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Squeeze Theorem
f is continuous at x=c if...
Intermediate Value Theorem
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)=k
Global Definition of a Derivative
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
nx^(n-1)
1
cf'(x)
f'(x)+g'(x)
f'(x)-g'(x)
cos(x)
-sin(x)
sec²(x)
-csc²(x)
sec(x)tan(x)
dy/dx
f'(g(x))g'(x)
Extreme Value Theorem
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.
Critical Number
If f'(c)=0 or does not exist, and c is in the domain of f, then c is a critical number. (Derivative is 0 or undefined)
Rolle's Theorem
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).
Mean Value Theorem
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.
First Derivative Test for local extrema
Point of inflection at x=k
Combo 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.
If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.
Horizontal Asymptote
x+c
sin(x)+C
-cos(x)+C
tan(x)+C
-cot(x)+C
sec(x)+C
-csc(x)+C
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
Fundamental Theorem of Calculus #2
Mean Value Theorem for integrals or the average value of a functions
ln(x)+C
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
ln(sinx)+C = -ln(cscx)+C
ln(secx+tanx)+C = -ln(secx-tanx)+C
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
If f and g are inverses of each other, g'(x)
Area under a curve
Formula for Disk Method
Axis of rotation is a boundary of the region.
Formula for Washer Method
Axis of rotation is not a boundary of the region.
Inverse Secant Antiderivative
Inverse Tangent Antiderivative
Inverse Sine Antiderivative
Derivative of eⁿ
ln(a)*aⁿ+C
Derivative of ln(u)
Antiderivative of f(x) from [a,b]
Opposite Antiderivatives
Antiderivative of xⁿ
Adding or subtracting antiderivatives
Constants in integrals
Identity function
D: (-∞,+∞)
R: (-∞,+∞)
R: (-∞,+∞)
Squaring function
D: (-∞,+∞)
R: (o,+∞)
R: (o,+∞)
Cubing function
D: (-∞,+∞)
R: (-∞,+∞)
R: (-∞,+∞)
Reciprocal function
D: (-∞,+∞) x can't be zero
R: (-∞,+∞) y can't be zero
R: (-∞,+∞) y can't be zero
Square root function
D: (0,+∞)
R: (0,+∞)
R: (0,+∞)
Exponential function
D: (-∞,+∞)
R: (0,+∞)
R: (0,+∞)
Natural log function
D: (0,+∞)
R: (-∞,+∞)
R: (-∞,+∞)
Sine function
D: (-∞,+∞)
R: [-1,1]
R: [-1,1]
Cosine function
D: (-∞,+∞)
R: [-1,1]
R: [-1,1]
Absolute value function
D: (-∞,+∞)
R: [0,+∞)
R: [0,+∞)
Given f(x):
Is f continuous @ C
Is f' continuous @ C
Is f continuous @ C
Is f' continuous @ C
Yes lim+=lim-=f(c)
No, f'(c) doesn't exist because of cusp
No, f'(c) doesn't exist because of cusp
Given f'(x):
Is f continuous @ c?
Is there an inflection point on f @ C?
Is f continuous @ c?
Is there an inflection point on f @ C?
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''
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''