chapter 4 (applications of derivatives) Flashcards
optimization problem steps
- identify 2 formulas
- solve for one of them and plug it into the other.
- take the derivative of the new function
- find the critical points
- this is the maximum or minimum
how to approximate F(x) given a
- plug in a to find its y coordinate
- take the derivative of the equation and plug in a to find the tangent line slope
- use the point slope formula and plug in x. this should allow you to get to y
rolles theorem
if the function is continuous and differential and f(a) and f(b) are equal then it is guaranteed there is a point with the slope of 0
finding guaranteed points with rolls theorem
- determine if it will work
- derive the equation and set equal to zero.
ta da
mean value thorem
if the function is continuous and differentiable from a to b then it is certain that there exists a point within the interval a to b that has a slope equal to the average slope
find values with mean value theorem
- see if it will work
- find the mean slope( y2-y1/x2-x1)
- derive and set equal to
lopitals o/o and inf/inf
- derive the top and bottom separately
2. take the limit of the top over the limit of the bottom
lopitals 0*inf
- make it into a fraction by placing one of the sides on the bottom to the -1 power.
- then treat like a normal lopitals
lopitals 0^inf inf^0 etc
- take the ln of BOTH sides
- use log rules to make it multiplication
- use the multiplication to find the limit.
- undo the log on the other side by taking e^ what you get
power rule for anti-derivatives
(x^p+1)/p+1
d/dx(sin ax)
acos ax
d/dx cos ax
-asin ax
d/dx tan ax
asec^2(ax)
d/dx cot ax
-acsc^2 ax
d/dx sec ax
a sec(ax) tan(ax)
d/dx csc ax
-a csc(ax) cot(ax)
int cos ax
(1/a)sin ax +C
int sin ax
-(1/a)cos ax +C
int sec^2 ax
(1/a)tan ax +C
int csc^2 ax
-(1/a)cot ax +C
int sec(ax) tan(ax)
(1/a)sec ax +C
int csc(ax) cot(ax)
-(1/a) csc ax +C
d/dx e^ax
ae^ax +C
d/dx b^x
(b^x)lnb +C
d/dx lnx
1/x +C
d/dx sin^-1 (x/a)
1/(sqrt(a^2-x^2)) +C
d/dx cos^-1 (x/a)
-1/(sqrt(a^2-x^2)) +C
d/dx tan^-1 (x/a)
a/(a^2+x^2) +C
d/dx cot^-1 (x/a)
-a/(a^2+x^2) +C
d/dx sec^-1 (x/a)
a/(xsqrt(x^2-a^2)) +C
d/dx csc^-1 (x/a)
-a/(xsqrt(x^2-a^2)) +C
int e^ax
(1/a)e^ax +C
int (b^x)
(1/lnb)b^x +C
int (1/x)
ln x +C
int 1/(sqrt(a^2-x^2))
sin^-1 (x/a) +C
int 1/(a^2+x^2)
(1/a) tan^-1 (x/a) +C
int 1/(xsqrt(x^2-a^2))
(1/a) sec^-1 (x/a) +C
how to find the integral equation at a point
solve for the indefinite integral and then set equal to the point to find c
how to find the critical point of a function
find teh derivative and set equal to zero
for a local extrema to exist what must the derivative at that point be
zero
a function that is continuous from a to b has what
a maximum and minimum on that interval
how to tell if a function is increasing or decreasing.
take the derivative and identify your interval.
if the function is negative in that interval then it is decreasing
if positive then it is increasing
if f’ goes from negative to positive at c
local minimum
if f’ goes from positive to negative at c
local maximum
inflection points
when the function switches fro concave up to concave down. found by setting the second derivative to 0
graphing steps
find both derivatives. 1. window 2. critical points and values 3. inflection point and values 4. x and y intercepts 5 asymptotes if any
if f has a slope of 0 what is the f’ graph doing
passing through 0
when the slope of the f’’ graph passes through zero, what is this
an inflection point
