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:

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’]/v², (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)

-csc² 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.