AP CALCULUS AB: Memory Check - Chapter 4

Mr. Baker's Memory Check! (,,> ᴗ <,,)

Definition of Derivative - Meaning of Derivative:

instantaneous rate of change

Definition of Derivative - Numerical Interpretation

Limit of the average rate of change over the interval from c to x as x approaches c

Definition of Derivative - Geometrical Interpretation of Derivative

Slope of the tangent line

Definition of Definite Integral - Meaning of Definite Integral

Product of (b-a) and f(x)

Definition of Definite Integral - Geometrical Interpretation of Definite Integral

Area under the curve between a and b

Verbal Definition of Limit

L is the limit of f(x) as x approaches c if and only if for any positive number epsilon, no matter how small, there is a positive number delta such that if x is within delta units of c (but not equal to c), then f(x) is within epsilon units of L.

The Limit Theorems (provided lim x→c f(x) and the lim x→c g(x) exists) - Limit of a Product of Functions

lim x→c [f(x) * g(x)] = lim x→c f(x) * lim x→c g(x)

The limit of a product equals the product of its limits.

The Limit Theorems (provided lim *x→c* ***f***(x) and the lim x→c g(x) exists) - Limit of a Sum of functions

**lim** ***x→c*** \[***f***(x) + ***g***(x)\] = lim ***x→c*** ***f***(x) + lim ***x→c*** ***g***(x)

The limit of a sum equals the sum of its limits.

The Limit Theorems (provided lim *x→c* ***f***(x) and the lim *x→c* ***g***(x) exists) - Limit of a Quotient of functions

**lim** x***→c*** \[***f***(x)/***g***(x)\] = lim ***x→c*** ***f***(x) / lim ***x→c*** ***g***(x) where lim ***x→c*** ***g***(x) ≠ 0

The limit of a quotient equals the quotient of its limits.

The Limit Theorems (provided lim *x→c* ***f***(x) and the lim *x→c* ***g***(x) exists) - Limit of a Constant Times a function

**lim** ***x→c*** \[k \* ***f***(x)\] = k \* lim ***x→c*** ***f***(x)

The limit of a constant times a function equals the constant times its limit.

The Limit Theorems (provided lim *x→c* ***f***(x) and the lim *x→c* ***g***(x) exists) -Limit of the identity function

**lim** ***x→c*** x=c

The limit of x as x approaches c is c.

The Limit Theorems (provided lim *x→c* ***f***(x) and the lim *x→c* ***g***(x) exists) -Limit of a constant function

If k is a constant, then **lim** ***x→*** k = k

The limit of the constant is the constant.

Property of Equal Left and Right Limits

**lim** ***x→c f***(x) exists if and only if **lim** ***x→c- f***(x) = lim ***x→c+ f***(x)

Definition of Continuity at a Point

***f*** is continuous at x = c if and only if:

1. ***f***(c) exists

2. **lim** ***x→c f***(x) exists, and

3. **lim** ***x→>c f***(x) = ***f***(c)

Horizontal Asymptote

If **lim** ***x→∞ f***(x) = L or **lim** ***x→-∞ f***(x) = L, then the line y = L is a horizontal asymptote.

Vertical Asymptote

If **lim** ***x→c f***(x) = ∞ or **lim** ***x→c f***(x) = -∞, then the line x = c is a vertical asymptote.

Intermediate Value Theorem (IVT)

If ***f*** is continuous for all x in the closed interval [a,b], and y is a number between ***f***(a) and ***f***(b)**, then there is a number c** in the open interval (a,b) for which ***f***(c)=y

Definition of a Derivative at a Point (**x=c form**)

***f*** ‘(c) = lim ***x→c*** \[***f***(x)-***f***(c)\]/\[x-c\]

Meaning: The instantaneous rate of change of ***f***(x)with respect to x at x=c

Definition of Derivative at a Point (**Δx** or h form)

***f*** '(x) = lim ***Δx→0*** **Δy/Δx = lim** ***Δx→0*** \[***f***(x+Δx)-***f***(x)\]/Δx = lim ***h→0*** \[***f***(x+h)-***f***(h)\]/h

Power Rule

If ***f*** (x) = xⁿ, where n is a constant, then ***f*** '(x) = nxⁿ⁻¹

Properties of Differentiation - Derivative of a Sum of Functions

If ***f***(x) = ***g***(x) + ***h***(x)**, then** ***f*** '(x) = ***g***'**(x) +** ***h***'(x).

The derivative of the Sum equals the Sum of the derivatives.

Properties of Differentiation - Derivative of a Constant Times a Function

If ***f***(x) = k \* ***g***(x)**, where k** is a constant, then ***f*** ’(x) = k \* ***g***’(x).

The derivative of a constant times a function equals the constant times the derivative

Properties of Differentiation - Derivative of a Constant Function

If ***f***(x) = C is a constant, then ***f*** ’(x) = 0.

The derivative of a constant is 0.

Chain Rule (dy/dx form)

dy/dx = dy/du \* du/dx

Chain Rule (***f***(x) form)

\[***f***(***g***(x))\]’ = ***f*** ’(***g***(x)) \* ***g***’(x)

Limit of (sin x) / x

lim *x→0* sin x/x = 1

Relationship between a Graph and its Derivatives Graph - Increasing/Decreasing

***f*** is increasing when ***f*** ’(x) > 0. ***f*** is decreasing when ***f*** ’(x) < 0.

Relationship between a Graph and its Derivatives Graph - Local maximum (or relative maximum)

occurs when ***f*** ’(x) changes from positive to negative at ***x*** **=** ***c***.

Relationship between a Graph and its Derivatives Graph - Local minimum (or relative minimum)

occurs when ***f*** ’(x) changes from negative to positive at ***x*** **=** ***c***.

The Calculus of Motion - Velocity

dx/dt, where x is the displacement

The Calculus of Motion - Acceleration

dv/dt = dx²/dt², where v is the velocity.

The Calculus of Motion - Distance

\[*displacement*\]

The Calculus of Motion - Speed

\[*velocity*\]

The Calculus of Motion - Speeding Up

occurs when velocity and acceleration are the same sign

The Calculus of Motion - Slowing Down

occurs when velocity and acceleration are the opposite signs

Equation of a Tangent Line

The equation of the line tangent to the graph of ***f*** at x = c is given by y = ***f***(c) + ***f*** ’(c)(x-c)

Product Rule

if y = uv, then y’ = u’v + uv’

Quotient Rule

if y = u/v, then y’ = \[*u’v* - uv’\]/*v*², (v ≠ 0)

Relationship between Differentiability and Continuity

If ***f*** is differentiable at x = c, then ***f*** is continuous at x = c.

Contrapositive: If ***f*** is not continuous at x =*c*, then ***f*** is not differentiable at x = c.

Derivative of an Inverse Function

The derivative of ***f*** ⁻¹(x) is 1/*f* ’(y)

Derivative of Trig Functions - d/dx(sin x)

cos x

Derivative of Trig Functions - d/dx(cos x)

\-*sin* x

Derivative of Trig Functions - d/dx(tan x)

sec² x

Derivative of Trig Functions - d/dx(sec x)

sec x tan x

Derivative of Trig Functions - d/dx(cot x)

\-*csc*² x

Derivative of Trig Functions - d/dx(csc x)

\-csc x cot x

Derivative of Inverse Trig Functions - d/dx(sin⁻¹ x)

1/*sqrt*(1-x²)

Derivative of Inverse Trig Functions - d/dx(cos⁻¹ x)

1/*sqrt*(1-x²)

Derivative of Inverse Trig Functions - d/dx(cos⁻¹ x)

\-1/*sqrt*(1-x²)

Derivative of Inverse Trig Functions - d/dx(tan⁻¹x)

1/(1+x²)

Derivative of Inverse Trig Functions - d/dx(cot⁻¹ x)

\-1/(1+x²)

Derivative of Inverse Trig Functions - d/dx(sec⁻¹ x)

1/\[|x|*sqrt*(x²-1)\]

Derivative of Inverse Trig Functions - d/dx(csc⁻¹ x)

\-1/\[|x|*sqrt*(x²-1)\]