Section 1
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-ln(cosx)+C = ln(secx)+C
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Cards (176)
Section 1
(50 cards)
-ln(cosx)+C = ln(secx)+C
hint: tanu = sinu/cosu
Mean Value Theorem for integrals or the average value of a functions
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)
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
-csc(x)+C
Point of inflection at x=k
Global Definition of a Derivative
0
Alternative Definition of a Derivative
f '(x) is the limit of the following difference quotient as x approaches c
sin(x)+C
ln(sinx)+C = -ln(cscx)+C
If f and g are inverses of each other, g'(x)
nx^(n-1)
tan(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).
Formula for Washer Method
Axis of rotation is not a boundary of the region.
uvw'+uv'w+u'vw
1
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.
sec²(x)
Fundamental Theorem of Calculus #2
-csc²(x)
1
sec(x)tan(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.
dy/dx
ln(secx+tanx)+C = -ln(secx-tanx)+C
Fundamental Theorem of Calculus #1
The definite integral of a rate of change is the total change in the original function.
x+c
L'Hopital's Rule
Horizontal Asymptote
Formula for Disk Method
Axis of rotation is a boundary of the region.
cf'(x)
ln(x)+C
-cos(x)+C
f'(g(x))g'(x)
cos(x)
Exponential growth (use N= )
f is continuous at x=c if...
Squeeze Theorem
f'(x)-g'(x)
Area under a curve
The position function OR s(t)
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.
-cot(x)+C
ln(cscx+cotx)+C = -ln(cscx-cotx)+C
sec(x)+C
f'(x)+g'(x)
First Derivative Test for local extrema
-sin(x)
Section 2
(50 cards)
Squaring function
D: (-∞,+∞) R: (o,+∞)
cos(3π/2)
0
cos(5π/4)
−√2/2
sin(5π/6)
1/2
Logistic function
D: (-∞,+∞) R: (0, 1)
sin(π/4)
√2/2
Constants in integrals
Cubing function
D: (-∞,+∞) R: (-∞,+∞)
sin(π/3)
√3/2
sin(3π/2)
−1
Cosine function
D: (-∞,+∞) R: [-1,1]
sin(π)
0
sin(π/2)
1
cos(4π/3)
−1/2
Inverse Tangent Antiderivative
cos(π/2)
0
Derivative of ln(u)
Adding or subtracting antiderivatives
sin(π/6)
1/2
sin(7π/6)
−1/2
sin(4π/3)
−√3/2
Inverse Secant Antiderivative
cos(7π/4)
√2/2
sin(5π/4)
−√2/2
cos(3π/4)
−√2/2
cos(π)
−1
Derivative of eⁿ
Antiderivative of xⁿ
Absolute value function
D: (-∞,+∞) R: [0,+∞)
sin(2π/3)
√3/2
cos(5π/3)
1/2
sin(3π/4)
√2/2
cos(π/3)
1/2
cos(2π)
1
cos(π/4)
√2/2
Square root function
D: (0,+∞) R: (0,+∞)
cos(π/6)
√3/2
Reciprocal function
D: (-∞,+∞) x can't be zero R: (-∞,+∞) y can't be zero
Natural log function
D: (0,+∞) R: (-∞,+∞)
cos(5π/6)
−√3/2
Antiderivative of f(x) from [a,b]
ln(a)*aⁿ+C
cos(7π/6)
−√3/2
Sine function
D: (-∞,+∞) R: [-1,1]
Inverse Sine Antiderivative
cos(11π/6)
√3/2
cos(2π/3)
−1/2
Exponential function
D: (-∞,+∞) R: (0,+∞)
Opposite Antiderivatives
Identity function
D: (-∞,+∞) R: (-∞,+∞)
Section 3
(50 cards)
What does the graph y = tan(x) look like?
f(x)=-lnx
Asymptote: x=0 Domain: (0, ∞)
What does the graph y = sec(x) look like?
f(x)=ln(x-2)
Asymptote: x=2 Domain: (2, ∞)
∫1/(a^2+x^2)dx=
(1/a)(tan⁻¹(x/a)+C
d/dx[e^x]=
e^x
d/dx[a^g(x)]=
g'(x)a^g(x)lna
d/dx[cos⁻¹x]=
-1/√(1-x^2)
sin(11π/6)
−1/2
What does the graph y = sin(x) look like?
f(x) = e^(x-2)
Asymptote: y=0 Domain: (-∞, ∞)
What does the graph y = cos(x) look like?
f(x) = e^(x) +2
Asymptote: y=2 Domain: (-∞, ∞)
d/dx[tanx]=
sec²x
sin(7π/4)
−√2/2
∫a^xdx=
(a^x)/lna+C
What does the graph y = cot(x) look like?
sin(5π/3)
−√3/2
∫1/(1+x^2)dx=
tan⁻¹x+C
d/dx[tan⁻¹x]=
1/(1+x^2)
What does the graph y = sin(x) look like?
What does the graph y = csc(x) look like?
Trig Identity: sin²x=
½(1-cos(2x))
What does the graph y = tan(x) look like?
Trig Identity: sin(2x)=
2sinxcosx
f(x)=e^(x+2)
Asymptote: y=0 Domain: (-∞, ∞)
d/dx[secx]=
secxtanx
Trig Identity: cos²x=
½(1+cos(2x))
f(x)= -2+lnx
Asymptote: x=0 Domain: (0, ∞)
d/dx[e^g(x)]=
g'(x)e^g(x)
Trig Identity: cos(2x)=
1-2sin²x = 2cos²x-1
f(x)=ln(x+2)
Asymptote: x=-2 Domain: (-2, ∞)
∫tanxdx=
ln|secx|+C
d/dx[a^x]=
a^x*lna
Integration by Parts: Choice of u
I = Inverse Trig Function L = Natural log (lnx) A = Algebraic Expression (x, x², x³...) T = Trig function (sinx, cosx) E = e^x
d/dx[cscx]=
-cscxcotx
d/dx[cotx]=
-csc²x
What does the graph y = cos(x) look like?
∫1/√(1-x^2)dx=
sin⁻¹x+C
What does the graph y = csc(x) look like?
What does the graph y = sec(x) look like?
f(x)=ln(-x)
Asymptote: x=0 Domain: (-∞, 0)
Trig Identity: 1=
cos²x+sin²x
∫1/xdx=
ln|x|+C
Trig Identity: sec²x=
tan²x+1
sin(2π)
0
∫e^xdx=
e^x+C
d/dx[sin⁻¹x]=
1/√(1-x^2)
∫secxdx=
ln|secx+tanx|+C
What does the graph y = cot(x) look like?
Section 4
(26 cards)