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

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1

Find the zeros

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

2

Show that f (x) is even

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

3

Show that f (x) is odd

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

4

Show that lim f (x) exists

x-> a

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

                                x->a-     x->a+      

5

Find lim f ( x ) , calculator allowed

                               x->a

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

6

Find lim f ( x ) , no calculator

                                    x->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

 

7

Find lim f ( x ) , calculator allowed

                               x →∞

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

8

Find lim f ( x ) , no calculator

                                    x →∞

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

2) slow/fast= 0

3) same/same= ratio of coefficients 

 

9

Find horizontal asymptotes of f (x)

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

                                    x →∞                 x → −∞

                                                          

10

Find vertical asymptotes of f (x)

Find where lim f ( x ) = ±∞

                      x->a±

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


 

11

Find domain of f (x)

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

12

Show that f (x) is continuous

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

13

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

Find derivative f ′(a ) = m

14

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

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

sometimes need to find b = f ( a )

15

Find equation of the line normal
(perpendicular) to f (x ) at ( a, b )

Same as above but m =

−1/f ′(a )

16

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

Find (f (b ) − f (a ))/(b-a)

 

 

17

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

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

18

Find the interval where f ( x ) is increasing

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.

19

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

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.

20

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

Find f ′(a )

21

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

Find v(t ) = s ′(t )

22

Find f ′( x ) by the limit definition

Frequently asked backwards

f '(x)= lim f ( x + h) − f ( x)/h                                or

          h->0

 

f '(a)= lim f(x)-f(a)/x-a

           x->a 

23

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

24

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

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

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

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

lim f '(x)=lim f '(x)

x->a-       x->a+