Section 1
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The Chain Rule: f'(g(x))g'(x)
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Cards (163)
Section 1
(50 cards)
The Chain Rule: f'(g(x))g'(x)
Squeeze Theorem
Define:
Rolle's Theorem
What theorem states that if we let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval)?
uvw'+uv'w+u'vw
Critical Number
If f'(c)=0 or does not exist, and c is in the domain of f, then c is a what? (Derivative is 0 or undefined)
Extreme Value Theorem
What theorem states that if f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval?
0
f '(x) is the limit of "[f(x)-f(c)]/[x-c]" (as x approaches c)
What is the Alternative Definition of a Derivative?
Alternative Definition of a Derivative
Define: f '(x) is the limit of the following difference quotient as x approaches c
sec(x)+C
Intermediate Value Theorem
What is the name of the theorem that states: "if f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)?"
f'(x)-g'(x)
The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line. f '(c) = [f(b) - f(a)]/[b - a]
Define the Mean Value Theorem.
-cos(x)+C
-cot(x)+C
1
Yes lim+=lim-=f(c) No, f'(c) doesn't exist because of cusp
Given f(x): Is f continuous @ C Is f' continuous @ C
1
tan(x)+C
Let c be a critical number of a function f that is continuous on the closed interval [a,b] that contains c. If f is differentiable on [a,b], then f(c) can be classified as follows... If f '(x) changes from a negative to a positive at c, then f(c) is a relative minimum of f. If f' (x) changes from a negative to a positive at c, then f(c) is a relative maximum of f
Define the First Derivative Test for local extrema.
cf'(x)
If k is in the domain of f If f ''(k)=0 or does not exist If f ''(x) changes sign @ x=k
When is x=k a point of inflection?
Define the Squeeze Theorem.
Suppose that g(x)≤f(x) and also suppose that {the limit of g(x) (as x goes to a)} = {the limit of h(x) (as x goes to a)} = L then {the limit of f(x) (as x goes to a) = L}
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.
Define the Extreme Value Theorem.
L'Hopital's Rule
Define:
1.) F(c) exists 2.) limit F(x) as x approaches c exists 3.) limit F(x) as x approaches c = F(c)
f is continuous at x=c if...
This is a graph of f'(x). Since f'(C) exists, differentiability implies continuouity, so Yes. Yes f' decreases on X<C so f''<0 f' increases on X>C so f''>0 A point of inflection happens on a sign change at f''
Given f'(x): Is f continuous @ c? Is there an inflection point on f @ C?
sec(x)tan(x)
The position function OR s(t)
Define:
Mean Value Theorem
What theorem states that the instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.
dy/dx
If we let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).
Define Rolle's Theorem.
x+c
the limit of {[f(x ⍖ Δx) - f(x)]/Δx} (as Δx approaches 0)
What is the Global Definition of a Derivative?
First Derivative Test for local extrema
Define:
sec²(x)
sin(x)+C
Combo Test (Second Derivative Test) for local extrema
If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c. If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.
-sin(x)
The definite integral of a rate of change is the total change in the original function. ∫ₐᵇ f(x)dx = F(b) - F(a)
What is the 1st fundamental theorem of Calculus?
-16t² ⍖ v₀t ⍖ s₀
What is the position function OR s(t)
cos(x)
-csc²(x)
-csc(x)+C
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)
Define the Intermediate Value Theorem.
Horizontal Asymptote
Type of Asymptote?
f'(x)+g'(x)
Global Definition of a Derivative
Define:
nx^(n-1)
Section 2
(50 cards)
d/dx (∫ˣ sub-c) f(t)dt = f(x)
What is the second fundamental theorem of Calculus?
Squaring function
Define: D: (-∞,+∞) R: (o,+∞)
Identity function
Define: D: (-∞,+∞) R: (-∞,+∞)
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
Inverse Tangent Antiderivative
Define:
Derivative of ln(u)
Define:
0
cos(3π/2)
Formula for Disk Method
Assume the axis of rotation is a boundary of the region, and define:
1
cos(2π)
Formula for Washer Method
Assume the axis of rotation is not a boundary of the region, and define:
Logistic function
Define: D: (-∞,+∞) R: (0, 1)
g'(x)
Assume f and g are inverses of each other.
ln(secx+tanx)+C = -ln(secx-tanx)+C
Opposite Antiderivatives
Define:
Fundamental Theorem of Calculus #2
√3/2
cos(π/6)
1/2
cos(5π/3)
ln(x)+C
Derivative of eⁿ
Define:
1/2
sin(π/6)
1/2
cos(π/3)
Square root function
Define: D: (0,+∞) R: (0,+∞)
Inverse Sine Antiderivative
Define:
Constants in integrals
Define:
ln(sinx)+C = -ln(cscx)+C
Sine function
Define: D: (-∞,+∞) R: [-1,1]
√2/2
cos(π/4)
Natural log function
Define: D: (0,+∞) R: (-∞,+∞)
Antiderivative of xⁿ
Define:
Greatest integer function
Define: D: (-∞,+∞) R: (-∞,+∞)
√3/2
cos(11π/6)
Exponential function
Define: D: (-∞,+∞) R: (0,+∞)
−√3/2
cos(5π/6)
Mean Value Theorem for integrals or the average value of a functions
Define this statement:
Absolute value function
Define: D: (-∞,+∞) R: [0,+∞)
Inverse Secant Antiderivative
Define:
Antiderivative of f(x) from [a,b]
Define:
-1
cos(π)
Cubing function
Define: D: (-∞,+∞) R: (-∞,+∞)
Area under a curve
Define:
Exponential growth
Define: N=
0
cos(π/2)
Reciprocal function
Define: D: (-∞,+∞) x can't be zero R: (-∞,+∞) y can't be zero
√2/2
cos(7π/4)
Cosine function
Define: D: (-∞,+∞) R: [-1,1]
-1/2
cos(2π/3)
ln(a)*aⁿ+C
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
Adding or subtracting antiderivatives
Define:
−√2/2
cos(3π/4)
Section 3
(50 cards)
√3/2
sin(2π/3)
If you want to find the X intercept, cover up the Y as well as its multiple (i.e. if the y were a 6y, then both the 6 and the y would be covered up), and solve for X, and visa versa.
Describe the cover up method to finding an intercept when the equation of a line is in standard form.
√3/2
sin(π/3)
√2/2
sin(3π/4)
The object has changed direction.
What is indicated by v(t) changing sign?
v(t) > 0
What is v(t) when an object is moving right?
The object is speeding up.
What is indicated by an a(t) and a v(t) that have the same sign?
The object is slowing down.
What is indicated by an a(t) and a v(t) that have different signs?
vu'+uv'
d/dx[uv]=
The object is stopped,
What is indicated by a v(t) that equals zero?
−√3/2
sin(4π/3)
The object is moving right.
What is indicated by a v(t) that is greater than zero?
y = tan(x)
What is this a graph of?
(vu'-uv')/v^2
d/dx[u/v]=
v(t) changes sign
What happens when an object changes direction?
a(t) and v(t) have different signs
What is noteworthy about a(t) and v(t) when an object is slowing down?
[s(b)-s(a)] / (b - a)
Express Average Velocity.
y = cos(x)
What is this a graph of?
Vertical because there isn't a Y intercept
Is X equals C horizontal or vertical? Why?
vu'+uv'
Express the Product Rule.
1/2
sin(5π/6)
secxtanx
d/dx[secx]=
√2/2
sin(π/4)
v(t) < 0
What is v(t) when an object is moving left?
0
What is v(t) when an object is stopped?
Horizontal because there isn't a X intercept
Is Y equals C horizontal or vertical? Why?
−1/2
sin(11π/6)
The object is moving left.
What is indicated by a v(t) that is less than zero?
−√2/2
sin(5π/4)
m equals y2 minus y1 over x2 minus x1
What is the point slope formula?
(Change in Position)/(Change in Time)
Average Velocity
y = sin(x)
What is this a graph of?
−√3/2
sin(5π/3)
1
sin(π/2)
v(t)
d/dt[s(t)]=
−1/2
sin(7π/6)
0
sin(π)
sec²x
d/dx[tanx]=
(Change in Velocity)/(Change in Time)
Average Acceleration
a(t) and v(t) have the same sign
What is noteworthy about a(t) and v(t) when an object is speeding up?
s(b) - s(a)
Express Displacement.
low d-hi minus high d-lo over low squared
What is the catchy way of remembering the Quotient Rule?
−1
sin(3π/2)
e^x
d/dx[e^x]=
a(t)
d/dt[v(t)]=
-csc²x
d/dx[cotx]=
0
sin(2π)
−√2/2
sin(7π/4)
1 (This is a Trig identity)
cos²x+sin²x
-cscxcotx
d/dx[cscx]=
Section 4
(13 cards)