1.1 Rates of Changes
What is the formula of the average rate of change
f(a+h)-f(a)
h
1.1 Rates of Changes
what is the average rate of change called
the difference quotient
1.1 Rates of Changes
How do you find the average velocity.
f(a+h)-f(a)
h
displacement/time
1.2 The Limits of a Function & One Sided Limit
Basic of limits
If f(x)=k then lim f(x) = ?
…………………..x->c
k
1.2 The Limits of a Function & One Sided Limit
Basic Limits
If f(x) = xn then lim f(x)=?
—————————————–x->c
cn<su/p>
1.2 The Limits of a Function & One Sided Limit
What does lim f(x)=L mean?
———————-x->c+
The right sided limits
x approaches c from values greater than c
1.2 The Limits of a Function & One Sided Limit
What does lim f(x)=L mean?
———————-x->c-
The left sided limits
x approaches c from values smaller than c
1.2 The Limits of a Function & One Sided Limit
when does the limit exist
when the right sided limit and the left sided limit both exist and are equal to eachotehr
1.3 Calculating Limits Using the Limit Laws
What are the 6 limit laws
- limit of f(x)+/-g(x) = limit of f(x) =/- limit of g(x)
- limit of f(x) x g(x) = limit of f(x) x limit of g(x)
- limit of f(x) / g(x) = limit of f(x) / limit of g(x)
- limit of kf(x) = k times limit of f(x)
- limit of f(x)k = limit of f(x) to the power to k
- limit of nth root of f(x) = nth root of the limit of f(x)
1.3 Calculating Limits Using the Limit Laws
Special Trigonometric Limits
1. lim sinx/x = ?
—x->0
1
1.3 Calculating Limits Using the Limit Laws
2. lim (1-cosx)/x = ?
—x->0
0
1.3 Calculating Limits Using the Limit Laws
5. lim x/sinx = 1
—x->0
1
1.3 Calculating Limits Using the Limit Laws
3. lim sin(ax)/ax = ?
—x->0
1
1.3 Calculating Limits Using the Limit Laws
4. lim sin(ax)/sin(bx) = ?
—x->0
a/b
1.3 Calculating Limits Using the Limit Laws
lim sin(1-x)/(1-x)=?
x->1
1
1.3 Calculating Limits Using the Limit Laws
What are the ways to solve Composite Function Limit Problems
lim g(f(x))
x->c
if the function of f(x) when going toward c from up then it is the right limit of g(x) and if f(x) is going toward c from down it is the left limit of g(x)
1.4 Properties and Continuity and Intermediate Value Theorem
What is the definition of continuity at c
- f(c) exists
- limit of f(x) exists
- the limit of f(x) is equal to f(c)
1.4 Properties and Continuity and Intermediate Value Theorem
What is the Intermediate Value Theorem
If f is a continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
1.5 Limits and Asymptotes
lim f(x) = infinity
x->c
What does this mean
f(x) approaches infinity as x approaces c
1.5 Limits and Asymptotes
If f(x) approaches plus minus infinity as x approaches then the line x=c is a __________________ of graph f
vertical asymptote
1.5 Limits and Asymptotes
the graph of a rational function given by y=f(x)/g(x) has a vertical asymptote at where?
at x values that make g(x) 0. f(x) can’t also be 0 at this value
1.5 Limits and Asymptotes
Horizontal asymptote: a line y=b is a horizontal asymptote of the graph of a function y = f(x) if ??
limit f(x) to infinity or negative infinity is b
1.6 기타 guitar
what are the 3 strategies for finding basic limits
- start with direct substitution
- divide out common factors
- rationalize the denominator
1.6 기타 guitar
The squeeze theorem
If a function is squeezed between two other functions as shown in the figure below, we can derive the following theorem
suppose h(x) is smaller or equal to g(x) and g(x) smaller or equal to f(x) what is the result.
g(x) is the same as the limit of h(x) f(x) on x=a (a is when f(x) and h(x))
