Math 1153

Math 1153

1. Limits

   • Continunity
     • Lim(x-a) f(x) = f(a) : a function is continuous at number a if a is in the domain and the limit value of f(x) as x approaches a is equal to the function value f(a)
       • Conditions
         • 1) lim(x-a) f(x) Exists (Therefore, the right hand limit and left hand limit at x=a exists and are equal)
         • 2) F(a) exists (a is defined and in the domain)
         • 3) Lim(x-a) f(x) = f(a) (the limit value is equal to the function value)
     • Continuity on an interval: a function is continuous in an interval if f(x) is continuous at every x in the domain within the interval
     • ALL ELEMENTARY FUNCTIONS ARE CONTINUOUS AT EVERY POINT IN THEIR DOMAIN
     • Intermediate Value Therome: given y = f(x) is continuous from [a,b], for any value of L between f(a) and f(b), there exists c such that f(c) = L
       • Does the root/value exist vs what is it's uniqueness?
       • ROOTS/ZEROS: given y = f(x), c is called a zero, root, or x-intercept of f(x) if f(c) = 0
         • Use the IVT to determine where roots are/between which numbers, or if they exists
   • Given y = f(x), the limit value of f as x approaches a (denoted by lim(x-a) f(x) , is a unique real number L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a, but not equal to a
     • Limits at Infinity
       • Lim(x-∞)f(x) = L : The limit value of f(x) as x becomes large is equal to L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently large.
       • Lim(x-(-∞))f(x) = L: The limit value of f(x) as x becomes small (negative large) is equal to L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently small (negative large)
         • Horizontal Asymptote is equal to L
       • Lim(x-a)f(x) = ∞: The limit value of f(x) as x approaches a is large if we can make the values of f(x) arbitrarily large by taking x to be sufficiently close to a, but not equal to a.
       • Lim(x-a)f(x) = (-∞): The limit value of f(x) as x approaches a is small (negative large) if we can make the values of f(x) arbitrarily small by taking x to be sufficiently close to a, but not equal to a.
         • Verticle asymptote is equal to a
     • Theorem of Limits: Lim(x-a)f(x) = L if and only if lim(x-a+)f(x) = lim(x-a-)f(x) = L
       • Right Hand Limit: The right hand limit value of f(x) as x approaches a from the right is equal to L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a, but more than a.
       • Left Hand Limit: The left hand limit value of f(x) as x approaches a from the left is equal to L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a, but less than a.
     • Limit Laws
       • Calculating Limits: The order of operations with limits can be reversed and arrive at the same answer
         • Sum Law: The limit of the sum or two functions is equal to the sum of the limits
           • lim( f(x) + g(x) ) = lim f(x) + lim g(x)
         • Difference Law: The limit of the difference of two functions i equal to the difference of the limits
           • lim( f(x) - g(x) ) = lim f(x) - lim g(x)
         • Constant Multiple Law: The limit of a function multiplied by a constant is equal to the constant multiplied by the limit of the function
           • lim (cf(x)) = c lim f(x)
         • Product Law: The limit of the product of two functions is equal to the product of the limits
           • lim( f(x)*g(x) ) = lim f(x) * lim g(x)
         • Quotient Law: The limit of a quotient of two functions is equal to the quotient of the two limits
           • lim( f(x)/g(x) ) = lim f(x) / lim g(x)
         • Square Root Law: The limit of a square root function is equal to the square root of the limit
           • lim √f(x) = √lim f(x)
         • Power Law: The limit of an exponential the function is equal to the limit of the function to the power of the exponent
           • lim f(x)^2 = (lim f(x))^2
         • Constant Law: The limit of a constant is equal to itself. (The constant function is continuous, so at any value of x, the limit value is the constant)
           • lim c = c
     • Indeterminate Form: The value is undecided and cannot be determined in its current form (denoted by ->0/->0, ∞/∞, -∞/∞, ∞-∞).
       • Restate equation using conjugaes, expanding, factoring, or rewriting using exponent laws
         • Factoring: factoring means to rewrite a polynomial expression as a product
         • Expanding: expanding means to rewrite a polynomial expression as a sum or difference
         • Conjugates: a pair of binomials that are the same except for the signs between the terms are opposite
     • Undefined: A/B is undefined when A is not equal to 0 and B = 0

2. Derivatives

   • Derivative: Given y = f(x), the derivative of f with respect to x, denoted by dy/dx (or f'(x)), if defined as follows; (dy/dx = lim(h-o) f(x+h)-f(x)/h) ) or (dy/dx = lim(x-a) f(x)-f(a)/x-a))
   • The derivative is equal to the limit of the difference quotient
   • The derivative is in an INTERMEDIATE FORM
   • The derivative represents the limit of the difference quotient at an arbitrary point (x, f(x))
     • The limit of the difference quotient represents the slope of the tangent line of y = f(x) as an arbitrary point (x, f(x))
       • S.O.T = V.O.D: The slope of the tangent line at a specific point (a, f(a)) is equal to the value of the derivative
         • Differentiability
           • Given y = f(x), f is differentiable at x =a if dy/dx at x =a exists, that is, if lim(h-0) f(a+h)-f(a)/h exists, that is the lim(h-0+) and lim(h-0-) exists and othe are equal.
             • given y = f(x), f is differentiable at x = a if the value of the derivative (slope of the tangent) exists, that is if the limit of the difference quotient at a exists, the is if the right hand limit and left hand limit of the difference quotient at a are equal
           • Theorm: If f is differentiable at x=a, then it is continuous at x=a
             • Differentiable functions are always continuous, but not all continuous functions are differentiable
             • For a value to be differentiable, it also must exists in the domain of the derivative (therefore, the slope of the tangent must exists)
           • How are Graphs not differentiable?
             • The secant lines will not divert into a single tangent (limit values of difference quotient does not exist)
             • Slope is undefined and does not exist
   • Point of Tangency: The point where the function intersects(touches) the tangent line

3. Additional Definitions

   • Domain: all values of x where the function f(x) is defined / exists
   • Elementary Functions: Polynomial, Exponential, Radial, Rational, Logarimetic, Trigonometic, Inverse, Inverse Trig
   • Absolute Value Function: If a value is positive, then its absolute value is itself. If a value is negative, then its absolute value is the negative of itself
   • Rational function: a rational function is a polynomial over a non-zero polynomial

4. Secant Lines: The slope of a line connecting two points on y = f(x). Defined by m = ∆y/∆x, m= f(x)-f(a)/x-a, m = f(x+h)-f(x)/(x+h)-x

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