Final exam Flashcards
2.1
average velocity
change in distance/ change in time
2.1
instantaneous velocity
1st derivative of a position equation
acceleration
2nd derivative of position function
2.1
slope using two points
Y2-Y1/X2-X1
Point slope form
(Y-Y1)=m(X-X1)
2.2
If a limit as X approaches 3+ = 4 and the limit as X approaches 3- = -4 does the limit exist?
no the limit does not exist, must agree on both sides
definition of a vertical asymptote
limit as x approaches a of f(x) = positive or negative infinity
a non zero number/ 0 =
+- infinity
2.5
what are the three points needed for a function to be continunous?
- f(a) is defined
- limit as x approaches a = f(a)
- the limit as x approaches a exsits (cant be one sided)
domain of polynomials
All reals
domain of a rational function
where the denominator does not = 0
domain of trig functions
all reals
domain of exponential functions
ex: e^x, 2^x, 7^x
all reals
domain of logarithmic functions
ex: ln, log
(0, infinity)
2.5
Intermediate value theorum
on a closed interal from [a,b] a continuous function will have a value for f(c) somewhere bc/ it is continuous
domain of odd and even root functions
odd (all reals)
even [o, infinity)
2.7
definition of a derivative
- lim as h approaches 0 f(a+h) -f(a)/h = f(x)-f(a)/x-a
- the slope of the tangent line
2.8
What does it mean to be differentiable?
if a function is differentibale at a point that means the derivative exists there and it is continuous
not differentiable where the derivative cannot exist
2.8
True or false? if x= a is cont then f is differentiable at x=a
false, you cant swap the sentence and still have it be true
3.1
derivative of b^x=
b^x *ln(b)
3.5
Implicit differentation
taking the derivative on both sides with the assumption that y is a function/ y= f(x)
3.10
differentials
dy=f’(x)* dx
use this for measurement error problems
3.10
surface area of a cube
6a^2
4.1
extreme value therorum
if f(x) is continous on a closed interval then f(x0 has an absolute max and min on that interval
on closed intervals you can choose the ends
4.1
the point is a critical point if…
f’(c) = 0 or f’(c) DNE
4.1
if f(x) has a local max or local min at x=c then f’(c) = 0 or f’(c) DNE
fermats theorum
CRITICAL POINT DOES NOT =. LOCAL MIN/MAX
4.2
Rolles theorum
- f(a) =f(b)
- f(x) is cont on (a,b)
- f(x) is differentiable on (a,b)
- f’(c) = 0
use to prove at most one solution
4.2
Mean Value Theorum
- f is cont on (a,b)
- f is differentiable on (a,b)
- f’(c) = f(b)-f(a)/b-a
average rate of change from a to b
4.3
Inflection points
where concavity changes from up or down or vice versa
4.4
what should you do if you get the indeterminate form 0/0 or infinity/infinity
take the derivative of the top and bottom separatley
L’hopital
4.4
What should you do if you get the indeterminate form 0 * infinity
f(x)/ (1/g(x))
4.4
what should you do if you get the indeterminate form infinity - infinity
find a common denominator or factor
4.4
what should you do if you get the indeterminate form of 0^0, infinity^0 or 1^infinity
e^ln(fx) and bring the limit up
4.5
how do you find the x and y intercepts of a function?
x int: 0= f(x)
y-int: y= f(0)
4.5
how do you determine symmetry for a function?
odd symmetry: f(-x)= -(f(x))
even: f(-x)=f(x)
4.9
Never forget to add what after the most general antiderivative?
+ C
Derivative of a log
log(x) base a
1/x*ln(a)
5.1
What does it mean to use right end points?
to use the right most values, leaving out the lowest value
- ex: of your points are 0, 1/2, 1 and 3/2 for right most you wouldnt use 0
5.2
What does it mean to use leftmost end points?
use the left most values and the lowest value
- ex: if your values are 0,2,4,6 then you wouldnt use 6
5.1
How do you find your change in x?
[a,b] w/ n number of rectangles
b-a/n
5.3
What does the Fundamental theorum of calculus let you do?
if f is cont on [a,b] then the intetral from a-b on f(x) is any antiderivatve F(b)- F(a)
definition of a absolute maximum
x=b is a local max of f(x) if f(b)> f(x) for any x near b
definition of an absolute minimum
x=a is an absolute minimum if f(a)<f(x) for all x in the domain of f
5.3
FTC pt 2
if f(x) is cont then the integral from a to b is = any antiderivative F(b)-F(a)
5.5
What should you never do when using the u sub rule for integrals?
u=x
it wont do anything
