ap calculus - semester review

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

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For my AP Calculus class.

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Page 1: AP Calculus - Semester Review

Bland introduction.

By: Chris Wilson

Friday, March 29, 13

Page 2: AP Calculus - Semester Review

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

Page 3: AP Calculus - Semester Review

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

Page 4: AP Calculus - Semester Review

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

Page 5: AP Calculus - Semester Review

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

Page 6: AP Calculus - Semester Review

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

Page 7: AP Calculus - Semester Review

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

Page 8: AP Calculus - Semester Review

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

Page 9: AP Calculus - Semester Review

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

Page 10: AP Calculus - Semester Review

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

Page 11: AP Calculus - Semester Review

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

Page 12: AP Calculus - Semester Review

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

Page 13: AP Calculus - Semester Review

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

Page 14: AP Calculus - Semester Review

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

Page 15: AP Calculus - Semester Review

Thanks for watching!

Friday, March 29, 13