Section 1
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Rolle's Theorem
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Last updated
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Date created
Mar 1, 2020
Cards (66)
Section 1
(50 cards)
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).
sec(x)tan(x)
Formula for Disk Method
Axis of rotation is a boundary of the region.
Horizontal Asymptote
Mean Value Theorem for Derivatives
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.
Mean Value Theorem for integrals or the average value of a functions
L'Hopital's Rule
f'(x)+g'(x)
If f and g are inverses of each other, 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.
Derivative of ln(u)
Opposite Antiderivatives
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
ln(x)+C
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
1
ln(sinx)+C = -ln(cscx)+C
Point of inflection at x=k
tan(x)+C
1
-cot(x)+C
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)
Formula for Washer Method
Axis of rotation is not a boundary of the region.
First Derivative Test for local extrema
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
-csc²(x)
Fundamental Theorem of Calculus #2
nx^(n-1)
Inverse Sine Antiderivative
2nd derivative test
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.
Inverse Tangent Antiderivative
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
ln(secx+tanx)+C = -ln(secx-tanx)+C
Area under a curve
Global Definition of a Derivative
Inverse Secant Antiderivative
x+c
sec(x)+C
f'(x)-g'(x)
sec²(x)
sin(x)+C
-sin(x)
-csc(x)+C
cos(x)
Derivative of eⁿ
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
Antiderivative of f(x) from [a,b]
f'(g(x))g'(x)
-cos(x)+C
0
Section 2
(16 cards)