AB/BC Calculus Exam – Review Sheet A. Precalculus Type problems
A1
When you see the words … Find the zeros of f ( x ) .
A2
Find the intersection of f ( x ) and g( x ) .
A3
Show that f ( x ) is even.
A4
Show that f ( x ) is odd.
A5
Find domain of f ( x ) .
A6
Find vertical asymptotes of f ( x ) .
A7
If continuous function f ( x ) has f ( a) < k and f (b) > k , explain why there must be a value c such that a < c < b and f (c ) k .
This is what you think of doing
=
B. Limit Problems
B1
When you see the words … Find lim f ( x ) .
B2
Find
This is what you think of doing
x "a
lim f
x "a
( x ) where f ( x ) is a
piecewise function. B3
Show that f ( x ) is continuous.
B4
Find
B5
Find horizontal asymptotes of f ( x ) .
lim f
x "#
( x ) and xlim f ( x ) .
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"$#
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B6 BC
When you see the words … f ( x ) lim
Find
x "0
g( x )
( x ) 0 and xlim g( x ) Find lim f ( x ) # g( x ) 0( ) x if lim f
=
x "0
B7 BC
This is what you think of doing
=
0
"0
=
± !
"0
B8 BC
Find
B9 BC
Find
lim f
( x ) # g( x )
lim f
( x)
x "0
x "0
g( x) =
=
#
$#$
0
1 or 0 or #
0
C. Derivatives, differentiability, and tangent lines
C1
When you see the words … Find the derivative of a function using the derivative definition.
C2
Find the average rate of change of f of f on [a, b].
C3
Find the instantaneous rate of change of f of f at x at x = = a.
C4
Given a chart of x of x and and f ( x ) and selected values of x of x between between a and b, approximate f "(c ) where c is a value between a and b. Find the equation of the tangent line to f at at ( x1, y1 ) .
C5
C6
Find the equation of the normal line to f at at ( x1, y1 ) .
C7
Find x Find x-values -values of horizontal tangents to f.
C8
Find x Find x-values -values of vertical tangents to f. to f.
C9
Approximate the value of f ( x1 + a) if
This is what you think of doing
you know the function goes through point ( x1, y1 ) .
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C10
When you see the words … Find the derivative of f ( g( x )) .
C11
The line y = mx + b is tangent to the graph of f ( x ) at
C12
This is what you think of doing
( x , y ) . 1
1
Find the derivative of the inverse to f ( x ) at x a . =
C13
Given a piecewise function, show it is differentiable at x a where the function rule splits. =
D. Applications of Derivatives
D1
When you see the words … Find critical values of f ( x ) .
D2
Find the interval(s) where f ( x ) is increasing/decreasing.
D3
Find points of relative extrema of f ( x ) .
D4
Find inflection points of f ( x ) .
D5
Find the absolute maximum or minimum of f ( x ) on [a [a, b].
D6
Find range of f ( x ) on
D7
Find range of f ( x ) on [a [a, b]
D8
Show that Rolle’s Theorem holds for f ( x ) on [a [a, b].
D9
Show that the Mean Value Theorem holds for f ( x ) on [a [a, b].
D10
Given a graph of f "( x ) , determine intervals where f ( x ) is increasing/decreasing.
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This is what you think of doing
("# #) . ,
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D11
When you see the words … Determine whether the linear approximation for f ( x1 + a) over-
This is what you think of doing
estimates or under-estimates f ( x1 + a) . D12
Find intervals where the slope of f ( x ) is increasing.
D13
Find the minimum slope of f ( x ) on [a, b]. b].
E. Integral Calculus When you see the words …
E1
This is what you think of doing
b
Approximate
" f ( x ) dx using left a
Riemann sums with n rectangles. E2
b
Approximate
" f ( x ) dx using right a
Riemann sums with n rectangles. E3
b
Approximate
" f ( x ) dx using midpoint a
Riemann sums. E4
b
Approximate
" f ( x ) dx using a
trapezoidal summation. E5
a
Find
" f ( x ) dx where a
< b
.
b
E6
x
Meaning of " f ( t ) dt . a
E7
b
Given
b
" f ( x ) dx , find " [ f ( x ) a
E8
E9
E10
+
k ] dx .
a
Given the value of F ( a) where the antiderivative of f is F is F , find F (b) . Find
Find
d dx d dx
x
" f (t ) dt . a g ( x)
" f (t ) dt . a
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When you see the words …
E11 BC
This is what you think of doing
"
Find
# f ( x ) dx . 0
E12 BC
Find
" f ( x ) # g( x ) dx
F. Applications of Integral Calculus
F1
When you see the words … Find the area under the curve f ( x ) on
This is what you think of doing
the interval [a [a, b]. F2
Find the area between f ( x ) and g( x ) .
F3
Find the line x line x = = c that divides the area under f ( x ) on [a [a, b] into two equal areas. Find the volume when the area under f ( x ) is rotated about the x the x-axis -axis on the interval [a [a, b]. Find the volume when the area between f ( x ) and g( x ) is rotated about the x the x-axis. -axis. Given a base bounded by [a, b] the cross f ( x ) and g( x ) on [a sections of the solid perpendicular to the x the x-axis -axis are squares. Find the volume. Solve the differential equation dy f ( x ) g( y ) . dx Find the average value of f ( x ) on [a, b].
F4
F5
F6
F7
=
F8
F9
Find the average rate of change of F "( x ) on [ t 1, t 2 ] .
F10
y is y is increasing proportionally to y. to y.
F11
F12 BC
Given
Find
dy dx
, draw a slope field.
" ax
dx 2
+ bx + c
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F14 BC
When you see the words … Use Euler’s method to approximate f (1.2) given a formula for dy , ( x 0, y 0 ) and " x 0.1 dx Is the Euler’s approximation an overo veror under-approximation?
F15 BC
A population P population P is is increasing logistically.
F16 BC
Find the carrying capacity of a population growing logistically.
F17 BC
Find the value of P of P when when a population growing logistically is growing the fastest. Given continuous f ( x ) , find the arc length on [a [a, b]
F13 BC
This is what you think of doing
=
F18 BC
G. Particle Motion and Rates of Change
G1
G2
When you see the words … Given the position function s( t ) of a
particle moving along a straight line, find the velocity and acceleration. Given the velocity function v ( t ) and s(0) , find s( t ) .
G3
Given the acceleration function a( t ) of a particle at rest and s(0) , find s( t ) .
G4
Given the velocity function v ( t ) , determine if a particle is speeding up or slowing down at t = k . Given the position function s( t ) , find the average velocity on [ t 1, t 2 ] .
G5
G6
This is what you think of doing
Given the position function s( t ) , find the instantaneous velocity at t k . =
G7
G8
Given the velocity function v ( t ) on [t 1, t 2 ] , find the minimum acceleration of a particle. Given the velocity function v ( t ) , find the average velocity on [ t 1, t 2 ] .
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G9
G10
G11
When you see the words … Given the velocity function v ( t ) , determine the difference of position of a particle on [ t 1, t 2 ] .
This is what you think of doing
Given the velocity function v ( t ) , determine the distance a particle travels on [ t 1, t 2 ] . t 2
Calculate
" v (t ) dt without a t 1
calculator. G12 Given the velocity function v ( t ) and s(0) , find the greatest distance of the particle from the starting position on [0,t 1] . G13
G14
The volume of a solid is changing at the rate of … b
The meaning of # R"( t )
dt .
a
G15
Given a water tank with g with g gallons gallons initially, filled at the rate of F ( t ) gallons/min and emptied at the rate of E ( t ) gallons/min on [ t 1, t 2 ] a) The amount of water in the tank at t = = m minutes. b) the rate the water amount is changing at t = m minutes and c) the time t when when the water in the tank is at a minimum or maximum.
H. Parametric and Polar Equations - BC When you see the words …
H1
H2
Given x
Given x
=
=
f ( t ), y g( t ), find
This is what you think of doing dy
=
f ( t ), y g( t ), find
dx
.
d 2 y
=
dx
2
.
H3
Given x f ( t ) y g( t ) find arc length on [ t 1, t 2 ] .
H4
Express a polar equation in the form of r f (" ) in parametric form.
=
,
=
,
=
H5
Find the slope of the tangent line to r f (" ) . =
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H6
When you see the words … Find horizontal tangents to a polar curve r f (" ) .
This is what you think of doing
=
H7
Find vertical tangents to a polar curve r f (" ) . =
H8
Find the area bounded by the polar curve r f (" ) on [" 1," 2 ] . =
H9
Find the arc length of the polar curve r f (" ) on [" 1," 2 ] . =
I. Vectors and Vector-valued functions - BC
I1
When you see the words … Find the magnitude of vector v
I2
Find the dot product:
I3
The position vector of a particle moving in the plane is r( t ) x ( t ) y ( t ) . Find a) the velocity =
I4
,
,
,
= 0, find the position vector. Given the velocity vector v ( t ) x ( t ) y ( t ) , when does the =
I7
,
v v 1 2
particle at time t. Given the velocity vector v ( t ) x ( t ) y ( t ) and position at time t =
I6
"
,
v v 1 2
vector and b) the acceleration vector. The position vector of a particle moving in the plane is r( t ) x ( t ) y ( t ) . Find the speed of the =
I5
,
u u 1 2
This is what you think of doing
,
particle stop? The position vector of a particle moving in the plane is r( t ) x ( t ) y ( t ) . Find the distance the =
,
particle travels from
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t 1 to t 2
.
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Stu Schwartz
J. Taylor Polynomial Approximations - BC
J1
J2
J3
J4
When you see the words … Find the nth degree Maclaurin polynomial to f ( x ) .
Find the nth degree Taylor polynomial to f ( x ) centered at x = x = c. Use the first-degree Taylor polynomial to f ( x ) centered at x at x = = c to approximate f ( k ) and determine whether the approximation is greater than or less than f ( k ) . Given an nth degree Taylor polynomial for f for f about x about x = = c, find n f (c ) f "(c ) f ""(c ) f ( ) (c ) . ,
J5
J6
J7
This is what you think of doing
,
,
... ...
,
Given a Taylor polynomial centered at c, determine if there is enough information to determine if there is a relative maximum or minimum at x at x = = c. Given an nth degree Taylor polynomial for f for f about x about x = = c, find the Lagrange error bound (remainder). Given an nth degree Maclaurin polynomial P polynomial P for f for f , find the f (k ) " P ( k ) .
K. Infinite Series - BC
K1
When you see the words … Given a , determine whether the sequence a converges.
This is what you think of doing
n
n
K2
Given a , determine whether the series a could converge. n
n
K3
Determine whether a series converges.
K4
Find the sum of a geometric series.
K5
Find the interval of convergence of a series.
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When you see the words …
K6
K7
K8
K9
1+
1 2
1 +
3
1
+ ... +
n
f ( x ) = 1 + x
f ( x ) = x "
f ( x ) = 1 "
This is what you think of doing
+
x
+
2
2!
2
+
2
3
3!
x
x
+
x
3
x
"
4
4!
+
3!
5
5!
x
x
"
n
n!
+
...
7
7!
x
... +
x
+
...
6
6!
+
...
K10
f ( x ) = 1 + x + x
K11
Given a formula for the nth derivative of f ( x ) . Write the first four terms and the general term for the power series for f ( x ) centered at x at x = = c.
K12
Let S 4 be the sum of the first 4 terms of an alternating series for f ( x ) .
2
+
x
3
+
... + x
n
+
....
Approximate f ( x ) " S 4 . K13
Write a series for expressions like
www.MasterMathMentor.com
x
e
2
.
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Stu Schwartz
AB/BC Calculus Exam – Review Sheet – Solutions A. Precalculus Type problems
A1
When you see the words … Find the zeros of f ( x ) .
A2
Find the intersection of f ( x ) and g( x ) .
A3
Show that f ( x ) is even.
A4
Show that f ( x ) is odd.
A5
Find domain of f ( x ) .
A6
Find vertical asymptotes of f ( x ) .
A7
If continuous function f ( x ) has f ( a) < k and f (b) > k , explain why there must be a value c such that a < c < b and f (c ) k .
This is what you think of doing Set function equal to 0. Factor or use quadratic equation if quadratic. Graph to find zeros on calculator. Set the two functions equal to each other. Find intersection on calculator.
Show that f (" x ) f ( x ) . This shows that the graph of f of f is symmetric to the y the y-axis. -axis. Show that f (" x ) " f ( x ) . This shows that the graph of f of f is symmetric to the origin. Assume domain is ("# #) . Restrict domains: denominators " 0, square roots of only non-negative numbers, logarithm or natural log of only positive numbers. Express f ( x ) as a fraction, express numerator and denominator in factored form, and do any cancellations. Set denominator equal to 0. This is the Intermediate Value Theorem. =
=
,
=
B. Limit Problems
B1
When you see the words … Find lim f ( x ) . x "a
This is what you think of doing Step 1: Find f (a) . If you get a zero in the denominator,
Step 2: Factor numerator and denominator of f ( x ) . Do any cancellations and go back to Step 1. If you still get a zero in the denominator, the answer is either !, -!, or does not exist. Check the signs of lim f ( x ) and lim f ( x ) for equality. x "a #
B2
Find
lim f
x "a
( x ) where f ( x ) is a
lim
x "a #
f ( x ) =
+
lim x "a
+
f ( x ) by plugging in a to
f ( x ), x < a and f ( x ), x > a for equality. If they are not equal, the
piecewise function. B3
Determine if
x "a
Show that f ( x ) is continuous.
limit doesn’t exist. Show that 1) lim f ( x ) exists x "a
2) f (a) exists 3) B4
Find
lim f
x "#
( x ) and xlim f ( x ) . "$#
lim f
x "a
( x)
=
f ( a)
Express f ( x ) as a fraction. fraction. Determine location of the highest power: Denominator: lim f ( x ) lim f ( x ) 0 =
x "#
=
x "$#
Both Num and Denom: ratio of the highest power coefficients Numerator: lim f ( x ) = ±# (plug in large number) x "#
B5
Find horizontal asymptotes of f ( x ) .
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lim f ( x )
and lim
x "#
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x "$#
f ( x ) Stu Schwartz
B6 BC
When you see the words … f ( x ) lim
Find
x "0
g( x )
if lim f x "0
B7 BC
Find
This is what you think of doing Use L’Hopital’s Rule:
lim f
x "0
( x)
=
( x)
0 and lim g x "0
( x ) # g( x )
=
(
0
=
0
)
± !
lim x "0
f ( x )
=
g( x )
Express
f #( x )
lim
g#( x )
x "0
( )
g x
1 =
and apply L’Hopital’s L ’Hopital’s rule.
1
( )
g x
B8 BC
Find
B9 BC
Find
lim f
( x ) # g( x )
lim f
( x)
x "0
x "0
g( x) =
=
$#$
0
#
1 or 0 or #
0
Express f ( x ) " g( x ) with a common denominator and use L’Hopital’s rule. Take the natural log of the expression and apply L’Hopital’s rule, remembering to take the resulting answer and raise e to that power.
C. Derivatives, differentiability, and tangent lines
C1 C2 C3 C4
C5 C6
When you see the words … Find the derivative of a function using the derivative definition.
This is what you think of doing f ( x + h ) # f ( x ) f ( x ) # f (a) Find lim or lim h"0 x " a h x # a Find the average rate of change of f of f f (b) " f (a) Find on [a [a, b]. b"a Find the instantaneous rate of Find f "( a) change of f of f at x at x = = a. Straddle c, using a value of k " c and a value of Given a chart of x of x and and f ( x ) and f (k ) $ f ( h ) selected values of x of x between between a and h # c. f "(c ) # b, approximate f "(c ) where c is a k $ $ h
value between a and b. Find the equation of the tangent line to f to f at at ( x1, y1 ) .
Find slope m
Find the equation of the normal line to f to f at at ( x1, y1 ) .
Find slope m"
y " y 1
C8 C9
Find x Find x-values -values of horizontal tangents to f. to f. Find x Find x-values -values of vertical tangents to f. to f. Approximate the value of f ( x1 + a) if you know the function goes through point ( x1, y1 ) .
f "( x i ) . Then use point slope equation:
(
m x " x 1
=
) #1
=
y " y 1
C7
=
=
f $( x i )
(
m x " x 1
. Then use point slope equation:
)
Write f "( x ) as a fraction. Set numerator of f "( x )
=
0 .
Write f "( x ) as a fraction. Set denominator of f "( x ) Find slope m y " y 1
=
(
=
=
0 .
f "( x i ) . Then use point slope equation:
m x " x 1
for y at at ) . Evaluate this line for y
x = x + a 1
. Note:
The closer a is to 0, the better the approximation will be. Also note that using concavity, it can be determine if this value is an over or under-approximation for f ( x1 + a) .
C10
Find the derivative of f ( g( x )) .
This is the chain rule. You are finding f "( g( x ))# g"( x ) .
C11
The line y = mx + b is tangent to
Two relationships are true: 1) The function f function f and and the line share the same slope at x 1: m f "( x1 ) 2) The function f function f and and the line share the same y same y-value -value at x 1.
the graph of f ( x ) at
( x , y ) . 1
1
=
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Stu Schwartz
C12
When you see the words … This is what you think of doing Find the derivative of the inverse to Follow this procedure: 1) Interchange x Interchange x and and y y in in f ( x ) . f ( x ) at x a . 2) Plug the x the x-value -value into this equation and solve for y for y (you (you may need a calculator to solve graphically) dy 3) Using the equation in 1) find implicitly. dx dy 4) Plug the y the y-value -value you found in 2) to dx Given a piecewise function, show it First, be sure that f ( x ) is continuous at x a . Then take the is differentiable at x a where the derivative of each piece and show that lim f $( x) = lim f $( x ) . x "a x "a function rule splits. =
C13
=
=
#
+
D. Applications of Derivatives
D1 D2
D3
When you see the words … Find critical values of f ( x ) .
Find the interval(s) where f ( x ) is increasing/decreasing. Find points of relative extrema of f ( x ) .
This is what you think of doing Find and express f "( x ) as a fraction. Set both numerator and denominator equal to zero and solve. Find critical values of f "( x ) . Make a sign chart to find sign of f "( x ) in the intervals bounded by critical values.
Positive means increasing, negative means decreasing. Make a sign chart of f "( x ) . At x At x = = c where the derivative switches from negative to positive, there is a relative minimum. When the derivative switches from positive to negative, there is a relative maximum. To actually find the point, evaluate f (c ) . OR if f "(c ) 0 , then if f ""(c ) > 0 , there is a relative minimum at x at x = = c. If f ""(c ) < 0 , there is a nd relative maximum at x at x = = c. (2 Derivative test). Find and express f ""( x ) as a fraction. Set both numerator and denominator equal to zero and solve. Make a sign chart of f ""( x ) . Inflection points occur when f ""( x ) witches from positive to negative or negative to positive. Use relative extrema techniques to find relative max/mins. Evaluate f Evaluate f at at these values. values. Then examine f ( a) and f (b) . The largest of these is the absolute maximum and the smallest of these is the absolute minimum Use relative extrema techniques to find relative max/mins. Evaluate f Evaluate f at at these values. values. Then examine f ( a) and f (b) . Then examine lim f ( x ) and lim f ( x ) . =
D4
Find inflection points of f ( x ) .
D5
Find the absolute maximum or minimum of f ( x ) on [a [a, b].
D6
Find range of f ( x ) on
("# #) . ,
x "#
x "$#
D7
Find range of f ( x ) on [a [a, b]
Use relative extrema techniques to find relative max/mins. Evaluate f Evaluate f at at these values. values. Then examine f ( a) and f (b) . Then examine f ( a) and f (b) .
D8
Show that Rolle’s Theorem holds for f ( x ) on [a [a, b].
Show that f that f is is continuous and differentiable on [a [a, b]. If [a, b] such that f "(c ) f ( a) f (b) , then find some c on [a
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=
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=
0.
Stu Schwartz
D9
Show that the Mean Value Theorem holds for f ( x ) on [a [a, b].
Show that f that f is is continuous and differentiable on [a [a, b]. If [a, b] such that f ( a) f (b) , then find some c on [a =
f "(c )
D10
D11
=
f (b) # f ( a)
b#a Make a sign chart of f "( x ) and determine the intervals
Given a graph of f "( x ) , determine intervals where f ( x ) is increasing/decreasing. Determine whether the linear approximation for f ( x1 + a) over-
where f "( x ) is positive and negative. Find slope m y " y 1
estimates or under-estimates f ( x1 + a) .
=
(
=
f "( x i ) . Then use point slope equation:
m x " x 1
for y at at ) . Evaluate this line for y
If f ""( x1) > 0 , f is is concave up at
x
x = x + a 1
.
and the linear
1
approximation is an underestimation for f ( x1 + a) . f ""( x1) < 0 , f is is concave down at
x
and the linear
1
approximation is an overestimation for f ( x1 + a) . D12
Find intervals where the slope of f ( x ) is increasing.
D13
Find the minimum slope of f ( x ) on [a, b]. b].
Find the derivative of f "( x ) which is f ""( x ) . Find critical values of f ""( x ) and make a sign chart of f ""( x ) looking for positive intervals. Find the derivative of f "( x ) which is f ""( x ) . Find critical values of f ""( x ) and make a sign chart of f ""( x ) . Values of x where x where f ""( x ) switches from negative to positive are potential locations for the minimum slope. Evaluate f "( x ) at those values and also f "( a) and f "(b) and choose the least of these values.
E. Integral Calculus When you see the words …
E1
b
Approximate
" f ( x ) dx using left a
This is what you think of doing # b " a & A = % ([ f ( x 0 ) + f ( x1 ) + f ( x 2 ) + ... + f ( x n "1 )] $ n '
Riemann sums with n rectangles. E2
# b " a & ([ f ( x1) + f ( x 2 ) + f ( x 3 ) + ... + f ( x n )] $ n '
b
Approximate
" f ( x ) dx using right
A = %
a
Riemann sums with n rectangles. E3
b
Approximate
" f ( x ) dx using a
midpoint Riemann sums. E4
Typically done with a table of points. Be sure to use only values that are given. If you are given 7 points, you can only calculate 3 midpoint rectangles.
# b " a & ([ f ( x 0 ) + 2 f ( x1) + 2 f ( x 2 ) + ... + 2 f ( x n "1) + f ( x n )] $ 2n '
b
Approximate
" f ( x ) dx using
A = %
a
This formula only works when the base of each trapezoid is the same. If not, calculate the areas of individual trapezoids.
trapezoidal summation. E5
a
Find
" f ( x ) dx where a b
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a
< b
.
" f ( x) dx b
b
=
# " f ( x ) dx a
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Stu Schwartz
When you see the words …
E6
x
Meaning of " f ( t ) dt . a
E7
b
Given
" f ( x ) dx , find
This is what you think of doing The accumulation function – accumulated area under function f starting at some some constant a and ending at some variable x variable x.. b
" [ f ( x )
a
+
b
b
a
a
k ] dx = " f ( x ) dx + " k dx
a
b
" [ f ( x )
+
k ] dx .
a
E8
Given the value of F ( a) where the antiderivative of f is F is F , find F (b) .
b
Use the fact that
" f ( x ) dx
=
F (b) # F ( a) so
a b
F (b) = F ( a) + " f ( x ) dx . Use the calculator to find the a
E9 E10 E11 BC
Find Find
d dx d dx
x
" f (t ) dt . a g ( x)
=
d
" f (t ) dt .
dx
a
"
Find
definite integral. x d " f (t ) dt f ( x ). The 2nd Fundamental Theorem. dx a
# f ( x ) dx .
Find
" f ( x ) # g( x ) dx
" f (t ) dt
=
f ( g( x )) # g$( x ). The 2nd Fundamental Theorem .
a
"
# f ( x ) dx 0
0
E12 BC
g ( x)
h
=
# f ( x ) dx
lim
h $"
=
lim F h $"
0
( h ) % F (0) .
If u-substitution doesn’t work, try integration by parts: # u " dv uv $ # v " du =
F. Applications of Integral Calculus
F1 F2
When you see the words … Find the area under the curve f ( x ) on the interval [a [a, b].
This is what you think of doing
" f ( x ) dx
Find the area between f ( x ) and g( x ) .
Find the intersections, a and b of f ( x ) and g( x ) . If
b
a
b
f ( x ) " g ( x ) on [a,b ] , then area A
=
# [ f ( x ) " g( x )] dx . a
F3
F4
F5
Find the line x line x = = c that divides the area under f ( x ) on [a [a, b] into two equal areas. Find the volume when the area under f ( x ) is rotated about the x the x-axis -axis on the interval [a [a, b]. Find the volume when the area between f ( x ) and g( x ) is rotated about the x the x-axis. -axis.
c
b
" f ( x ) dx " f ( x ) dx =
a
c
b
or
c
" f ( x ) dx 2 " f ( x ) dx =
a
a b
2
Disks: Radius = f ( x ) : V " # [ f ( x )] dx =
a
Washers: Outside radius = f ( x ) . Inside radius = g( x ) . Establish the interval where f ( x ) " g( x ) and the values of b
=
=
a
www.MasterMathMentor.com
(
2
a and b, where f ( x ) g ( x ) . V " $ [ f ( x )] # [ g( x )]
- 15 -
2
) dx
Stu Schwartz
F6
F7
When you see the words … Given a base bounded by f ( x ) and g( x ) on [a [a, b] the cross
sections of the solid perpendicular to the x the x-axis -axis are squares. Find the volume. Solve the differential equation dy f ( x ) g( y ) . dx Find the average value of f ( x ) on [a, b]. =
F8
F9
Find the average rate of change of F "( x ) on [ t 1, t 2 ] .
This is what you think of doing
Base = f ( x ) " g( x ). Area = b
F12 BC
F13 BC
Find
dy dx
, draw a slope field.
" ax
dx 2
+ bx + c
Use Euler’s method to approximate f (1.2) given a formula for dy , ( x 0, y 0 ) and " x 0.1 dx Is the Euler’s approximation an overo veror under-approximation?
2
b
" f ( x ) dx F avg d
=
a
b# a
t 2
# F "( x) dx
dt t
dy dt
Given
[ f ( x ) " g( x )] .
Separate the variables: x variables: x on on one side, y side, y on on the other with the dx and dx and dy in the numerators. Then integrate both sides, remembering the +C +C , usually on the x the x-side. -side.
=
F11
=
a
t 2 $ t 1
y is y is increasing proportionally to y. to y.
2
2
Volume = # [ f ( x ) " g( x )] dx
1
F10
base
=
( ) $ F "( t )
F " t 2
1
t 2 $ t 1
ky which translates to y Ce =
kt
Use the given points and plug them into
dy
, drawing little dx lines with the calculated slopes at the point. Factor ax 2 + bx + c into non-repeating factors to get dx " (mx + n )( px + q) and use Heaviside method to create partial fractions and integrate each fraction. dy dy = (" x ), y new = y old + dy dx
=
F14 BC
F15 BC F16 BC F17 BC
A population P population P is is increasing logistically. Find the carrying capacity of a population growing logistically. Find the value of P of P when when a population growing logistically is growing the fastest.
F18 BC
Given continuous f ( x ) , find the arc length on [a [a, b]
www.MasterMathMentor.com
2
dy
Look at sign of
d y
in the interval. This gives and 2 dx dx increasing/decreasing and concavity information. Draw a picture to ascertain the answer. dP dt dP dt
).
(
)
kP C " P
=
kP C " P
=
0 # C
=
P .
2
dP dt
(
=
=
(
)
kP C " P # Set
b
L = #
1+
[ f "( x )]
2
d P 2
dt
=
0
dx
a
- 16 -
Stu Schwartz
G13 G14
When you see the words … The volume of a solid is changing at the rate of …
dV =
...
dt
This gives the accumulated change of R( t ) on [a [a, b].
b
The meaning of # R"( t )
This is what you think of doing
dt .
b
b
# R"( t )
a
( ) $ R(a) or R(b) = R(a) + # R"( t ) dt
dt = R b
a
G15
Given a water tank with g with g gallons gallons initially, filled at the rate of F ( t ) gallons/min and emptied at the rate of E ( t ) gallons/min on [ t 1, t 2 ] a) The amount of water in the tank at t = m minutes. b) the rate the water amount is changing at t = m minutes and c) the time t when when the water in the tank is at a minimum or maximum.
a
m
a) g + # [F (t ) " E ( t )] dt 0
d
b)
m
# [ F ( t ) " E (t )] dt
dt
=
F ( m) " E (m)
0
c) set F ( m) " E ( m) = 0, solve for m, and evaluate evaluate m
g + # [F ( t ) " E ( t )] dt at values of m and also the endpoints. 0
H. Parametric and Polar Equations - BC When you see the words …
H1
H2
H3
Given x
Given x
=
=
f ( t ), y g( t ), find
dy
=
f ( t ), y g( t ), find =
dx
.
d 2 y dx 2
dy
=
=
d " dy %
.
Given x f ( t ) y g( t ) find arc length on [ t 1, t 2 ] . ,
=
dx
This is what you think of doing dy dt dx dt
x
f ( t ), y g( t ), find
=
=
Express a polar equation in the form of r f (" ) in parametric form.
=
$
'
dx # dx &
=
'
dt # dx & dx dt
t 2
,
(
L =
t 1
H4
d y dx 2
$
d " dy %
2
" dx % $ ' # dt &
x r cos" =
=
2
2
+
" dy % $ ' dt # dt &
f (" ) cos"
y r sin " =
=
f (" ) sin "
=
H5
Find the slope of the tangent line to r f (" ) . =
H6
Find horizontal tangents to a polar curve r f (" ) . =
H7
Find vertical tangents to a polar curve r f (" ) . =
H8
Find the area bounded by the polar curve r f (" ) on [" 1," 2 ] . =
H9
Find the arc length of the polar curve r f (" ) on [" 1," 2 ] . =
x r cos" y r sin " # =
x
=
=
r cos" y
Find where x
=
A
1 =
# r d " " 2 2
" 1
" 2
s=
when
" 2
# [ f (" )] 2
d " "
cos " # r cos"
0
r sin " #
0
2
d " "
" 1
2
$ [ f (" )] [ f #(" )] +
- 18 -
d " " dx
r sin "
1 =
dx
when
cos " = 0 r cos"
" 2
" 1
www.MasterMathMentor.com
=
=
r sin "
r sin " = 0
r cos" y
Find where
=
dy
dy
2
" 2
d " " =
$ " 1
2
% dr ( r + ' * d " " & d " " ) 2
Stu Schwartz
I. Vectors and Vector-valued functions - BC
I1
When you see the words … Find the magnitude of vector v v1, v 2 .
I2
Find the dot product:
I3
The position vector of a particle moving in the plane is r( t ) x ( t ) y ( t ) . Find a) the =
I4
I7
"
a) v ( t ) b) a( t )
+ v
2 2
,
= uv + u v 1 1 2 2
v v 1 2
x "( t ) y "( t )
=
,
x ""( t ) y ""( t )
=
,
Speed = v ( t )
2
[ x "( t )] [ y "(t )]
=
2
+
- a scala scalarr
,
Use v ( t )
,
,
particle stop? The position vector of a particle moving in the plane is r( t ) x ( t ) y ( t ) . Find the distance =
2
1
u u 1 2
time t = = 0, find the position vector. Given the velocity vector v ( t ) x ( t ) y ( t ) , when does the =
v
,
,
v v 1 2
,
=
the particle at time t. Given the velocity vector v ( t ) x ( t ) y ( t ) and position at =
I6
"
v
velocity vector and b) the acceleration vector. The position vector of a particle moving in the plane is r( t ) x ( t ) y ( t ) . Find the speed of =
I5
,
u u 1 2
This is what you think of doing
s( t ) = " x (t ) dt + " y ( t ) dt + C
(0) to find C , remembering that it is a vector.
s
=
()
0 " x t
=
()
0 AND AND y t
t 2
Distance
=
2
=
# [ x "( t )] [ y "( t )]
2
+
0
dt
t 1
,
the particle travels from
t 1 to t 2
.
J. Taylor Polynomial Approximations - BC
J1
When you see the words … Find the nth degree Maclaurin polynomial to f ( x ) .
This is what you think of doing f ""(0) 2 Pn ( x ) = f (0) + f "(0) x + x + f """(0) 3!
J2
Find the nth degree Taylor polynomial to f ( x ) centered at x = x = c.
2! ( n)
x 3 + ... +
Use the first-degree Taylor polynomial to f ( x ) centered at x = x = c to approximate f ( k ) and determine whether the approximation is greater than or less than f ( k ) .
www.MasterMathMentor.com
(0)
n!
Pn ( x ) = f (c ) + f "(c )( x # c ) + f """(c )( x # c ) 3!
J3
f
3
+
... +
x
n
f ""(c )( x # c )
f (
2
2! n)
(c)( x # c )
+
n
n! Write the first-degree TP and find f ( k ) . Use the signs of f "(c ) and f ""(c ) to determine increasing/decreasing and
concavity and draw your line (1st degree TP) to determine whether the line is under the curve (under-approximation) or over the curve (over-approximation).
- 19 -
Stu Schwartz
J4
When you see the words … Given an nth degree Taylor polynomial for f for f about x about x = = c, find ( n) f (c ) f "(c ) f ""(c ) f (c ) ,
,
,
... ...
,
This is what you think of doing f (c ) will be the constant term in your Taylor polynomial (TP) f "(c ) will be the coefficient of the x the x term term in the TP. f ""(c ) f
J5
J6
2! (n)
will be the coefficient of the
(c )
2
x
will be the coefficient of the
term in the TP. n
term in the TP. n! Given a Taylor polynomial centered If there is no first-degree x first-degree x-term -term in the TP, then the value of c at c, determine if there is enough about which the function is centered is a critical value. Thus information to determine if there is the coefficient of the x 2 term is the second derivative divided a relative maximum or minimum at by 2! Using the second derivative test, we can tell whether x = x = c. there is a relative maximum, minimum, or neither at x at x = = c. n 1 Given an nth degree Taylor f ( ) ( z) n 1 of z is is some number Rn ( x ) = x " c . The value of z polynomial for f for f about x about x = = c, find (n + 1)! the Lagrange error bound st n 1 between x between x and and c. f ( ) ( z) represents the ( n + 1) derivative of (remainder). z . This usually is given to you. Given an nth degree Maclaurin This is looking for the Lagrange error – the difference between polynomial P polynomial P for f for f , find the the value of the function at x k and the value of the TP at x k . f (k ) " P ( k ) . x
+
+
+
J7
=
=
K. Infinite Series - BC
K1
When you see the words … Given a , determine whether the sequence a converges. Given a , determine whether the series a could converge. n
a
n
This is what you think of doing converges if lim a exists. n "#
n
n
K2
n
n
K3
Determine whether a series converges.
If
lim a
=
n
n "#
0 ,
the series could converge. If
lim a $ 0 , n "#
n
the
series cannot converge. (n (nth term test). Examine the nth term of the series. Assuming it passes the nth term test, the most widely used series forms and their rule of convergence are: "
#
Geometric:
n
ar
- converges if
r < 1
n= 0
"
p-series: p-series:
1
#n
p
- converges if p if p > > 1
n 1 =
#
Alternating:
$ ("1)
n
a
n
- converges if
0<
a
n +1
< a
n
n =1
"
Ratio:
#
a
n
n= 0
K4
Find the sum of a geometric series.
"
# n= 0
K5
lim
n "#
a
n +1
<1
a
n
a
n
ar
- converges if
=
1 $ r
Find the interval of convergence of a If not given, you will have to generate the nth term formula. series. Use a test (usually the ratio test) to find the interval of convergence and then check out the endpoints.
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- 20 -
Stu Schwartz
When you see the words …
K6 K7 K8 K9 K10 K11
K12
K13
1+
1 2
1 +
3
1
+ ... +
n
f ( x ) = 1 + x f ( x ) = x " f ( x ) = 1 "
This is what you think of doing The harmonic series – divergent.
+
x
+
2
2!
+
2
3
3!
x
x 2
+
f ( x ) = 1 + x + x
x
+
x
"
4
4! 2
+
3!
5
5!
x
x 3
"
x
n!
+
...
+
+
+
... + x
n
+
....
Given a formula for the nth derivative of f ( x ) . Write the first four terms and the general term for the power series for f ( x ) centered at x at x = = c. Let S 4 be the sum of the first 4 terms of an alternating series for f ( x ) . Approximate f ( x ) " S 4 . Write a series for expressions like x
e
2
.
=
f ( x )
...
x
f ( x ) e f ( x )
...
6
6! 3
n
7
7!
x
... +
x
f ( x )
=
sin x
=
cos
x
1 =
1 " x
f ( x ) = f (c ) + f "(c )( x # c ) + f """(c )( x # c )
3
... +
+
3!
f ""(c )( x # c )
2
+
2!
f (
n)
(c)( x # c )
n
n!
+
...
th
This is the error for the 4 term of an alternating th series which is simply the 5 tern. It will be positive since you are looking for an absolute value. Rather than go through generating a Taylor polynomial, use the fact that if
( )
x
f ( x ) e , then f x =
x
f ( x ) = e
( )
f x
www.MasterMathMentor.com
("1,1)
Con Conve verge rgent nt :
2
=
= 1+
x
e
- 21 -
x
+
2
= 1+
x
2
x
x 2
e . So
2
2 2
=
+
+
x
x
3
3!
4
2
+
+
x
x
4
4! 6
3!
+
+
x
... +
x
n
n!
8
4!
+
... +
+
x
... and
2n
n!
+
...
Stu Schwartz
