Studied by 84 peopleAP 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
f(c) exists
lim x→c f(x) exists, and
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
f is differentiable on the open interval (a,b), and
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
f is differentiable on the open interval (a, b), and
f is continuous on the closed interval [a, b], and
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.
Note
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