M449 – AP Calculus AB
Flash Card Project Quarter 2
Flash Card Project
Quarter 2
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Rationale:
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These flash cards must be turned in to Mrs. Sagerer on or before Wednesday, January 9th .
LATE WORK is NOT accepted.
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The flash cards should be clipped together or rubber banded together. You can take a trip to Home
To help you prepare for the AP exam (and for each chapter test), you will need to create a set of
flash cards for yourself. Throughout the year, when you have a few minutes, you can flip through the cards to
review important concepts that we have learned during the year. This way you will not forget these important
ideas for the final or for the AP test. NOTE: It will be to your advantage to save these note card projects till after
the AP exam.
Depot or Lowes and purchase a key ring. You could then hole punch a corner of each index card and
slip the key ring through the hole. Another idea is to purchase a plastic index card holder. I have
seen these at Staples and Walgreens.
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You should create your flash cards on 3x5 or 4x6 index cards.
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The project is worth 50 quiz points.
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You must include a flash card for each topic below:
1) Chain Rule (one note card)
If f is differentiable at g(x) and g is differentiable at x, then the composition
 f  g   x   f  g  x   is differentiable at x, meaning
 f  g   x  
d
dy dy du
f  g  x    f   g  x    g  x  or


dx
dx du dx
2) Implicit Differentiation (one note card)
a) differentiate both sides of eqn wrt x
b) collect the terms with dy/dx on one side
c) factor out dy/dx
d) solve for dy/dx
3) Absolute Extreme Values (Absolute or global extrema) (one note card)
Let f be a function with domain D. The value f(c) is
a) an absolute max value iff f(x)  f(c) for all x in D
b) an absolute min value iff f(x)  f(c) for all x in D
4) Extreme Value Theorem (one note card)
If f is continuous on a closed interval [a,b] then f has both an absolute max and min
on the interval.
5) Local Extreme Values (Relative extrema) (one note card)
o
o
o
f(c) is a local max at x=c iff f (x)  f(c) for all x in some open interval containing c
f(c) is a local min at x=c iff f (x)  f (c) for all x in some open interval containing c
If a function f has a local max or min value at point c in the domain of f and f (c)
exists, then f (c) =0.
12/14/2012
M449 – AP Calculus AB
Flash Card Project Quarter 2
6) Critical Point (c,f(c)) (one note card)
 A point in the domain D of f(x) where f (c) =0 or f (c) DNE is a critical point
of f(x).
 The y value is called the critical value.
 The x value is called the critical number.
7) Mean Value Theorem (two note cards)
 If f(x) is continuous at every point of the closed interval [a,b] and
differentiable at every point of its interior (a,b), then there is at least one point
c in (a,b) at which
f(b)  f(a)
f   c 
ba
 Graphically, this theorem states that there exists an x value called c in the
interval (a,b) such that the slope of the tangent line at x=c is parallel to the
secant line connecting the endpoints of the interval (a,b). Include a picture of
this.
FYI: This theorem connects instantaneous rate of change and average rate of change.
8) Increasing/Decreasing (one note card)
Let f be continuous on [a,b] and differentiable on (a,b).
If f   0 at each point of (a,b), then f increases on [a,b].
If f   0 at each point of (a,b), then f decreases on [a,b].
9) Monotonic Functions (one note card)
A function is monotonic on an interval if it is either increasing or decreasing on the
entire interval. (Ex. y  x 3 increases from   ,   , so y(x) is monotonic.)
10) First Derivative Test (one note card)
Let f(x) be a continuous function. At a critical point (c,f(c)):
If f  changes sign from pos to neg at c then f has a local max at x=c.
If f  changes sign from neg to pos at c then f has a local min at x=c.
If f  does not change sign at c, then f has no local max/min at x=c.
11) Finding Absolute Max/Min (one note card)
To find the absolute extrema of the continuous function f (x) on the interval [a,b] use the
following process:
1. Find all critical numbers of f (x) in [a,b] .
2. Evaluate f (x) at values found in Step 1.
3. Evaluate f (a) and f (b).
4. Identify the abs. max. (largest function value) and the abs. min.(smallest function
value) from the evaluations in Steps 2 & 3.
12/14/2012
M449 – AP Calculus AB
Flash Card Project Quarter 2
12) Concavity (two note cards)
The graph of a differentiable function y=f(x) is
Concave up on an interval I if y  is increasing
Concave down on an interval I if y  is decreasing
If f(x) is a twice differentiable function, then f(x) is
Concave up on an interval I where y   0
Concave down on an interval I where y   0
13) Point of Inflection (one note card)
A point where f(x) is twice differentiable and the concavity changes is a point of inflection.
(Remember you need to give coordinates since it is a point.)
14) To Find Points of Inflection (one note card)
The possible points of inflection are where f   0 or f  DNE.
15) Second Derivative Test (one note card)
If f (c)  0 and f   0 , then f has a local max at x=c.
If f (c)  0 and f   0 , then f has a local min at x=c.
If f (c)  0 and f   0 or DNE, go back to the First Derivative Test.
16) Graph of f(x) to Graph of f (x) (one note card)
 Horizontal tangent on f(x) is zero on f (x)
 Point of inflection on f(x) is max/min on f (x)
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Positive tangent line slopes on f(x) are positive y values on f (x)
Negative tangent line slopes on f(x) are negative y values on f (x)
17) Graph of f (x) to Graph of f(x) (one note card)
 Zero on f (x) is horizontal tangent on f(x)
 Max/min on f (x) is point of inflection on f(x) is
 Positive y values on f (x) are positive tangent line slopes on f(x)
 Negative y values on f (x) are negative tangent line slopes on f(x)
18) Linearization (one note card)
If f is differentiable at x=a, then the equation of the tangent line
L(x)  f   a  x  a   f  a  defines the linearization of f at a. This equation is used to
approximate y values for f(x).
19) Optimization Problems (one note card)
Steps to Solving Optimization Problems
 Step 1: Identify all quantities to be determined.
 Step 2: Write a primary eqn for quantity to be maximized or minimized
 Step 3: Reduce the primary eqn to one eqn having a single independent variable. (You
may need to use a secondary eqn.)
 Step 4: Determine the domain of the eqn. (Valid possible answers.)
 Step 5: Find the derivative of the equation from Step 3.
 Step 6: Find the critical numbers from Step 5.
 Step 7: Determine the desired max/min value by calculus techniques (i.e. using the first
derivative test or second derivative test).
12/14/2012
M449 – AP Calculus AB
Flash Card Project Quarter 2
20) Related Rates (one note card)
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Sketch picture.
Identify known/unknown quantities.
Write down equation relating quantities.
Differentiate with respect to t using implicit differentiation (i.e. add on a derivative
every time you differentiate a function of t).
Plug in known quantities and solve for the unknown quantity.
21) Differentials (one note card)
Let y = f(x) be a differentiable function. The differential dy is dy  f   x  dx . In other
words, implicitly differentiate y = f(x) and multiply both sides by dx.
22) Inverse Functions (one note card)
Functions f(x) and g(x) are inverse functions or inverses of each other if
 f  g   x   x and  g  f   x   x .
23) Exponential and Logarithmic Properties (eleven note cards)
log b
a) e0  1
i) log a b 
log a
b) log1  0
c) log a a  1
j) Graph of y  ex
d) log a a x  x
e) log a  xy   log a x  log a y
x
f) log a    log a x  log a y
y
g) y  a x  x  loga y
k) Graph of y  log x
h) a loga x  x
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