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Section 1

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Special Cases/Theorem 1.9

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Calculus Math

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Mar 14, 2020

Cards (16)

Section 1

(16 cards)

Special Cases/Theorem 1.9

Front

lim x->0 (sinx/x)=1 lim x->0 (1-cosx/x)=0 lim x->0 (1+x)^1/x = e

Back

Formal Definition of a Derivative

Front

f'(x) = limit as delta x approaches 0 ((f(x+delta x)-f(x))/(delta x) provided the limit exists. For all x for which this limit exists, f' is a function of x.

Back

Two types of discontinuity

Front

Removable- hole in the graph; limit exists non-removable- gap in picture/vertical asymptote; limit does not exist

Back

Concave Down

Front

because f¨(x)<0 in these in these intervals

Back

Find vertical asymptote of limit approaching infinity.

Front

LH and RH limits (infinity, negative infinity, DNE)

Back

Continuity

Front

A function f is continuous at C when these 3 conditions are met

Back

Relative Max

Front

If f'(x) changes from + to - and f'(x)=0.

Back

Squeeze Theorem

Front

If f(x) ≤ g(x) ≤ h(x) for all x in an open interval containing c, except possibly at c itself, and if lim x->c h(x)=L=lim x->c g(x) then, lim x->c f(x)=L

Back

The Mean Value Theorem

Front

If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that: f'(c) = (f(b)-f(a))/(b-a)

Back

Difference between f(c) and lim x-->c

Front

f(c) is its y-value, while the limit is where it ends up

Back

Concave Up

Front

because f¨(x)>0 in these in these intervals

Back

Find horizontal asymptote of limit approaching infinity.

Front

1. top degree bigger = no limit 2. top and bottom same = ratio of coefficients 3. bottom degree bigger = 0

Back

Relative Min

Front

If f'(x) changes from - to + and f'(x)=0.

Back

Intermediate Value Theorem

Front

If f is continuous on [a,b], f(a) doesn't equal f(b), and k is any number between f(a) and f(b), then there exists at least one number c, such that f(c)=k

Back

How to tell if continuous:

Front

1. f(c) is defined 2. lim x->c f(x)= "exists" 3. The left hand and right hand limits match

Back

Definition of differentiability

Front

1. Must be continuous 2. Cannot be a sharp point

Back