Calculus 1 Section 2

Trenton Schrimer

Instantaneous rate of change

the limit as x2 approaches x1 f(x2) - f(x1) over x2 - x1

average speed is equal to

distance travelled over time travelled

what rate of change will give us the closest estimate of the instantaneous speed?

whichever is closest to the time slot of the instantaneous speed

what should you do instead of taking a window close to the instantaneous speed?

make a table

What is each column for in the table for finding instantaneous speed?

first is the input, the second is the function, and the third is the average rate of change between the whole number and another number

what is the limit of the instantaneous speed?

the number that approaches arbitrarily close.

What is a limit?

limit as x approaches a f(x) = b if and only if f(x) gets arbitrarily close to b as x gets arbitrarily close to a (x does not equal a)

what is the important caveat in the definition of a limit?

f(x) must stay close to b, it can’t keep bouncing away from it forever.

what is the definition of a “one-sided” limit?

the limit as x approaches a from the right/left f(x) = b if and only if f(x) gets arbitrarily close to b as it approaches a from the right/left

what has to happen for the “two-sided” limit to exist?

the one-sided limits have to exist and be equal to each other

What is the definition of a positive infinite limit?

the limit as x approaches a f(x) is equal to infinity if and only if f(x) grows arbitrarily positively large as x approaches a

What is the definition of a negative infinite limit?

the limit as x approaches a f(x) is equal to negative infinity if and only if f(x) grows arbitrarily negatively large as x approaches a

If the limit as x approaches a f(x) is equal to negative and positive infinity…

then f(x) has a vertical asymptote at x = a

What is the upshut for limits?

always try plugging in f(a), if it is well defined you are done. If not, the limit may still exist, but there is more work to do

what three algebraic methods should you try for solving limits?

• try factoring

• multiply the top and bottom by the congejule

• if you have fractions in fractions, clear the little fractions

What is the definition of the Squeeze Theorem?

Suppose f(x) is less than or equal to g(x) is less than or equal to h(x) for all x in an open interval (a,b) containing c. Suppose also that the limit as x approaches c of f(x) = L= the limit as x approaches c of h(x), then the limit as x approaches c of g(x) = L

What is the trick for solving a function using the Squeeze Theorem?

Set the part of the function that makes it go crazy (sin/cos) equal to 1 and -1. Those are your two confining functions.

what is the Arch length equivalent to?

the corresponding radian measure/angle

fact of Sin

as the limit of theta approaches 0, sin theta over theta is = to 1

Intuitively: What does it mean to say a function is “continuous” over an interval of x values?

y makes no instantaneous jumps in values as x varies, and it doesn’t suddenly disappear or go to infinity within the int

What are the three main ways continuity of a function breaks down?

• hole in the graph

• jump in the graph

• infinite asymptote

What is the definition of Continuity?

y = f(x) is continuous at x = a if and only if f(a) exists, the limit as x approaches a exists, and they are equal. We say that f(x) is continuous on (a,b) if f(x) is continuous at every point in (a,b)

What about continuity in closed intervals [ ]?

we say that the interval [a,b] is continuous if every point including a and b are continous, using the previous definition.

What about continuity for intervals that contain end points?

we loosen the definition so that all points within [a,b] must be well defined and the limit as x approaches a from the right must exist and be continuous with the right side, and the limit as x approaches b from the left must exist and be continuous with the left side

What are the classifications for the three main kinds of discontinuities?

• “removable” discontinuity

• “jump” discontinuity

• “infinite” discontinuity

More on Removable Discontinuities

the limit as x approaches a of f(x) exists, but either f(a) doesn’t exist or it is not equal to the limit as x approaches a of f(x). we can redefine f(a) to equal the limit as x approaches a of f(x) and it makes the graph continuous at x = a

More on Jump Discontinuities

Both one-sided limits exist, but they are not equal to each other. There is no way to define f(a) to make this discontinuity go away

More on Infinite Discontinuities

There is an infinite limit involved. And its really bad and tricky to deal with later on in calculus

What is the Intermediate Value Theorem?

if f(x) is continuous on [a,b] and if y0 is any value in between f(a) and f(b), then there is some x0 within [a,b] such that f(x0) = y0. In other words if the graph of f(x) is connected over [a,b] it reaches every y-value between f(a) and f(b) at least once.