square Root Law
lim x→a ⁿ√x = ⁿ√a where "n" is a positive integer If "n" is even, then a>0
Finding a Tangent line at a point
m = lim f(x) - f(a) x→a x-a m=slope
Formula for a tangent line at (a,f(a))
m = lim f(x) - f(a) x→a x-a m=slope
Defn. # 1 Derivatives
m= lim f(a+h) - f(a) x→a h
List the four functions that are continuous at every number in their domains
polynomials rational root trigonometric
The tangent line represents...
the instantaneous rate of change of the function at that one point.
Formula for the equation of a tangent line
y-y₁ = m (x-x₁) then simplify into y=mx+b slope intercept form
Continuous on an Interval
ƒ is continuous at every number in the interval
Theorem 9 if g is continuous at a and f is continuous at g(a), then the composition function f ° g is continuous at a.
(f ° g) (x) = f(g(x))
Find the Derivitive of f(x) = x²-8x=9 at the number a, and use it to find the equation of the tangent line to the curve at the point (3,-6).
1. Find the Derivitive use the equation f'(a)=lim f(a+h) -f(a) h→0 h to find the slope (m) of the tangent line 2. Use the equation of a tangent line y-y₁ = m (x-x₁) to solve, then simplify into the slope intercept form y = mx+ b
Three requirements if ƒ is continuous at "a"
1. f(a) is defined ("a" is in the domain of ƒ) 2. lim x→a f(x) exists 3. lim x→a f(x) = f(a)
Rational Functions Continuous?
On its Domain (wherever it is defined.)
The Precise Definition of a Limit (Proof)
Show that |f(x) -L|<ε if |x-a|<δ Assume that |x-a|< value found for ε that = δ
Theorem 10 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 the f(c) = N.
Infinite Discontinuity
The function at the singular point goes to infinity in different�directions on the two sides. ��
Definition of Tangent Line
The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope m = lim f(x) - f(a) x→a x-a provided that this limit exists.
Jump Discontinuity
Value of the function jumps from one piece of the graph to the other. Also the discontinuity where both right and left limit exist, but are not equal to each other.
A secant line is...
a straight line joining two points on a function.
Theorem 1 Two sided limits
a two sided limit exists only if the left limit and the right limit are the same
