Section 1
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Point of inflection at x=k
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Cards (145)
Section 1
(50 cards)
Point of inflection at x=k
0
sec(x)+C
Exponential growth (use N= )
Mean Value Theorem for integrals or the average value of a functions
ln(secx+tanx)+C = -ln(secx-tanx)+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).
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
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
The position function OR s(t)
f'(x)-g'(x)
uvw'+uv'w+u'vw
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.
dy/dx
Formula for Disk Method
Axis of rotation is a boundary of the region.
f is continuous at x=c if...
-csc²(x)
Global Definition of a Derivative
1
f'(g(x))g'(x)
L'Hopital's Rule
x+c
cos(x)
sec²(x)
-cot(x)+C
Horizontal Asymptote
ln(x)+C
ln(sinx)+C = -ln(cscx)+C
-cos(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)
-csc(x)+C
sin(x)+C
cf'(x)
f'(x)+g'(x)
nx^(n-1)
Area under a curve
-sin(x)
Squeeze Theorem
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
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.
Fundamental Theorem of Calculus #2
If f and g are inverses of each other, g'(x)
Formula for Washer Method
Axis of rotation is not a boundary of the region.
sec(x)tan(x)
1
First Derivative Test for local extrema
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
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.
Section 2
(50 cards)
d/dx[e^g(x)]=
g'(x)e^g(x)
f(x)=ln(x-2)
Asymptote: x=2 Domain: (2, ∞)
Opposite Antiderivatives
f(x)=ln(x+2)
Asymptote: x=-2 Domain: (-2, ∞)
f(x)=-lnx
Asymptote: x=0 Domain: (0, ∞)
Exponential function
D: (-∞,+∞) R: (0,+∞)
Adding or subtracting antiderivatives
d/dx[secx]=
secxtanx
Constants in integrals
What does the graph y = cot(x) look like?
d/dx[sin⁻¹x]=
1/√(1-x^2)
What does the graph y = tan(x) look like?
∫e^xdx=
e^x+C
What does the graph y = csc(x) look like?
Sine function
D: (-∞,+∞) R: [-1,1]
Derivative of eⁿ
∫a^xdx=
(a^x)/lna+C
What does the graph y = sin(x) look like?
f(x)=e^(x+2)
Asymptote: y=0 Domain: (-∞, ∞)
Antiderivative of xⁿ
f(x)=ln(-x)
Asymptote: x=0 Domain: (-∞, 0)
Natural log function
D: (0,+∞) R: (-∞,+∞)
Squaring function
D: (-∞,+∞) R: (o,+∞)
∫1/xdx=
ln|x|+C
Identity function
D: (-∞,+∞) R: (-∞,+∞)
Square root function
D: (0,+∞) R: (0,+∞)
d/dx[a^x]=
a^x*lna
Inverse Secant Antiderivative
d/dx[cscx]=
-cscxcotx
f(x) = e^(x-2)
Asymptote: y=0 Domain: (-∞, ∞)
What does the graph y = sec(x) look like?
d/dx[tan⁻¹x]=
1/(1+x^2)
d/dx[cos⁻¹x]=
-1/√(1-x^2)
f(x) = e^(x) +2
Asymptote: y=2 Domain: (-∞, ∞)
d/dx[e^x]=
e^x
Cubing function
D: (-∞,+∞) R: (-∞,+∞)
f(x)= -2+lnx
Asymptote: x=0 Domain: (0, ∞)
Absolute value function
D: (-∞,+∞) R: [0,+∞)
Inverse Sine Antiderivative
d/dx[tanx]=
sec²x
d/dx[cotx]=
-csc²x
Derivative of ln(u)
Antiderivative of f(x) from [a,b]
Inverse Tangent Antiderivative
Cosine function
D: (-∞,+∞) R: [-1,1]
d/dx[a^g(x)]=
g'(x)a^g(x)lna
Logistic function
D: (-∞,+∞) R: (0, 1)
ln(a)*aⁿ+C
Reciprocal function
D: (-∞,+∞) x can't be zero R: (-∞,+∞) y can't be zero
What does the graph y = cos(x) look like?
Section 3
(45 cards)