Find the zeros - Henry County School...
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Transcript of Find the zeros - Henry County School...
To find the zeros...
Set function = 0
Factor or use quadratic equation if quadratic.
Graph to find zeros on calculator.
Slope of f (x) is increasing
Find the derivative of f ’(x) = f “ (x)
Set numerator and denominator = 0
to find critical points
Make sign chart of f “ (x)
Determine where it is positive
Minimum value of a function
Make a sign chart of f ‘( x)
Find all relative minimums
Plug those values into f (x)
Choose the smallest
Minimum slope of a function
Make a sign chart of f ’(x) = f ” (x)
Find all the relative minimums
Plug those back into f ‘ (x )
Choose the smallest
Find inflection points
Express f “ (x) as a fraction
Set numerator and denominator = 0
Make a sign chart of f “ (x)
Find where it changes sign
( + to - ) or ( - to + )
Find the absolute minimum of f(x)
a) Make a sign chart of f ’(x)
b) Find all relative minima (or maxima)
c) Plug those values into f (x)
d) Find f (a) and f (b) -----endpoints!
e) Choose the smallest (largest for max)
of c) and d)
You think…
When you see…
Show that a piecewise
function is differentiable
at the point a where the
function rule splits
Show a piecewise function is
differentiable at x=a
First, be sure that the function is continuous at
ax .
Take the derivative of each piece and show that
xfxfaxax
limlim
Given v(t), determine if the particle is
speeding up at t=a
Find v (k) and a (k).
Multiply their signs.
If positive, the particle is speeding up.
If negative, the particle is slowing down
Show that Rolle’s Theorem holds on
[a,b]
Show that f is continuous and differentiable
on the interval
If
f a f b , then find some c in
a , b
such that
f c 0.
Show that the MVT holds on [a,b]
Show that f is continuous and differentiable
on the interval.
Then find some c such that
f c f b f a
b a.
Find the domain of f(x)
Assume domain is
, .
Domain restrictions: non-zero denominators,
Square root of non negative numbers,
Log or ln of positive numbers
Find the range of f(x) on [a,b]
Use max/min techniques to find relative
max/mins.
Then examine
f a , f b
Use max/min techniques to find relative
max/mins.
Then examine
limx
f x .
Find the range of f(x) on ,
Derivative of the inverse of f(x) at x=a
If x=a and y=b on the inverse, then
find the derivative of f(x) at y=b and
find its reciprocal
You think…
When you see…
Find the line x = c that
divides the area under
f(x) on [a, b] into two
equal areas
.y = mx+b is tangent to f(x) at (a,b)
Two relationships are true.
The two functions share the same
slope ( xfm )
and share the same y value at 1x.
Area using midpoint rectangles
Typically done with a table of values.
Be sure to use only values that are
given.
If you are given 6 sets of points, you can
only do 3 midpoint rectangles.
Area using trapezoids
nn xxxxxbase
A 1210 2...222
This formula only works when the base is the
same.
If not, you have to do individual trapezoids
Solve the differential equation...
Separate the variables –
x on one side, y on the other.
The dx and dy must all be upstairs..
Meaning of the integral of f(t) from a to x
The accumulation function –
accumulated area under the
function xf
starting at some constant a
and ending at x
Given v(t), find minimum acceleration
First find the acceleration tvta
Then minimize the acceleration by
examining ta .
Approximate f(0.1) using tangent line
to f(x) at x = 0
Find the equation of the tangent line to f
using 11 xxmyy
where 0fm and the point is 0,0 f .
Then plug in 0.1 into this line.
Be sure to use an approximation sign.
You think…
When you see…
Given the value of F(a)
and the fact that the
anti-derivative of f is F,
find F(b)
Given F(a) and the that the
anti-derivative of f is F, find F(b)
Usually, this problem contains an antiderivative
you cannot take. Utilize the fact that if xF
is the antiderivative of f,
then aFbFdxxF
b
a
.
Solve for bF using the calculator
to find the definite integral
Given area under a curve and vertical
shift, find the new area under the curve
dxkdxxfdxkxf
b
a
b
a
b
a
Given a graph of f ‘(x) , find where f(x) is
increasing
Make a sign chart of xf
Determine where xf is positive
You think…
When you see…
Given v(t) and s(0), find the
greatest distance from the
origin of a particle on [a, b]
Given v(t) and s(0), find the greatest distance from
the origin of a particle on [a, b]
Generate a sign chart of tv to find
turning points.
Integrate tv using 0s to find the
constant to find ts .
Find s(all turning points) which will give
you the distance from your starting point.
Adjust for the origin.
When you see…
Given a water tank with g gallons
initially being filled at the rate of
F(t) gallons/min and emptied at
the rate of E(t) gallons/min on
, find 1 2[ , ]t t
You think…
When you see…
Given a chart of x and f(x)
on selected values between
a and b, estimate where
c is between a and b.
'( )f x
Draw a slope field of dy/dx
Use the given points
Plug them into dx
dy,
drawing little lines with the
indicated slopes at the points.
You think…
When you see…
Find the volume if the
region between the
curve f(x) and a given
horizontal line (y=k) is
rotated about that line
Volume generated by rotating region
between f(x) and y=k about the line y=k
creating a disk
kxfand
horizontalisrotationofaxisthethatgiven
dxkxfV
b
a
)(
))(( 2
You think…
When you see…
Find the volume if the
region between the
curves f(x) and g(x) is
rotated about a
horizontal axis of rotation
Volume generated by rotating region
between f(x) and g(x) about a given
horizontal axis of rotation creating a washer
)()(
]))(())([( 22
xgxfand
horizontalisrotationofaxisthethatgiven
dxaxisxgaxisxfV
b
a
You think…
When you see…
Find the volume of a
known cross-section cut
perpendicular to the x-
axis with a given side on
the region specified by
f(x) and g(x)
Volume of known cross-sections cut
perpendicular to the x-axis on region
between f(x) and g(x) where typically
f(x)-g(x) is the base of the figure
b
adxbasebasetheondiametersemicircle
b
adxbasebasetheonhyprtisosceles
b
adxbasebasetheonlegrtisosceles
b
adxbasebasetheonlequilatera
dxb
abasebasetheonsquares
2)(8
:,
2)(4
1:,
2)(2
1:,
2)(4
3:
2)(: