Calculus Mrs. Dougherty’s Class
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3 Big Calculus Topics Limits Derivatives Integrals
Chapter 2
2.1 Limits and continuity
Limits can be found Graphically
Limits can be found Graphically Numerically
Limits can be found Graphically Numerically By direct substitution
Limits can be found Graphically Numerically By direct substitution By the informal definition
Limits can be found Graphically Numerically By direct substitution By the informal definition By the formal definition
Limits Informal Def.
Limits Informal Def. Given real numbers c and L, if the values f(x) of a function approach or equal L
Limits Informal Def. Given real numbers c and L, if the values f(x) of a function approach or equal L as the values of x approach ( but do not equal c),
Limits Informal Def. Given real numbers c and L, if the values f(x) of a function approach or equal L as the values of x approach ( but do not equal c), then f has a limit L as x approaches c.
Limits notation
LIFE IS GOOD
Theorem 1 Constant Function f(x)=k Identity Function f(x)=x
Theorem 2 Limits of polynomial functions can be found by direct substitution.
Properties of Limits
If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c
Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Sum Rule: lim [f(x) + g(x)]= lim f(x) +lim g(x)=L1 + L2
Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Difference Rule: lim [f(x) - g(x)]= L1 - L2
Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Product Rule: lim [f(x) * g(x)]= L1 * L2
Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Constant multiple Rule: lim c f(x) = c L1
Properties of Limits If lim f(x) = L 1 and lim g(x) = L2 x-> c x -> c Quotient Rule: lim [f(x) / g(x)]= L1 / L2, L1=0 NOT
Theorem 3 Many ( not all ) limits of rational functions can be found by direct substitution.
Right-hand and Left-hand Limits
Theorem 4 A function, f(x), has a limit as x approaches c
Theorem 4 A function, f(x), has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist
Theorem 4 A function, f(x), has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal.
Calculus 2.2 Continuity
Definition f(x) is continuous at an interior point of the domain if
Definition f(x) is continuous at an interior point of the domain if lim f(x) = f(c ) x->c
Definition f(x) is continuous at an endpoint of the domain if
A “continuous” function is continuous at each point of its domain.
Definition Discontinuity If a function is not continuous at a point c, then c is called a point of discontinuity.
Types of Discontinuities Removable
Types of Discontinuities Removable Non-removable A) jump
Types of Discontinuities Removable Non-removable A) jump B) oscillating
Types of Discontinuities Removable Non-removable A) jump B) oscillating C) infinite
Test for Continuity
y=f(x) is continuous at x=c iff 1.
Test for Continuity y=f(x) is continuous at x=c iff 1. f(c) exists
Test for Continuity y=f(x) is continuous at x=c iff 1. f(c) exists 2. lim f(x) exists x-> c
Test for Continuity y=f(x) is continuous at x=c iff 1. f(c) exists 2. lim f(x) exists x -> c 3. f(c ) = lim f(x) x -> c
Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x)
Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x) 2. f(x) – g(x)
Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x) 2. f(x) – g(x) 3. f (x) g(x)
Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x) 2. f(x) – g(x) 3. f (x) g(x) 4. k g(x)
Theorem 5 Properties of Continuous Functions If f(x) and g(x) are continuous at c, then 1. f(x)+g(x) 2. f(x) – g(x) 3. f (x) g(x) 4. k g(x) 5. f(x)/g(x), g(x)/=0 are continuous
Theorem 6 If f and g are continuous at c, Then g f and f g are continuous at c
Theorem 7 If f(x) is continuous on [a,b], then f(x) has an absolute maximum,M, and an absolute minimum,m, on [a,b].
Intermediate Value Theorem for continuous functions A function that is continuous on [a,b] takes on every value between f(a) and f(b).
Calculus 2.3 The Sandwich Theorem
If g(x) < f(x) < h(x) for all x /=c and lim g(x) = lim h(x) = L, then lim f(x) = L.
Use sandwich theorem to find lim sin x x->0 x
Sandwich theorem examples So you can see the light.
Calculus 2.4 Limits Involving Infinity
Limits at + infinity are also called “end behavior” models for the function.
Definition y=b is a horizontal asymptote of f(x) if
Horizontal Tangents Case 1 degree of numerator < degree of denominator
Case 2 degree of numerator = degree of denominator
Case 3 degree of numerator > degree of denominator
Theorem Polynomial End Behavior Model
Calculus 2.6 The Formal Definition of a Limit
Now this is mathematics!!!