Section 1
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uvw'+uv'w+u'vw
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Cards (64)
Section 1
(50 cards)
uvw'+uv'w+u'vw
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.
Derivative of eⁿ
sec(x)+C
L'Hopital's Rule
-cot(x)+C
ln(sinx)+C = -ln(cscx)+C
-sin(x)
Squeeze Theorem
f'(x)-g'(x)
f'(x)+g'(x)
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
Fundamental Theorem of Calculus #2
ln(x)+C
First Derivative Test for local extrema
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)
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
-csc(x)+C
The position function OR s(t)
ln(secx+tanx)+C = -ln(secx-tanx)+C
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
f is continuous at x=c if...
If f and g are inverses of each other, g'(x)
x+c
-csc²(x)
Derivative of ln(u)
f'(g(x))g'(x)
ln(a)*aⁿ+C
1
cf'(x)
sec(x)tan(x)
Horizontal Asymptote
1
dy/dx
cos(x)
tan(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
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.
sin(x)+C
nx^(n-1)
Mean Value Theorem for integrals or the average value of a functions
Global Definition of a Derivative
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).
Point of inflection at x=k
sec²(x)
-cos(x)+C
0
Exponential growth (use N= )
Section 2
(14 cards)