Section 1
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Fundamental Theorem of Calculus #2
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Section 1
(50 cards)
Fundamental Theorem of Calculus #2
1
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.
uvw'+uv'w+u'vw
dy/dx
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)
sec(x)tan(x)
cf'(x)
Horizontal Asymptote
sin(x)+C
f'(x)+g'(x)
Formula for Disk Method
Axis of rotation is a boundary of the region.
sec(x)+C
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
tan(x)+C
Mean Value Theorem for integrals or the average value of a functions
f is continuous at x=c if...
-cot(x)+C
-csc²(x)
f'(g(x))g'(x)
1
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
First Derivative Test for local extrema
-sin(x)
ln(secx+tanx)+C = -ln(secx-tanx)+C
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
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.
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).
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
Formula for Washer Method
Axis of rotation is not a boundary of the region.
ln(sinx)+C = -ln(cscx)+C
f'(x)-g'(x)
Area under a curve
Squeeze Theorem
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
sec²(x)
L'Hopital's Rule
The position function OR s(t) with constant acceleration of -32ft/s^2
Exponential growth (use N= )
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.
-csc(x)+C
Point of inflection at x=k
If f and g are inverses of each other, g'(x)
nx^(n-1)
0
x+c
ln(x)+C
-cos(x)+C
Global Definition of a Derivative
cos(x)
Section 2
(25 cards)