Transcript When you see…

When you see…
Find the zeros
You think…
To find the zeros...
Set function = 0
Factor or use quadratic equation if quadratic.
Graph to find zeros on calculator.
When you see…
Find equation of the line
tangent to f(x) at (a, b)
You think…
Equation of the tangent line
Find f   x 
Let m  f   a 
Use  y  b   m  x  a 
When you see…
Find equation of the line
normal to f(x) at (a, b)
You think…
Equation of the normal line
Find f   x 
1
Let m 
f a
Use  y  b   m  x  a 
When you see…
Show that f(x) is even
You think…
Even function
Show that
f x  f  x
The graph of f  x  is
symmetrical about the y  axis,
for example, f  x   cos x
When you see…
Show that f(x) is odd
You think…
Odd function
Show that
f x   f  x
The function f  x  is
symmetrical about the origin,
for example, f  x   sin x
When you see…
Find the interval where
f(x) is increasing
You think…
f(x) increasing
Find f   x 
Find values of x where f   x   0 or D.N.E.
Use sign chart to evaluate changes in f   x 
Cite interval(s) where f   x   0
Either  a, b  or a  x  b
When you see…
Find the interval where the
slope of f (x) is increasing
You think…
Slope of f (x) is increasing
Find f   x 
 slope of f   x  
Find values of x where f   x   0 or D.N.E.
Use sign chart to evaluate changes in f   x 
Cite interval(s) where f   x   0
Either  a, b  or a  x  b
When you see…
Find the minimum value
of a function
You think…
Minimum value of a function
Find f   x 
Find values of x where f   x   0 or D.N.E.
Use sign chart to find where f   x  changes from  to +
Evaluate f  x  at these points and at the endpoints
Select the least value of f  x 
When you see…
Find the minimum slope
of a function
You think…
Minimum slope of a function
Find f   x 
Find values of x where f   x   0 or D.N.E.
Use sign chart to find where f   x  changes from  to +
Evaluate f   x  at these points and at the endpoints
Select the least value of f   x 
When you see…
Find critical numbers
You think…
Find critical numbers
Find f   x 
Find values of x where f   x   0 or D.N.E.
These values of x are the critical values
When you see…
Find inflection points
You think…
Find inflection points
Find f   x 
Find values of x where f   x   0 or D.N.E.
Use sign chart to find where f   x  changes
from  to + or + to 
Evaluate y  f  x  at these values of x.
The found points  x, y  are the inflection points
When you see…
Show that lim f ( x ) exists
x a
You think…
Show lim f ( x ) exists
x a
Show that
lim f  x  = lim+ f
x a
x a
x
When you see…
Show that f(x) is
continuous at x = a
You think…
.
f(x) is continuous
Show that
1 lim f  x  exists, previous slides
x a
2  Find f  a 
3 If lim f  x   f  a  , f  x  is continuous at x  a
x a
When you see…
Find vertical
asymptotes of f(x)
You think…
Find vertical asymptotes of f(x)
Express f  x  as a rational function
Eliminate any removable discontinuities
Find the values of x for which the denominator  0
These values of x are the locations of the vertical
asymptotes
When you see…
Find horizontal
asymptotes of f(x)
You think…
Find horizontal asymptotes of f(x)
Find lim f  x  and lim f  x 
x 
x 
If either or both limits are a finite number(s),
then y  lim f  x  and/or y  lim f  x 
x 
are/is horizontal asymptote(s)
x 
When you see…
Find the average rate of
change of f(x) at [a, b]
You think…
Average rate of change of f(x)
Average rate of change of f  x  from a to b
m
f b  f  a 
ba
When you see…
Find the instantaneous
rate of change of f(x)
at a
You think…
Instantaneous rate of change of f(x)
Find f   x  and
evaluate at x  a
When you see…
Find the average value
of f  x  on  a , b 
You think…
Average value of the function
f  x average
1 b

a f  x  dx
ba
When you see…
Find the absolute
minimum of f(x) on [a, b]
You think…
Find the absolute minimum of f(x)
1 Find f   x 
2  Find all critical points, values of x such that f   x   0 or D.N.E.
3 Find all critical points where f   x  changes from  to +
4  Find f  x  for these points as well as f  a  and f  b 
5  Select the least value found
When you see…
Show that a piecewise
function is differentiable
at the point a where the
function rule splits
You think…
Show a piecewise function is
differentiable at x=a
Show that
lim f  x   lim f  x 
xa 
x a 
then show that
lim f   x   lim f   x 
xa

x a

When you see…
Given s(t) (position
function), find v(t)
You think…
Given position s(t), find v(t)
Find v t  
s  t 
When you see…
Given v(t), find how far a
particle travels on [a, b]
You think…
Given v(t), find how far a particle
travels on [a,b]
distance 
b
a
v  t  dt
When you see…
Find the average
velocity of a particle
on [a, b]
You think…
Find the average velocity on [a,b]
s b  s  a 
1 b
average velocity 
a v  t  dt 
ba
ba
When you see…
Given v(t), determine if a
particle is speeding up at
t=a
You think…
Given v(t), determine if the particle is
speeding up at t=a
Find v  a  and a  a 
Multiply their signs
If positive, speed is increasing,
If negative, speed is decreasing
When you see…
Given v(t) and s(0),
find s(t)
You think…
Given v(t) and s(0), find s(t)
 v  t  dt  s  t   s  0 
t
0
s  t   s  0    v  t  dt
t
0
When you see…
Show that Rolle’s
Theorem holds on [a, b]
You think…
Show that Rolle’s Theorem holds on
[a,b]
Show that f is continuous and differentiable
on the interval
a  f 
b , then find some c in a , b 
If f 
such that f c   0.



When you see…
Show that the Mean
Value Theorem holds
on [a, b]
You think…
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
.
When you see…
Find the domain
of f(x)
You think…
Find the domain of f(x)
Assume domain is ,.
Domain restrictions: Zero denominators,
 of negative numbers,
Square roots
Log or ln of non-positive numbers
When you see…
Find the range
of f(x) on [a, b]
You think…
Find the range of f(x) on [a,b]
Use max/min techniques to find relative
max/mins.
Then examine f a , f b 

When you see…
Find the range
of f(x) on ( , )
You think…
Find the range of f(x) on    ,  
Use max/min techniques to find relative
max/mins.
i m f x.
Then examine xl

When you see…
Find f ’(x) by definition
You think…
Find f ‘( x) by definition
f x   lim
h0
f x   lim
xa
f x  h   f  x 
h
f x   f a 
x  a
or
When you see…
Find the derivative of
the inverse of f(x) at x = a
You think…
Derivative of the inverse of f(x) at x=a
 x  g  x
f b   a
f b   c
f
1
1
g a  
c
When you see…
y is increasing
proportionally to y
You think…
.
y is increasing proportionally to y
dy
 ky
dx
dy
  kdx

y
y  Ce
kt
When you see…
Find the line x = c that
divides the area under
f(x) on [a, b] into two
equal areas
You think…
Find the x=c so the area under f(x) on
[a, b] is divided equally
c
a
f  x  dx 
b
c
f  x  dx
evaluate the integrals and solve for c
When you see…
d x
a f  t  dt 
dx
You think…
Fundamental Theorem
Part II of the Fundamental Theorem of Calculus
d x
a f  t  dx  f  x 
dx
When you see…
d u
a f  t  dt 
dx
You think…
Fundamental Theorem, again
Part II of the Fundamental Theorem of Calculus
d u
du
a f  t  dt  f  u 
dx
dx
When you see…
The rate of change of
population is …
You think…
Rate of change of a population
dP
 ...
dt
When you see…
The line y = mx + c is
tangent to f(x) at (a, b)
You think…
.
y = mx+b is tangent to f(x) at (a,b)
Two relationships are true:
f a  m
f a  y a  b
When you see…
Find area using left
Riemann sums
You think…
Area using left Riemann sums
ba
Given equal intervals of h 
and f  a   y0 and f  b   yn
n
b
a f  x  dx  h  y0  y1  y2    yn 1 
For unequal intervals, use
b
a
f  x  dx   x1  y0   x2  y1   x3  y2     xn  yn 1  
When you see…
Find area using right
Riemann sums
You think…
Area using right Riemann sums
ba
Given equal intervals of h 
and f  a   y0 and f  b   yn
n
b
a f  x  dx  h  y1  y2  y3    yn 
For unequal intervals, use
b
a
f  x  dx   x1  y1   x2  y2   x3  y3     xn  yn  
When you see…
Find area using
midpoint rectangles
You think…
Area using midpoint rectangles
Typically done using a table of values
Be sure to use only values from the table
If seven pairs of data are given, then only three
midpoints are possible
When you see…
Find area using
trapezoids
You think…
Area using trapezoids
h
 f  x  dx   y0  2 y1  2 y2    2 yn1  yn 
2
a
b
ba
This only applies for equal intervals where h 
n
For unequal intervals, use
b
1
 f  x  dx   x1  y0  y1   x2  y1  y2     xn  yn 1  yn  
2
a
When you see…
Solve the differential
equation …
You think…
Solve the differential equation...
Separate the variables –
x on one side, y on the other.
The dx and dy must both be in the
numerator.
Integrate both sides, add C to the right.
Use initial condition to evaluate C
Write the answer
When you see…
Meaning of
 f  t  dt
x
a
You think…
Meaning of the integral of f(t) from a to x
This is the accumulation function,
it gives the accumulated change in
F  t  on the interval [a, x].
When you see…
Given a base, cross
sections perpendicular to
the x-axis that are
squares
You think…
Square cross sections perpendicular to
the x-axis
The base of the solid is typically the region between
two curves, f  x  and g  x  , where f  x  > g  x  .
The volume of the solid is:
V
b
a
 f  x  - g  x   dx
2
When you see…
Given a base, cross
sections perpendicular to
the x-axis that are semicircles
You think…
Semicircular cross sections
perpendicular to the x-axis
The base of the solid is typically the region between
two curves, f  x  and g  x  , where f  x  > g  x  .
The volume of the solid is:
V

2
 f  x - g  x 

 dx
2


2
b
a
When you see…
Find where the tangent
line to f(x) is horizontal
You think…
Horizontal tangent line
Write f   x  as a rational function
Set the numerator equal to zero
Solve for the unknown variable
Verify the value is within the range or domain
When you see…
Find where the tangent
line to f(x) is vertical
You think…
Vertical tangent line to f(x)
Write f   x  as a rational function
Set the denominator equal to zero
Solve for the unknown variable
Verify the value is within the range or domain
When you see…
Find the minimum
acceleration given v(t)
on [a, b]
You think…
Given v(t), find minimum acceleration
Find a  t   v  t 
Find j  t   a   t 
Equate j  t  to zero
Find which value(s) of t have j  t  changing from  to +
Find a  t  at these values of t and compare to a (t ) at
the end points and select the least value
When you see…
Approximate the value
f(0.1) of by using the
tangent line to f at x = 0
You think…
Approximate f(0.1) using tangent line
to f(x) at x = 0
Find f  0  and f   0 
Find the equation for the tangent line using: y  f  0   f   0  x
y  0.1  f  0   f   0  0.1
When you see…
Given the value of F(a)
and the fact that the
anti-derivative of f is F,
find F(b)
You think…
Given F(a) and the that the
anti-derivative of f is F, find F(b)
Usually involves a function that is not readily integrated.
Use the Fundamental Theorem of Calculus
b
a
f  t  dt  F  b   F  a   F  b   F  a   ab f  t  dt
Solve using numerical integration on calculator.
When you see…
Find the derivative of
f(g(x))
You think…
Find the derivative of f(g(x))
d
f  g  x   f  g  x  g x 
dx
When you see…
Given  f  x  dx , find   f  x   k  dx
b
a
b
a
You think…
Given area under a curve and vertical
translation, find the area under the new
curve
  f  x   k  dx   f  x  dx  k  dx
b
a
b
a
b
a
When you see…
Given a graph of f '( x)
find where f(x) is
increasing
You think…
Given a graph of f ‘(x) , find where f(x) is
increasing
Examine the graph and identify all intervals on which the graph
has positive values
When you see…
Given v(t) and s(0), find the
greatest distance from the
origin of a particle on [a, b]
You think…
Given v(t) and s(0), find the greatest distance from
the origin of a particle on [a, b]
Find v  t 
Find each time t for which v  t   0
Evaluate s  t   s  0   0t v  t  dt for each of these times
Compare to initial value and select the greatest value
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
[t1 , t2 ] , find
a) the amount of water in
the tank at m minutes
You think…
Amount of water in the tank at t minutes
g
t2
t1
 F  t   E  t   dt
b) the rate the water
amount is changing
at m
You think…
Rate the amount of water is
changing at t = m
d m
0  F  t   E  t   dt  F  m   E  m 
dt
c) the time when the
water is at a minimum
You think…
The time when the water is at a minimum
F  m   E  m  = 0,
Compare amounts at each time m and the endpoints.
Select the time at which the amount is least.
When you see…
Given a chart of x and f(x)
on selected values between
a and b, estimate f '( x) where
c is between a and b.
You think…
Estimating f ′ (c)
Straddle c, using a value k greater
than c and a value h less than c.
so
f k   f h 
f c  
k  h
When you see…
dy
Given dx , draw a
slope field
You think…
Draw a slope field of dy/dx
Use the given points
dy
dx
Plug them into
,
drawing little lines with the
indicated slopes at the points.
When you see…
Find the area between
curves f(x) and g(x) on
[a,b]
You think…
Area between f(x) and g(x) on [a,b]
A    f  x   g  x   dx ,
b
a
assuming f (x) > g(x)
When you see…
Find the volume if the
area between the curves
f(x) and g(x) is
rotated about the x-axis
You think…
Volume generated by rotating area between
f(x) and g(x) about the x-axis
V 
b
a
 f  x     g  x    dx


2
assuming f (x) > g(x).
2