Exam 3 Flashcards
Extreme Value Theorem
a function that is continuous on a closed interval [a,b] has an absolute maximum value and an absolute minimum value on that interval
Anti Derivative Rules
power rule
sum and multiplication rules
derivatives of
cosx
sinx
tanx
secx
cotx
cscx
-sin x
cos x
sec^2 x
sec x tan x
-csc^2 x
-csc x cot x
derivatives of
inverse
sin x
tan x
sec x
sin -1 x= 1/ √1-u^2 x du/dx
tan -1 x= 1/1+u^2 x du/dx
sec -1 x=1/ |u|√u^2-1 x du/dx
how to find local minimum
first derivative test
find critical points by setting first derivative to 0.
Plug in these points into the original function to find the maximum and minimum values
how to find absolute maximum
first derivative test
find critical points by setting first derivative to 0.
Plug in these points and endpoints into the original function to find the maximum and minimum values
steps to draw curve
- find the x and y intercept
for x set to zero and solve for y set x=0 - Find all asymptotes (only for rational fx)
vertical asymptote if denominator=0
horizontal asymptote if n<m>m no HA if n=m y= n/m if n is one more than m then there is slant asymptote (use long division)</m> - 1st derivative test
- 2nd derivative test
- make sign table
- list all possible points
- graph
how to find critical points and where function is increasing and decreasing
set first derivative to 0 to find the critical points
create sign table to see if function is increasing or decreasing
plug in critical points into first derivative to see if increasing or decreasing
how to find intervals of concavity
use second derivative test
set 2nd derivative to 0 find possible points of inflection
use these points to create sign table and plug in values into second derivative to see if concave up or down, if zero this is inflection point
state the mean value theorem
if f is continuous on the closed interval [a,b] and differentiable on (a,b), then there is at least one point c in (alb) such that
f(b)-f(a)/b-a= f’(c)
to verify the mean value theorem plug in alb into function to find the slope
equal the derivative of the function to the slope to get the points where the derivative is equal to the slope.
steps to solve optimization problem
identify constraint
create formula
differentiate constraint to one variable to plug into the formula
identify intervals of interest such as critical points and end points
use calculus to find absolute max and min (first derivative test)
plug in values to OG function to find absolute max/min
what are formulas of area for square circle rectangle and triangle
volume of box and cylinder
A=a^2 square
A=wxl rectange
A= pi r^2 circle
A= bxh/2 triangle
V= L x W x H box
V= pi r^2 H cylinder
pythagorean theorem
a^2+b^2=c^2
Formula for linear approximation
L(x)=f(a)+f’(a)(x-a)
identify the function, choose an a
plug a into the formula
then plug in given value
how to use L hopitals rule and on what forms
can be used for 0/0 or infinity/infinity
forms infinity-infinity and infinity times 0 must be manipulated to get the above forms in order to use
f(x)/g(x)=f’(x)/g’(x)
for f(x)^g(x)= 0^0,infinity^0, 1^infinity
try taking the limit of e^ln of the function
