Chapter 1 Flashcards
How to find the tangent line from secant lines
- find the slope of each secant line
- average the slopes of the two closest points
How to find the area of a bounded region
- width=given
- height=sin new width
- multiply and add all together
Use the rectangles in each group to approximate the area of the region bounded by y=5/x, y=0, x=1, and x=5
- start with 5/1+5/2+5/3+5/4 since there are four rectangles
- Since there are now 8 rectangles, do (1/2)(5/1)+(1/2)(5/1.5)+… etc until you get to and including 4.5
How would you describe the instantaneous rate of change of an automobiles position on the highway?
The instantaneous rate of change for a car is the distance that it is traveling every instant of time, in other words its instantaneous velocity. The instantaneous rate of change should be a function of time that describes the car’s velocity every instant of time.
Distance formula
SQRT (x2-x1)^2 + (y2-y1)^2
complete the table and use the results to estimate the limit.
- complete the table
- average the two closest
create a table of values for the function and use the result to estimate the limit
add/subtract .001 then proceed to drop a decimal place
use the graph to find the limit at a specific point
- go based off graph, not point
limit DNE if
- behavior differs from right and left (piece wise)
- unbounded behavior (1/x)
- oscillating (sin 1/x)
f(1) vs limf(x)–>1
f(1) can be the point but limit is not
use the graph of f to identify the values of c with limf(x)–>c exists
limf(x) exists for all points on the graph except where c=
double brackets mean
discrete, use int in calculator
estimate the limit (1+x)^1/x x–>0 or other weird ones
use a table
if f is undefined at x=c, then the limit of f(x) as x approaches c does not exist
false, duh
if the limit of f(x) as x approaches c is 0, then there must exist a number k such that f(k)
false, The existence of nonexistence of f at x=c has no bearing on the existence of the limit of f(x) as x approaches c. The limit of f(x) at x=c is the value that f(x) moves arbitrarily close to as x moves close to c.
if f(c)=L, then limf(x)x–>c=L
false
if limc(x)x–>0=L, then f(c)=L
false
is limSQRT0 as x–>0=0 a true statement
no because x can’t approach 0 from the left
For what value h do the rectangle and the triangle have the same area?
bh=b(1-h/2) / 2
finding limit with square root in numerator
- rationalize the numerator
- multiply numerator and denominator by opposite sign thing
- if you end up with SQRT in denominator, its ok but make sure to simplify it!!!
weird delta x
distribute
limits of trig functions
limx–>c of trigx=trigc
special trig functions
- limx–>0 of sinx / x=1
- limx–>0 of (1-cosx)/x=0
when doing limits of trigs
- always bring numerators out to see if theres a special trig
- rearrange tan and stuff
lim x–>0 of tan^2theta / theta
ends up being (sinx)(sinx/x)(1/cosx^2)
f(x+deltax)-f(x)/delta x
- plus in x+deltax into f(x) and subtract f(x)
if its not working out then
-0 might be -1 in numerator
approaching
- factor, simplify, then plug
- plug in close values, USUALLY EXISTS
limx–>-3- x/SQRTx^2-9
limit does not exists
abs/same thing but not abs value
-1 if approaching from left, 1 if approaching from right
limit cotx and sec x when x approaches pi/pi/2
DNE because doesn’t exist from left and right
int values
- if approaching, plug in
- if x–> specific value, then try one off from both sides to see if it exists
