Brainscape's Knowledge GenomeTM


Top Flashcards (Certified)
All Flashcards

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Show that f (x) is even

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Show that f (x) is odd

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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+

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Find lim f ( x ) , calculator allowed

x →∞

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Find horizontal asymptotes of f (x)

A

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

x →∞ x → −∞

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

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

A

Find derivative f ′(a ) = m

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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 )

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Find equation of the line normal

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

A

Same as above but m =

−1/f ′(a )

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

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

A

Find (f (b ) − f (a ))/(b-a)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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) .

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

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

A

Find f ′(a )

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

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

A

Find v(t ) = s ′(t )

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

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

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
Given a graph of f ′( x ) , find where f (x ) is increasing
Determine where f ′( x ) is positive (above the x-axis.)
26
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.
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
27
Given a graph of f ′( x ) , find where f (x ) has a relative maximum.
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.
28
Given a graph of f ′( x ) , find where f (x ) is concave down.
Identify where f ′(x ) is decreasing.
29
Given a graph of f ′( x ) , find where f (x ) has point(s) of inflection.
Identify where f ′(x ) changes from increasing to decreasing or vice versa.
30
Show that a piecewise function is differentiable at the point a where the function rule splits
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 ## Footnote lim f '(x)=lim f '(x) x-\>a- x-\>a+
31
Given a graph of f ( x ) and h ( x ) = f^-1(x) ## Footnote find h ' ( 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)
32
Given the equation for f ( x ) and h ( x ) = f ^-1( x ) , find h ' ( a )
Understand that the point ( a, b ) is on h ( x ) so the point ## Footnote ( b, a ) is on f ( x ) . So find b where f ( b ) = a h '(a) = 1/f '(b)
33
Given the equation for f ( x ) , find its derivative algebraically.
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
Given a relation of x and y, find dy/dx algebraically.
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
Find the derivative of f ( g ( x ))
Chain Rule f ′( g ( x )) ⋅ g ′( x )
36
Find the minimum value of a function on [a, b]
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
Find the minimum slope of a function on [a, b]
Solve f " ( x ) = 0 or DNE, make a sign chart, find sign ## Footnote 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
Find critical values
Express f ′( x ) as a fraction and solve for numerator and denominator each equal to zero.
39
Find the absolute maximum of f ( x )
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
Show that there exists a c in [a, b] such that f '(c) = 0
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 ) , ## Footnote then there is some c in [k , j] such that f ′(c)= 0.
41
Show that there exists a c in [a, b] such that ## Footnote f '(c) = m
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
Find range of f (x ) on [a, b]
Use max/min techniques to find values at relative max/mins. Also compare f ( a ) and f ( b ) (endpoints)
43
Find range of f (x ) on (− ∞, ∞ )
Use max/min techniques to find values at relative max/mins. Also compare lim f (x ). ## Footnote x →±∞
44
Find the locations of relative extrema of f ( x ) given both f ' ( x ) and f " ( x ) . ## Footnote Particularly useful for relations of x and y where finding a change in sign would be difficult.
Second Derivative Test Find where f ' ( x ) = 0 OR DNE then check the value of ## Footnote 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
Find inflection points of f (x ) algebraically.
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
Show that the line y = mx + b is tangent to f ( x ) at ( x1 , y1 )
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
Find any horizontal tangent line(s) to f ( x ) or a relation of x and y.
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
Find any vertical tangent line(s) to f ( x ) or a relation of x and y.
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
Approximate the value of f (0.1) by using the tangent line to f at x = 0
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
Find rates of change for volume problems.
Write the volume formula. Find dV/dt. Careful about product/chain rules. Watch positive (increasing measure)/negative (decreasing measure) signs for rates.
51
Find rates of change for Pythagorean Theorem problems.
52
Find the average value of f (x ) on [a, b]
53
Find the average rate of change of f ( x ) on [a, b]
f (b) − f ( a )/ b-a
54
``` Given v(t ) , find the total distance a particle travels on [a, b] ```
55
``` Given v(t ) , find the change in position a particle travels on [a, b] ```
56
``` Given v(t ) and initial position of a particle, find the position at t = a. ```
57
f (x )
58
f ( g ( x) ) g '( x)
59
Find area using left Riemann sums
A = base[x0 + x1 + x 2 + ... + x n −1] Note: sketch a number line to visualize
60
Find area using right Riemann sums
A = base[x1 + x 2 + x3 + ... + x n] Note: sketch a number line to visualize
61
Find 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. Note: sketch a number line to visualize
62
Find area using trapezoids
63
Describe how you can tell if rectangle or trapezoid approximations over- or under- estimate area.
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
65
Given dy/dx, draw a slope field
Use the given points and plug them into dy/dx, drawing little lines with the indicated slopes at the points.
66
y is increasing proportionally to y
67
Solve the differential equation …
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
Find the volume given a base bounded by f ( x ) and g ( x ) with f (x ) \> g ( x ) and ## Footnote cross sections perpendicular to the x-axis are squares
69
Given the value of F (a ) and F ' ( x ) = f ( x ) , ## Footnote find F (b )
70
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
Given v(t ) and s (0 ) , find the greatest distance from the origin of a particle on [a, b]
Solve v ( t ) = 0 OR DNE . Then integrate v(t ) adding ## Footnote s (0 ) to find s (t ) . Finally, compare s(each candidate) and s(each endpoint). Choose greatest distance (it might be negative!)
72
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 ## Footnote a) the amount of water in the tank at m minutes
73
b) the rate the water amount is changing at m
74
c) the time when the water is at a minimum
75
76
``` Find the volume of the area between f ( x ) and g ( x ) with f ( x ) \> g ( x ) , rotated about ``` ## Footnote the x-axis.
77
Given v(t ) and s (0 ) , find s (t )
78
``` Find the line x = c that divides the area under f ( x ) on [a, b] to two equal areas ```
79
Find the volume given a base bounded by f ( x ) and g ( x ) with f (x ) \> g ( x ) and ## Footnote cross sections perpendicular to the x-axis are semi-circles

AP Calculus AB (2 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2025 Bold Learning Solutions. Terms and Conditions