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AP Calculus AB > When You See...Think > Flashcards

When You See...Think Flashcards

1
Q

Find the zeros

A

Find roots. Set function = 0, factor or use quadratic
equation if quadratic, graph to find zeros on calculator

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2
Q

Show that f (x) is even

A

Show that f (−x) = f ( x )
symmetric to y-axis

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3
Q

Show that f (x) is odd

A

Show that f (−x) = − f (x) OR f (x) = − f (−x)
symmetric around the origin

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4
Q

Show that lim f (x) exists

x-> a

A

Show that lim f (x )= lim f (x ); exists and are equal

x->a- x->a+

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5
Q

Find lim f ( x ) , calculator allowed

x->a

A

Use TABLE [ASK], find y values for x-values close to a
from left and right

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6
Q

Find lim f ( x ) , no calculator

x->a

A

Substitute x = a

1) limit is value if b/c, incl. 0/c=0; c cannot equal 0.
2) DNE for b/0
3) 0/0 DO MORE WORK!
a) rationalize radicals
b) simplify complex fractions
c) factor/reduce
d) know trig limits
1. lim sinx/x= 1

        x-\>0 

     2. lim 1-cosx/x= 0

         x-\>0 

   e) piece-wise function: check if RH = LH at break
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7
Q

Find lim f ( x ) , calculator allowed

x →∞

A

Use TABLE [ASK], find y values for large values of x,
i.e. 999999999999

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8
Q

Find lim f ( x ) , no calculator

x →∞

A

Ratios of rates of changes
1) fast/slow= DNE

2) slow/fast= 0
3) same/same= ratio of coefficients

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9
Q

Find horizontal asymptotes of f (x)

A

Find lim f ( x ) and lim f ( x )

x →∞ x → −∞

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10
Q

Find vertical asymptotes of f (x)

A

Find where lim f ( x ) = ±∞

                  x-\>a±

1) Factor/reduce f (x ) and set denominator = 0
2) ln x has VA at x = 0

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11
Q

Find domain of f (x)

A

Assume domain is (−∞, ∞). Restrictable domains:
denominators ≠ 0, square roots of only non-negative
numbers, log or ln of only positive numbers, real-world
constraints

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12
Q

Show that f (x) is continuous

A

Show that…

1) lim f(x) exists (limf(x)=limf(x))

                    x-\>a                   x-\>a-    x-\>a+

2) f (a) exists
3) lim f ( x ) = f (a )

      x→a
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13
Q

Find the slope of the tangent line to f (x ) at
x = a.

A

Find derivative f ′(a ) = m

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14
Q

Find equation of the line tangent to f ( x ) at

( a, b )

A

f ′(a ) = m and use y − b = m ( x − a )

sometimes need to find b = f ( a )

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15
Q

Find equation of the line normal

(perpendicular) to f (x ) at ( a, b )

A

Same as above but m =

−1/f ′(a )

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16
Q

Find the average rate of change of f ( x ) on
[a, b]

A

Find (f (b ) − f (a ))/(b-a)

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17
Q

Show that there exists a c in [a, b] such that
f (c) = n

A

Intermediate Value Theorem (IVT)
Confirm that f ( x ) is continuous on [a, b] , then show that
f (a) ≤ n ≤ f (b) .

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18
Q

Find the interval where f ( x ) is increasing

A

Find f ′(x ) , set both numerator and denominator to zero
to find critical points, make sign chart of f ′( x ) and
determine where f ′( x ) is positive.

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19
Q

Find interval where the slope of f (x ) is
increasing

A

Find the derivative of f ′( x ) = f ′′( x ) , set both numerator
and denominator to zero to find critical points, make
sign chart of f ′′( x ) and determine where f ′′( x ) is
positive.

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20
Q

Find instantaneous rate of change of f (x ) at
a

A

Find f ′(a )

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21
Q

Given s (t ) (position function), find v(t )

A

Find v(t ) = s ′(t )

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22
Q

Find f ′( x ) by the limit definition

Frequently asked backwards

A

f ‘(x)= lim f ( x + h) − f ( x)/h or

      h-\>0

f ‘(a)= lim f(x)-f(a)/x-a

       x-\>a
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23
Q

Find the average velocity of a particle on
[a, b]

A
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24
Q 
Given v(t ) , determine if a particle is
 speeding up at t = k
A

Find v ( k ) and a ( k ) . If signs match, the particle is

speeding up; if different signs, then the particle is
slowing down.

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25
Q

Given a graph of f ′( x ) , find where f (x ) is
increasing

A

Determine where f ′( x ) is positive (above the x-axis.)

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26
Q

Given a table of x and f ( x ) on selected
values between a and b, estimate f ′(c )
where c is between a and b.

A

Straddle c, using a value, k, greater than c and a value, h, less than c. So f ‘(c) ≈ f(k)-f(h)/ k-h

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27
Q

Given a graph of f ′( x ) , find where f (x ) has
a relative maximum.

A

Identify where f ′ ( x ) = 0 crosses the x-axis from above to below OR where f ′(x ) is discontinuous and jumps from above to below the x-axis.

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28
Q

Given a graph of f ′( x ) , find where f (x ) is
concave down.

A

Identify where f ′(x ) is decreasing.

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29
Q

Given a graph of f ′( x ) , find where f (x ) has
point(s) of inflection.

A

Identify where f ′(x ) changes from increasing to
decreasing or vice versa.

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30
Q

Show that a piecewise function is
differentiable
at the point a where the function rule
splits

A

First, be sure that the function is continuous at x = a by
evaluating each function at x = a. Then take the
derivative of each piece and show that

lim f ‘(x)=lim f ‘(x)

x->a- x->a+

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31
Q

Given a graph of f ( x ) and h ( x ) = f^-1(x)

find h ‘ ( a )

A

Find the point where a is the y-value on f ( x ) , sketch a
tangent line and estimate f ‘ ( b ) at the point, then

h ‘(a)= 1/f ‘(b)

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32
Q

Given the equation for f ( x ) and
h ( x ) = f ^-1( x ) , find h ‘ ( a )

A

Understand that the point ( a, b ) is on h ( x ) so the point

( b, a ) is on f ( x ) . So find b where f ( b ) = a

h ‘(a) = 1/f ‘(b)

33
Q

Given the equation for f ( x ) , find its
derivative algebraically.

A

1) know product/quotient/chain rules
2) know derivatives of basic functions
a. Power Rule: polynomials, radicals, rationals
b. e^x ; b^x
c. ln x;logb x
d. sin x;cos x; tan x
e. arcsin x;arccos x;arctan x;sin^−1 x; etc

34
Q

Given a relation of x and y, find dy/dx algebraically.

A

Implicit Differentiation
Find the derivative of each term, using
product/quotient/chain appropriately, especially, chain rule: every derivative of y is multiplied by dy/dx; then group all dy/dx terms on one side; factor out dy/dx and solve.

35
Q

Find the derivative of f ( g ( x ))

A

Chain Rule
f ′( g ( x )) ⋅ g ′( x )

36
Q

Find the minimum value of a function on
[a, b]

A

Solve f ′ ( x ) = 0 or DNE, make a sign chart, find sign
change from negative to positive for relative minimums
and evaluate those candidates along with endpoints backi nto f (x ) and choose the smallest. NOTE: be careful to confirm that f ( x ) exists for any x-values that make f ‘ ( x ) DNE.

37
Q

Find the minimum slope of a function on
[a, b]

A

Solve f “ ( x ) = 0 or DNE, make a sign chart, find sign

change from negative to positive for relative minimums
and evaluate those candidates along with endpoints back
into f ‘ ( x ) and choose the smallest. NOTE: be careful to

confirm that f ( x ) exists for any x-values that make
f “ ( x ) DNE.

38
Q

Find critical values

A

Express f ′( x ) as a fraction and solve for numerator and
denominator each equal to zero.

39
Q

Find the absolute maximum of f ( x )

A

Solve f ′ ( x ) = 0 or DNE, make a sign chart, find sign
change from positive to negative for relative maximums
and evaluate those candidates into f (x ) , also find

lim x->∞ f(x) and lim x->-∞; choose the largest.

40
Q

Show that there exists a c in [a, b] such that
f ‘(c) = 0

A

Rolle’s Theorem
Confirm that f is continuous and differentiable on the
interval. Find k and j in [a, b] such that f ( k ) = f ( j ) ,

then there is some c in [k , j] such that f ′(c)= 0.

41
Q

Show that there exists a c in [a, b] such that

f ‘(c) = m

A

Mean Value Theorem
Confirm that f is continuous and differentiable on the
interval. Find k and j in [a, b] such that

m= f(k)-f(j)/(k-j), then there is some c in [k,j] such that f ‘(c)=m

42
Q

Find range of f (x ) on [a, b]

A

Use max/min techniques to find values at relative
max/mins. Also compare f ( a ) and f ( b ) (endpoints)

43
Q

Find range of f (x ) on (− ∞, ∞ )

A

Use max/min techniques to find values at relative
max/mins. Also compare lim f (x ).

x →±∞

44
Q

Find the locations of relative extrema of
f ( x ) given both f ‘ ( x ) and f “ ( x ) .

Particularly useful for relations of x and y
where finding a change in sign would be
difficult.

A

Second Derivative Test
Find where f ‘ ( x ) = 0 OR DNE then check the value of

f “ ( x ) there. If f “ ( x ) is positive, f ( x ) has a relative

minimum. If f “ ( x ) is negative, f ( x ) has a relative

maximum.

45
Q

Find inflection points of f (x ) algebraically.

A

Express f ′′( x ) as a fraction and set both numerator and
denominator equal to zero. Make sign chart of f ′′( x ) to
find where f ′′( x ) changes sign. (+ to – or – to +)
NOTE: be careful to confirm that f (x ) exists for any x-
values that make f “ ( x ) DNE.

46
Q

Show that the line y = mx + b is tangent to
f ( x ) at ( x1 , y1 )

A

Two relationships are required: same slope and point of
intersection. Check that m = f ‘ ( x1 ) and that ( x1 , y1 ) is on both f (x ) and the tangent line.

47
Q

Find any horizontal tangent line(s) to f ( x )
or a relation of x and y.

A

Write dy/dx as a fraction. Set the numerator equal to zero.
NOTE: be careful to confirm that any values are on the
curve. Equation of tangent line is y = b. May have to find b.

48
Q

Find any vertical tangent line(s) to f ( x ) or a
relation of x and y.

A

Write dy/dx as a fraction. Set the denominator equal to zero.

NOTE: be careful to confirm that any values are on the
curve.
Equation of tangent line is x = a. May have to find a.

49
Q

Approximate the value of f (0.1) by using
the tangent line to f at x = 0

A

Find the equation of the tangent line to f using
y − y1 = m( x − x1 ) where m = f ′(0 ) and the point
is (0, f (0 )) . Then plug in 0.1 into this line; be sure to use
an approximate (≈ ) sign.
Alternative linearization formula:
y = f ‘ ( a )( x − a ) + f ( a )

50
Q

Find rates of change for volume problems.

A

Write the volume formula. Find dV/dt. Careful about product/chain rules. Watch positive (increasing measure)/negative (decreasing measure) signs for rates.

51
Q

Find rates of change for Pythagorean
Theorem problems.

A
52
Q

Find the average value of f (x ) on [a, b]

A
53
Q

Find the average rate of change of f ( x ) on
[a, b]

A

f (b) − f ( a )/ b-a

54
Q 
Given v(t ) , find the total distance a particle
 travels on [a, b]
A
55
Q 
Given v(t ) , find the change in position a
 particle travels on [a, b]
A
56
Q 
Given v(t ) and initial position of a particle,
 find the position at t = a.
A
57
Q
A

f (x )

58
Q
A

f ( g ( x) ) g ‘( x)

59
Q

Find area using left Riemann sums

A

A = base[x0 + x1 + x 2 + … + x n −1]
Note: sketch a number line to visualize

60
Q

Find area using right Riemann sums

A

A = base[x1 + x 2 + x3 + … + x n]
Note: sketch a number line to visualize

61
Q

Find area using midpoint rectangles

A

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.
Note: sketch a number line to visualize

62
Q

Find area using trapezoids

A
63
Q

Describe how you can tell if rectangle or
trapezoid approximations over- or under-
estimate area.

A

Overestimate area: LH for decreasing; RH for
increasing; and trapezoids for concave up
Underestimate area: LH for increasing; RH for
decreasing and trapezoids for concave down
DRAW A PICTURE with 2 shapes.

64
Q
A
65
Q

Given dy/dx, draw a slope field

A

Use the given points and plug them into dy/dx, drawing little lines with the indicated slopes at the points.

66
Q

y is increasing proportionally to y

A
67
Q

Solve the differential equation …

A

Separate the variables – x on one side, y on the other.
The dx and dy must all be upstairs. Integrate each side,
add C. Find C before solving for y,[unless ln y , then
solve for y first and find A]. When solving for y, choose
+ or – (not both), solution will be a continuous function
passing through the initial value.

68
Q

Find the volume given a base bounded by
f ( x ) and g ( x ) with f (x ) > g ( x ) and

cross sections perpendicular to the x-axis are
squares

A
69
Q

Given the value of F (a ) and F ‘ ( x ) = f ( x ) ,

find F (b )

A
70
Q
A

The accumulation function: net (total if f ( x ) is positive)
amount of y-units for the function f ( x ) beginning at
x = a and ending at x = b.

71
Q

Given v(t ) and s (0 ) , find the greatest
distance
from the origin of a particle on [a, b]

A

Solve v ( t ) = 0 OR DNE . Then integrate v(t ) adding

s (0 ) to find s (t ) . Finally, compare s(each candidate) and
s(each endpoint). Choose greatest distance (it might be
negative!)

72
Q

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 [0, b] , find

a) the amount of water in the tank at m
minutes

A
73
Q

b) the rate the water amount is changing
at m

A
74
Q

c) the time when the water is at a
minimum

A
75
Q
A
76
Q 
Find the volume of the area between f ( x )
 and g ( x ) with f ( x ) \> g ( x ) , rotated about

the x-axis.

A
77
Q

Given v(t ) and s (0 ) , find s (t )

A
78
Q 
Find the line x = c that divides the area
 under f ( x ) on [a, b] to two equal areas
A
79
Q

Find the volume given a base bounded by
f ( x ) and g ( x ) with f (x ) > g ( x ) and

cross sections perpendicular to the x-axis are
semi-circles

A

AP Calculus AB (2 decks)

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