Section 1
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Intermediate Value Theorem
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Cards (61)
Section 1
(50 cards)
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(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
1
limit sin(θ) when θ goes to zero
ln(sinx)+C = -ln(cscx)+C
-cos(x)+C
Average value of a functions
cos(x)
L'Hopital's Rule
-csc²(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.
0
limit 1- cos(θ) when θ goes to zero
sec²(x)
Point of inflection at x=k
quadratic function
D: (-∞,+∞) R: (o,+∞)
cf'(x)
If f and g are inverses of each other, g'(x)
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).
First Derivative Test for local extrema
f'(x)+g'(x)
Horizontal Asymptote
dy/dx
sec(x)tan(x)
ln(x)+C
Mean Value Theorem
The instantaneous rate of change will equal the average rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.
f'(g(x))g'(x)
uvw'+uv'w+u'vw
Derivative of ln(u)
-cot(x)+C
Identity function
D: (-∞,+∞) R: (-∞,+∞)
Derivative of eⁿ
-sin(x)
ln(a)*aⁿ+C
Definition of a Derivative
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
sec(x)+C
sin(x)+C
f is continuous at x=c if...
x+c
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
Exponential growth (use N= )
ln(secx+tanx)+C = -ln(secx-tanx)+C
-csc(x)+C
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.
tan(x)+C
nx^(n-1)
Critical Number
If f'(c)=0 or does not exist, and c is in the domain of f. (Derivative is 0 or undefined)
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
Fundamental Theorem of Calculus #2
f'(x)-g'(x)
Cubing function
D: (-∞,+∞) R: (-∞,+∞)
Section 2
(11 cards)