Ap calc BC

AROC

Slope between two points

y2-y1 / x2-x1

Instantaneous ROC

The slope of the tangent line at a specific point on a curve, representing the rate of change of a function at that point.

AROC except the points get infinitely close together (LIMIT!!)

Limit notation

The super script on a tells if you are approaching from left or right

- is left +is right

If the left and right limits ARE NOT EQUAL then the limit does not exist overall

Understanding limits graphically and tabularly

Graphs: look at points and their behavior near the limit, while in a table, analyze values approaching the limit from both sides (if they are really big or small its infinity, if they approach the same left and right the limit exists, etc.)

Limit Constant Rule

Limit Identity rule

If you plug c in for x

Limit Coefficient rule

Coefficient can come outside

Limit Sum/difference rule

Limit product rule

Limit quotient rule

Limit power rule

Limit root rule

Limit composite function rule

(do the limit of the INSIDE function and then just plug the answer into the outside fcn)

Limit Squeeze Theorem

Steps to solve limits algebraically

Direct substitution

• Get a number = plug it in and solve

  • Function or Piecewise

• Get indeterminate form = simplify algebraically

  • Factoring (polynomials)

  • Conjugates (radicals)

  • Common denominator (fractions)

Direct substitution for regular functions

plug into function and solve

Direct substitution for piecewise functions

Plug in for both the left and right side to find left/right hand limits. If they give the same value that is the answer for overall limit

Indeterminate limit algebra simplification: factoring for polynomials

When you plug into the polynomials on top and bottom you get 0/0

Factor both polynomials, and something should cancel out

Re-substitute value into limit and solve!

Indeterminate limit algebra simplification: conjugates for radicals

When you plug into top and bottom with the radical you get 0/0

Multiply on top and bottom by the expression with the radical BUT the signs are flipped (multiply by conjugate (conjugate of (a-b) is (a+b))

Something should cancel

Substitute and solve!

Indeterminate limit algebra simplification: Common Denominator for fractions

When you plug into the top and bottom including fractions you get 0/0

Multiply top and bottom of individual fractions and overall fraction to get common denominator

Simplify and cancel stuff

Substitute and solve!

Definition of continuity

1. Function exists

2. OVERALL limit exist

3. Function = limit

Types of Discontinuity

1. Removable discontinuity: the limit exists, but either f(a) is undefined or f(a) is not equal to the limit. Graphically: a hole (maybe with a misplaced dot). (Removeable)

2. Jump discontinuity: the left and right limits exist but are different. This occurs when the curve “breaks” and resumes at a different height. (Nonremovable)

3. Infinite discontinuity (also called essential/infinite discontinuity): the function becomes unbounded near the point, typically because there is a vertical asymptote. (Nonremovable)

How to tell algebraically if a discontinuity is removeable

If a factor cancels: the point that would go there is removable

Factors left in denominator: the point that goes there is nonremovable

Limits that solve to infinity

If the bottom is 0 after simplifying, there is an asymptote and it goes to infinity

To find out if it is positive or negative infinity you must find the top and bottom signs

On top plug in and solve

On bottom plug it .1 away from the number and see if it would be + or -

• (plug in .1 less for - left hand lims)

• (plug in .1 more for + right hand lims)

Solving for limits when x→infinity

Top degree = bottom degree

• Use leading coefficients of top and bottom to tell you the exact number

Top degree < bottom degree

• The limit approaches 0

Top degree > bottom degree

• Use polynomial long division to find and graph slant asymptote and then figure out end behavior based on slant

Intermediate value theorem

1. If a fcn is continuous

2. and f(a) DOES NOT EQUAL f(b)

3. then there is a value c such that if a < c < b then f(a) < f( c) < f(b)

Derivative of Constant

0

Derivative if a single variable (x)

1

Sum and Difference rule for derivatives

Power rule

Derivative sinx

cosx

Derivative of cosx

-sinx

derivative of tanx

sec²x

derivative of secx

secx • tanx

derivative of cscx

-cscx • cotx

derivative of cotx

-csc²x

Product rule

derivative of f(x) • g(x)

f’(x)•g(x) + f(x)•g’(x)

quotient rule

derivative of f(x) / g(x)

( f’(x)•g(x) - f(x)•g’(x) ) / (g(x))²

chain rule

derivative of f(g(x))

f’(g(x)) • g’(x)

Things to know for find tangent lines to curves

• Point slope form: (y-y1) = m(x-x1)

• Point (x1,y1) comes from the function values at the point

• Slope (m) comes from the value of the derivative at that point

• You can also convert this to standard form (y=mx+b) by multiplying m and adding y1 to both sides

Finding a normal line

• Normal lines can still be point slope form (y-y1)=m(x-x1)

• The point (x1,y1)

• The slope (m) is the perpendicular line to the tangent line

• This means the slope is the opposite sign reciprocal to the derivative value at the point

What are position velocity and acceleration of a particle in relation to each other

s(t)=position

v(t)=s’(t)=velocity

a(t)=s”(t)=acceleration

How to know which way a particle is moving

• If v(t) is positive, the particle moves up or right

• If v(t) is negative, the particle moves down or left

• If v(t) is 0 the particle is stopped

What is the speed of a particle in rectangular form

speed = l velocity l

How to tell if a particle is speeding up or slowing down

• Speeding up if a(t) and v(t) have the same sign

• Slowing down if a(t) and v(t) have opposite signs

Requirements for differentiability

• The derivative/slope must be the same value on the left and right

• Cannot have cusps/corners

• Cannot have vertical tangent

• Cannot have discontinuity

Steps for implicit differentiation

1. Differentiate BOTH sides with respect to independent variable (x)(Must obey chain rule with dy/dx)

2. Get all dy/dx terms to one side and others to opposite

3. Factor out dy/dx if necessary

4. Solve for dy/dx

Formula for inverse derivatives

[f^-1]’(x) = 1 / f’(f^-1(x))

Tips for derivatives of inverse fcns

• The derivative of the inverse function at x is the reciprocal of the derivative of the function at y.

• To find (f^-1(x)) take whatever x value is given, find what y is at that point and plug that y value in as f^-1(x)

derivative of (sin^-1(u))

u’ / √(1-u²)

derivative of (cos^-1(u))

-u’ / √(1-u²)

derivative of (tan^-1(u))

u’ / 1+u²

derivative of (cot^-1(u))

-u’ / 1+u²

derivative of (sec^-1(u))

u’ / |u| * √(u²-1)

derivative of (csc^-1(u))

-u’ / |u| * √(u²-1)