AP Calculus - Semester Review
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Transcript of AP Calculus - Semester Review
Bland introduction.
By: Chris Wilson
Friday, March 29, 13
Tangent Lines
These are the formulas one needs to know, including one you thought you’d escaped in Algebra.
When you’re given f(x) and your points for x, you just plug them in the appropriate formula.
Just plug and chug.
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.
Friday, March 29, 13
Limits - Graphical
Negative means approach from the left,Positive means approach from the right.
If the approach from both sides equal each other, that number is the limit asked for here.
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
Tangent Lines: Part 2
When you’re given s(t) and your points for t, you just plug them in the appropriate formula.
Just plug and chug.
These are the formulas one needs to know. Don’t they look familiar?Secant line formula = average velocity.
Tangent line formula = instantaneous velocity.
Friday, March 29, 13
Derivatives: Normal
Sunday, September 30, 12
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’ = 5cx4
Friday, March 29, 13
Derivatives: Quotient and Product Rules
Thursday, September 27, 12
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.
Product Rule:Derivative of the first times the second, plus the derivative of the
second times the first.
Quotient Rule:Derivative of the bottom times the top, minus the derivative of the top
times the bottom, all over the bottom squared.
Used for equations such as:
Used for equations such as:
y = (2x+1)(x2)
(2x+1)y =
(x2)
Friday, March 29, 13
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’
using derivates.Chain Rule:Take the derivative of the outside,
multiplied by the inside, multiplied by the inside, (etc., until out of
parentheses)
For our example problem, this is how you would use the chain rule to find f’(x):
3(inside)2
times(derivative of inside)
3(x3 - 3x2 + 2x - 1)2
times(3x2 - 6x + 2)
One generally needs to use the product rule
and/or the quotient rule within the chain rule.
Friday, March 29, 13
Derivatives: Implicit Differentiation
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.
Derivative of x2 = 2xDerivative of y3 = 3y2y’
Derivative of 5 = 0
For Implicit Differentiation, we take the derivative of the problem as it
stands (unlike solving for y in normal differentiation). It’s similar to chain
rule in some aspects.
Used for equations where you don’t want to (or cannot) solve for y.
2x + 3y2y’ = 0
Then solve for y’.
Friday, March 29, 13
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’
using derivates.y = (x2 + 1)1/2
First Derivative:y’ = 1/2(x2 + 1)-1/2(2x)
y’ = x(x2 + 1)-1/2
Second Derivative:y’’ = (x2 + 1)-1/2 + (-1/2(x2 + 1) -3/2)(x)
y’’ = 1/(x2 + 1) - x/(2√(x2 + 1)3)
To find the Second Derivative of any equation, you just take the derivative
of it once, and then take the derivative of that again.
Used for equations where they ask you for the second, third, fourth, etc. derivative.
Friday, March 29, 13
Inverse Functions
f(x) = 10x + 8Find f -1(x)
y = 10x + 8x = 10y + 810y = x - 8
y = x/10 - 4/5
Used for equations where they ask you for the inverse function (or you need the inverse function to get the
answer).
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.
Usually not as pleasant as my example problem.
Friday, March 29, 13
Continuity and Differentiability
The idea behind continuity is very simple: is the function continuous?
If you can draw the function without lifting up your pencil, it’s continuous.All three examples above are indeed
continuous.
While functions that are differentiable are also continuous, functions that are continuous are not necessarily differentiable.
The idea behind differentiability (with graphs) is:
1) the function must be continuousand 2) there must be no jagged
edges.Above graph’s I and II have jagged
points, so they’re not differentiable.Above graph III is a smooth line,
making it differentiable.
Friday, March 29, 13
Rolle’s Theorem
The idea behind Rolle’s Theorem is:1) the function must be differentiableand 2) therefore at some point, the
derivative of f(c) is equal to the slope between (a, f(a)) and (b, f(b)).
First, we need to know if f(0) = f(4).
f(0) = 0; f(4) = 0; f(0) = f(4).Then we plug 0 into f ’(x)
2x - 4 = 0; x = 2Is 2 on between 0 and 4? Yes.
Then x = 2 is the answer.
Friday, March 29, 13
Related Rates
The idea behind related rates is that if you take the derivative of a
formula, and you know the rate at which certain things are changing and
some values at a certain point in time, then you can solve for the rest
of the variables.
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.
Formula for this is abc = V, where a is the length, b is the width, c is the
depth, and V is the volume.
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’.
Friday, March 29, 13
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
Vo = Initial VelocitySo = Initial Velocity
t = Time
Friday, March 29, 13
Thanks for watching!
Friday, March 29, 13