For my AP Calculus class.

1. Bland introduction.
By: Chris Wilson
Friday, March 29, 13

2. Tangent Lines
These are the formulas one needs to know, including one you thought you’d
escaped in Algebra.
You’re finding the slope of a line tangent or secant to an equation. Using the point-
slope equation, you can then plot the line itself.
When you’re given f(x) and your points for x, you just plug
them in the appropriate formula.
Just plug and chug.
Friday, March 29, 13

3. Limits - Graphical
If the approach from both sides equal each
Negative means approach from the left, other, that number is the limit asked for here.
Positive means approach from the right.
Just like with greater than and greater than or equal to
number lines, an open circle means it is not included, a
filled-in circle means it is included.
Friday, March 29, 13

4. Tangent Lines: Part 2
These are the formulas one needs to know. Don’t they look familiar?
Secant line formula = average velocity.
Tangent line formula = instantaneous velocity.
When you’re given s(t) and your points for t, you just plug
them in the appropriate formula.
Just plug and chug.
Friday, March 29, 13

5. Derivatives: Normal
Y prime (Y’) represent the slope at
any point of an equation.
Rather than finding it algebraically as
we were before, we can just find Y’
using derivates.
y = c where c is any constant.
y’ = 0
y = cx where c is any constant.
y’ = c
y = cx5 where c is any constant.
y’ = 5cx 4
Friday, March 29, 13

6. Derivatives: Quotient and Product Rules
Y prime (Y’) represent the slope at
any point of an equation.
Rather than finding it algebraically as
we were before, we can just find Y’
using derivates.
Quotient Rule:
Derivative of the bottom times the
top, minus the derivative of the top
Product Rule: times the bottom, all over the
Derivative of the first times the bottom squared.
second, plus the derivative of the
second times the first.
Thursday, September 27, 12
Used for equations such as:
Used for equations such as: (2x+1)
y=
y = (2x+1)(x2) (x2)
Friday, March 29, 13

7. Derivatives: Chain Rule
Y prime (Y’) represent the slope at
any point of an equation.
Rather than finding it algebraically as
we were before, we can just find Y’
Chain Rule: using derivates.
Take the derivative of the outside,
multiplied by the inside, multiplied by For our example problem,
the inside, (etc., until out of this is how you would use
parentheses) the chain rule to find f’(x):
3(inside)2
times
One generally needs to (derivative of inside)
use the product rule
and/or the quotient rule 3(x3 - 3x2 + 2x - 1)2
within the chain rule. times
(3x2 - 6x + 2)
Friday, March 29, 13

8. Derivatives: Implicit Differentiation
Y prime (Y’) represent the slope at
any point of an equation.
Rather than finding it algebraically as
Derivative of = 2x x2 we were before, we can just find Y’
Derivative of y3 = 3y2y’ using derivates.
Derivative of 5 = 0
For Implicit Differentiation, we take
the derivative of the problem as it
2x + 3y2y’ = 0 stands (unlike solving for y in normal
differentiation). It’s similar to chain
Then solve for y’.
rule in some aspects.
Used for equations where you don’t want to (or
cannot) solve for y.
Friday, March 29, 13

9. Derivatives: Second Derivatives or more
Y prime (Y’) represent the slope at
any point of an equation.
Rather than finding it algebraically as
we were before, we can just find Y’
y = (x2 + 1)1/2
using derivates.
First Derivative:
y’ = 1/2(x2 + 1)-1/2(2x)
y’ = x(x2 + 1)-1/2
To find the Second Derivative of any
Second Derivative:
equation, you just take the derivative
y’’ = (x2 + 1)-1/2 + (-1/2(x2 + 1) -3/2)(x)
of it once, and then take the
y’’ = 1/(x2 + 1) - x/(2√(x2 + 1)3)
derivative of that again.
Used for equations where they ask you for the
second, third, fourth, etc. derivative.
Friday, March 29, 13

10. Inverse Functions
When asked to find f-1(x) or the inverse
of the function, set f(x) as y and then
switch the x’s and y’s. Then solve for y.
f(x) = 10x + 8
Find f -1(x) Used for equations where they ask
y = 10x + 8 you for the inverse function (or you
x = 10y + 8 need the inverse function to get the
10y = x - 8 answer).
y = x/10 - 4/5
Usually not as pleasant as my example problem.
Friday, March 29, 13

11. Continuity and Differentiability
The idea behind differentiability (with
graphs) is:
The idea behind continuity is very
1) the function must be continuous
simple: is the function continuous?
and 2) there must be no jagged
If you can draw the function without
edges.
lifting up your pencil, it’s continuous.
Above graph’s I and II have jagged
All three examples above are indeed
points, so they’re not differentiable.
continuous.
Above graph III is a smooth line,
making it differentiable.
While functions that are differentiable are also continuous,
functions that are continuous are not necessarily differentiable.
Friday, March 29, 13

12. Rolle’s Theorem
The idea behind Rolle’s Theorem is:
1) the function must be differentiable
and 2) therefore at some point, the
derivative of f(c) is equal to the slope First, we need to know if f(0) = f(4).
between (a, f(a)) and (b, f(b)). f(0) = 0; f(4) = 0; f(0) = f(4).
Then we plug 0 into f’(x)
2x - 4 = 0; x = 2
Is 2 on between 0 and 4? Yes.
Then x = 2 is the answer.
Friday, March 29, 13

13. Related Rates
The idea behind related rates is that
if you take the derivative of a
formula, and you know the rate at Formula for this is abc = V, where a is
which certain things are changing and the length, b is the width, c is the
some values at a certain point in depth, and V is the volume.
time, then you can solve for the rest
of the variables.
One would take the derivative of
abc = V and then plug in 6 for a, 4 for
b, 8 for c, and 3 for A’.
State what you know and what you don’t, take the derivative of
the formula (keeping in mind which variables are changing), and
then plug in what you know to find what you don’t.
Friday, March 29, 13

14. Straight-Line Motion
The big idea is that a(t), and that a’(t)
= v(t), a’’(t) = v’(t) = s(t).
With non-meter measurement, the
formula you need to know is:
s(t) = -16t2 + Vot + So
So = Initial Velocity
t = Time Vo = Initial Velocity
Friday, March 29, 13

15. Thanks for watching!
Friday, March 29, 13