Transcript Average Value of a Function

Calculus highlights
for AP/final review
Evaluating Limits
*Common methods for evaluating finite limits analytically (that
don’t work with direct substitution):
1) Factor:
x 2  a2
lim
 lim x  a   2a
xa x  a
xa
2) Rationalize:
2x  3
lim

2
x3/2 12x  16x  3
xa  a
1
1
lim
 lim

x0
x0
x
xa  a 2 a
2x  5  5
lim

x0
x
1
lim
lim
x0
x
x0
3) Clearing Fractions:
2
1

(x  a) a
2

(x  3)
3
x
1
1
 lim

x0 a(x  a)
2a

f (x)
f '(x)
 lim
4) L’Hopital’s Rule: lim
xc g(x)
xc g'(x)
(for indeterminate forms of continuous/differentiable
functions)
sin x
lim

x0
x
1 cos x
lim

x0
x
limits at infinity and
infinite limits
*Evaluate limits of rational functions as x approaches infinity by
using the rules of horizontal asymptotes:
ax  ...
lim m
 0,n  m
x bx  ...
n
ax  ... a
lim m
 ,n  m
x bx  ...
b
n
ax  ...
lim m
 ,n  m
x bx  ...
n
Continuity, differentiability, and limits
Limits: lim f (x) exists if
xc
lim f (x)  lim f (x)
xc
Continuity: f(x) is continuous if:
1)
lim f (x)exists
xc
2) f(c) is defined
3)
lim f (x)  f (c)
xc
xc
f (x  x)  f (x)
Differentiability: f(x) is differentiable if f '(x)  lim
x0
x
exists.
*If f(x) is differentiable, then it is continuous.
*f(x) is not differentiable at any sharp turns, i.e.,
is not differentiable at x=0.
f (x)  x
Which of the following are true for the following graph.
I.
lim f (x)exists for all values of c in the given domain
xc
II. f(x) is continuous on the given domain
III. f(x) is differentiable on the given domain
IV. f(c) is defined for all values of c in the given domain
Derivatives
*Power Rule:
d n
n1
 x   nx
dx
d
*Product Rule:  f (x)g(x)  f (x)g'(x)  g(x) f '(x)
dx


d
f
(x)
g(x)
f
'(x)

f
(x)g'(x)
*Quotient Rule:

2


dx  g(x) 
g(x)
*Chain Rule:
d
f (g(x))  f '(g(x))g'(x)

dx
*Trigonometric Functions:
d
sin x   cos x

dx
d
cos x    sin x

dx
d
2
tan x   sec x

dx
d
csc x    csc x cot x

dx
d
sec x   sec x cot x

dx
d
2
cot x    csc x

dx
*Exponential and Logarithmic Functions:
d x
x
 e   e
dx
d
1
ln x  

dx
x
*Implicit Differentiation: When differentiating a term of a
function that contains y, multiply by y’ or
dy
dx
Find
dy
dx
2 x
y
 3x  1
3x  1
y  cos 3x sin 4 x
4
2

y  tan (5x )
3
2
3xy  y  4x  6
2
2
y  3x e
2
x
y  ln x 
3/2
tangent lines
dy
*The derivative
or f '(x) represents the slope of the graph of
dx
f(x) at any given value of x.
*The equation of the tangent line to the graph of f(x) at a given
point (x, y) is found by finding f '(x) at that given value, then
plugging it into the point-slope equation for a line.
Find the linear approximation of f(0.2) at x=0 for
f (x)  3x 1
2
3
average rate of change and instantaneous
rate of change
*Average Rate of Change/Approximate Rate of Change:
f (b)  f (a)
ba
(slope formula)
*Instantaneous Rate of Change/Exact Rate of Change:
f '(x)
Intermediate value theorem, rolle’s
theorem, and mean value theorem
*Intermediate Value Theorem: If f(x) is continuous on [a, b] and
f(a)<k<f(b), then there exists at least one value of c in [a, b] such
that f(c)=k.
*Rolle’s Theorem: If f(x) is continuous and differentiable, and
f(a)=f(b), then there exists at least one value of c in (a, b) such that
f '(c)  0
*Mean Value Theorem: If f(x) is continuous and differentiable
then there exists a value c in (a, b) such that
f (b)  f (a)
f '(c) 
ba
related rates
*Process for solving related rates problems:
1) Write an equation to represent the problem.
2) Find the derivative (implicitly, with respect to t) for the
equation.
3) Plug in all known variables, including given rates.
4) Solve for the unknown.
Graphical analysis
*Increasing/Decreasing Behavior:
f(x) is increasing if
f '(x)  0
f(x) is decreasing if
f '(x)  0
*Concavity/Points of Inflection:
f(x) is concave upward if
f "(x)  0
f(x) is concave downward if
f(x) has a point of inflection if
f "(x)  0
f "(x)
changes sign
*Absolute/Global Extrema on [a, b]:
1) Find critical numbers (where
f '(x)  or
0
f '(x)
is undefined)
2) Plug in critical numbers and endpoints into f(x)
3) Find smallest y-value (absolute/global minimum)
Find largest y-value (absolute/global maximum)
*Relative/Local Extrema:
First Derivative Test:
1) Find critical numbers (where f '(x)  0 or f '(x) is
undefined)
2) When
f '(x) changes from positive to negative at a
critical number, there is a relative maximum at (x, f(x)).
When f '(x) changes from negative to positive at a critical
number, there is a relative minimum at (x, f(x)).
Second Derivative Test:
1) Find critical numbers
2) f(x) has a relative maximum if f "(x)  0 at a critical
number.
f(x) has a relative minimum if
number .
f "(x)  0 at a critical
For f (x)  2x 3  2x 2 12x  5 , find
(a) the intervals on which f(x) is increasing
or decreasing
(b) the intervals on which f(x) is concave
upward or concave downward
(c) the points of inflection of f(x)
(d) any relative extrema of f(x)
(e) the absolute extrema of f(x) on [-2, 2]
Optimization
*Process for solving optimization problems:
1) Draw and label a sketch, if applicable.
2) Write a primary equation to be optimized.
3) Use any secondary equations to rewrite the primary
equation in terms of one variable.
4) Apply First Derivative Test, Second Derivative Test, or
absolute extrema test to finding the maximum or minimum
value.
Riemann sums and
trapezoidal sums
*Riemann Sum: Approximates the area under a curve with a
finite number of rectangles that intercept the graph at their right
endpoint, left endpoint, or midpoint.
*Trapezoidal Sum: Approximates the area under a curve with a
finite number of trapezoids.
Given the following table of values, find R(3), L(3),
M(3), and T(3).
x
f(x)
-1
6
2
-4
 
7
2
10
0
Given f (x)  cos x 2 , find R(4), L(4), M(4), and T(4)
[0,2 ]
on the interval
Integrals
*Definition of a Definite Integral: When the limit as the number of
rectangles approaches infinity of a Riemann Sum is found, this
represents the area under the curve bound by the x-axis, or the
definite integral of the function on a given interval. A definite
integral is also used to find the “total amount accumulated” of
something given its rate of change.
n1
x
C
*Power Rule:  x dx 
n 1
n
*U-Substitution:

f (u)du  F(u)  C
*Exponential and Logarithmic Functions:
e
x
dx  e  C
x
1
 x dx  ln x  C
*Trigonometric Functions:
 sin x dx   cos x  C
 cos x dx  sin x  C
 tan x dx   ln cos x  C
 csc x dx   ln csc x  cot x  C
 sec x dx  ln sec x  tan x  C
 cot x dx  ln sin x  C
 sec x dx  tan x  C
 csc x cot x dx   csc x  C
 sec x tan x dx  sec x  C
 csc x dx   cot x  C
2
2
 x (5  3x )dx 
2
3
 tan 2x sec (2x)dx 
2

ln x 
5
3x

e
x
x
dx 
dx 
*Special Properties of Integrals:
a
 f (x)dx  0
a
a
b
b
a
 f (x)dx    f (x)dx
Fundamental theorems of
calculus
*First Fundamental Theorem of Calculus:
b
 f (x)dx  F(b)  F(a)
a
b
-or-
 f '(x)dx  f (b)  f (a)
a
b
-or- (with U-Substitution)
g(b)
 f (g(x))g'(x)dx  
a
g(a)
f (u)du
*Second Fundamental Theorem of Calculus:

d 
  f (t)dt   f (x)
dx  a

x
-or-
d 

dx 
g( x )

a

f (t)dt   f (g(x))g'(x)

6
Given
 f (x)dx  22 and F(3)=-10, find F(6).
3
e
x2
 lnt dt 
5
average value of a function
*Average Value of a Function:
b
1
f (x)dx

ba a
The following graph shows the number of cellphone
sales an AT&T representative makes at each hour of a
given workday. Find the average number of sales the
representative makes during [1, 8].
particle motion
*Position, Velocity, and Acceleration:
x '(t)  v(t)
v'(t)  a(t)
x"(t)  a(t)
*Speed:
 v(t)dt  x(t)  C
a(t)dt

v(t)

C

  a(t)dt  x(t)  C
speed  v(t)
*Total Distance Traveled:

v(t) dt
*A particle moves left when v(t)<0 and right when v(t)>0.
*A particle changes direction when v(t) changes sign.
*A particle stops when v(t)=0.
*A particle speeds up when v(t) and a(t) are the same sign and
slows down when v(t) and a(t) are opposite signs.
*A particle is farthest to the left when x(t) is minimized and
farthest to the right when x(t) is maximized.
solving differential equations through
separation of variables
dy
*Differential equations are equations containing
dx
*To solve a differential equation means to integrate to find the
original equation in terms of only x and y. We can do this by
first separating the variables, then integrating both sides.
Solve
dy
xy  2
dx
Area between two curves
*To find the area between two curves, integrate the difference
between the larger function and the smaller function (top
function - bottom function).
Volumes of revolution
*Disk Method: To find the volume of the solid formed by
b
revolving a region
about
a horizontal line that is adjacent to the
2
region, use
  R(x) dx
a
.
To find the volume of the solid formed by revolving a region about
b
a vertical line that
is adjacent to the region, use
2
  R(y) dy
a
.
*Washer Method: To find the volume of the solid formed by
revolving a regionb about a horizontal line that is not adjacent to
2
2

dx .

(R(x))

(r(x))
the region, use  
a
To find the volume of the solid formed by revolving a region
about a vertical line that is not adjacent to the region, use
b
  (R(y))2  (r(y))2 dy
a
Cross-sectional volumes
*To find the volume of the solid formed by lying cross-sections in
the form of a geometric shape perpendicular to the x-axis in a
b
bounded region, use
 A  f (x)  g(x)dx
, where A is the area
a
formula for the given geometric shape and f(x)-g(x) is the height
of the representative rectangle in the region. Be sure you consider
what quantity f(x)-g(x) represents in the area formula.