AP Calculus AB Chapter 5 Memory Sheet Check

AP Calculus AB Chapter 5 Memory Sheet Check (Mr. Wong & Mr. Baker)

**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) 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

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)

*sec*² 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(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/\[|x|*sqrt*(x²-1)\]

__Integral of a Constant times a Function__

If k is a constant, then **∫** k \* *f* (x)**dx** = k \* **∫** *f*(x)**dx**

The integral of a constant times a function equals the constant times the integral.

**Integral of a Sum of Functions**

∫\[f(x) + g(x)\]dx = ∫f(x)dx + ∫g(x)dx.

The integral of a sum equals the sum of the integrals.

**Derivative and Integrals of Logarithmic and Exponential Graphics** - d/dx(ln(x))

1/x

**Derivative and Integrals of Logarithmic and Exponential Graphics** - ∫ 1/x dx

ln|x| + C

**Derivative and Integrals of Logarithmic and Exponential Graphics** - d/dx(e^x)

e^x

**Derivative and Integrals of Logarithmic and Exponential Graphics** - ∫ e^x dx

e^x + C

**Derivative and Integrals of Logarithmic and Exponential Graphics** - d/dx(log b X)

1/ln(b) \* 1/x

**Derivative and Integrals of Logarithmic and Exponential Graphics** - d/dx(b^x)

b^x \* ln(b)

**Derivative and Integrals of Logarithmic and Exponential Graphics** - ∫ b^x dx

( 1/ln(b) \* b^x ) + C

**Integrals that Yield Trig Functions** - ∫ cos x dx

sin x + C

**Integrals that Yield Trig Functions** - ∫ sin x dx

\-cos x + C

**Integrals that Yield Trig Functions** - ∫ sec² x dx

tan x + C

**Integrals that Yield Trig Functions** - ∫ csc² x dx

\-cot x + C

**Integrals that Yield Trig Functions** - ∫ sec x tan x dx

sec x + C

**Integrals that Yield Trig Functions** - ∫ csc x cot x dx

\-csc x + C

**Integrals that Yield Inverse Trig Functions** - ∫ 1/sqrt(1-x²) dx

sin⁻¹ x + C

**Integrals that Yield Inverse Trig Functions** - ∫ -1/sqrt(1-x²) dx

cos⁻¹ x + C

**Integrals that Yield Inverse Trig Functions** - ∫ 1/(1+x²) dx

tan⁻¹ + C

**Integrals that Yield Inverse Trig Functions** - ∫ -1/(1+x²) dx

cot⁻¹ x + C

**Integrals that Yield Inverse Trig Functions** - ∫ 1/\[|x|sqrt(x²-1)\] dx

sec⁻¹ x + C

**Integrals that Yield Inverse Trig Functions** - ∫ -1/\[|x|sqrt(x²-1)\] dx

csc⁻¹ x + C

**Definition of Definite Integral**: a ^b∫ f(x)dx = lim Δx → 0 i=1 ^nΣ f(ci)Δx

__Verbally__

Limit of a Riemann Sum

**Definition of Definite Integral**: a ^b∫ f(x)dx = lim Δx → 0 i=1 ^nΣ f(ci)Δx

__Meaning of Definite Integral__

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

**Definition of Definite Integral**: a ^b∫ f(x)dx = lim Δx → 0 i=1 ^nΣ f(ci)Δx

__Geometrical Interpretation of Definite Integral__

Area under the curve between *a* and *b*

**Definition of Integrability**

*f* (x) is integrable on an interval if and only if *f* (x) is continuous on that interval

**Mean Value Theorem**

If

1. f is differentiable on the open interval (a,b), and

2. f is continuous on the close interval [a,b].

   then there is at least one number x = c (a,b) such that f’(c) = [f(b) - f(a)]/(b-a)

**Rolle’s Theorem**

If

1. f is differentiable on the open interval (a, b), and

2. f is continuous on the closed interval [a, b], and

3. f(a) = f(b) = 0

   then there is at least one number x = c (a, b) such that f’(c) = 0

**Fundamental Theorem of Calculus**

If g(x) = ∫ f(x) dx, then a ^b∫ f(x) dx = g(b) - g(a)

**Second Fundamental Theorem of Calculus**

If g(x) = a ^x∫ f(t) d(x), where a is a constant, then g’(x) = f(x)

**Properties of Definite Integrals** - __Reversal Of Limits__

a ^b∫ f(x) dx = -a ^b∫ f(x) dx

**Properties of Definite Integrals** - __Sum of Integrals with Same Integrand__

a ^b∫ f(x) dx = a ^c∫ f(x) dx + c ^b∫f(x) dx

**Properties of Definite Integrals** - __Symmetric Limits__

If *f* is odd , -a ^a∫ f(x) dx = 0. If *f* is even, then -a ^a∫ f(x) dx = 2 \[0 ^a ∫f(x) dx\]

**Properties of Definite Integrals** - __Integral of a Sum of Functions__

a ^b∫\[f(x) + g(x)\] dx = a^b∫ f(x) dx + a^b∫ g(x) dx

The integral of a sum equals the sum of the integrals.

**Properties of Definite Integrals** - __Integral of a Constant times a Function__

a ^b∫ k *f* (x) dx = k \[a ^b∫ f(x) dx\]

The integral of a constant times a function is the constant times the integral.