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127 Cards in this Set
- Front
- Back
- 3rd side (hint)
∫ 1/x dx
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ln|x|+ C
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d/dx (cu)
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cu'
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d/dx (u ± v)
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u' ± v'
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d/dx (uv)
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uv' + vu'
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Product Rule
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d/dx (u/v)
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(vu'- uv')/v^2
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Quotient Rule
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d/dx (c)
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0
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d/dx (u^n)
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nu^(n-1) * u'
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Power Rule
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d/dx (x)
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1
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d/dx (ln u)
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1/u * u'
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d/dx (e^u)
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e^u * u'
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d/dx (sin u)
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(cos u) u'
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d/dx (cos u)
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- (sin u) u'
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d/dx (tan u)
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(sec^2 u) u'
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psst!
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d/dx (cot u)
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- (csc^2 u) u'
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psst!
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d/dx (sec u)
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(sec u tan u) u'
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psst!
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d/dx (csc u)
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- (csc u cot u) u'
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psst!
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d/dx (arcsin u)
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u' / √(1 - u^2 )
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d/dx (arccos u)
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- u'/ √(1 - u^2 )
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d/dx (arctan u)
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u' / (1 + u^2)
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d/dx (arccot u)
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- u' / (1 + u^2)
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∫ kf(u) du
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k ∫ f(u) du
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Property
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∫〖f(u) ± g(u)〗 du
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∫〖f(u)〗du ± ∫〖g(u)〗du
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Property
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∫ du
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u + C
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∫〖u^n〗du
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u^(n + 1) / (n + 1) + C
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∫〖e^u〗du
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e^u + C
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∫〖sin u〗du
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- cos u + C
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∫〖cos u〗du
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sin u + C
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∫〖tan u〗du
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- ln|cos u |+ C
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∫〖cot u〗du
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ln|sin u |+ C
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∫〖sec u〗du
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ln|sec u tan u |+ C
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∫〖csc u〗du
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- ln|csc u cot u |+ C
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∫〖sec^2 u〗du
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tan u + C
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psst! backwards
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∫〖csc^2 u〗du
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- cot u + C
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psst! backwards
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∫〖sec u tan u〗du
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sec u + C
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psst! backwards
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∫〖csc u cot u〗du
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- csc u + C
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psst! backwards
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∫ 〖1/√(a^2- u^2 ) 〗du
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arcsin u/a + C
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∫ 1/(a^2 + u^2 ) du
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1/a arctan u/a + C
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Definition of a Derivative
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lim(h→0)〖(f(x+h)- f(x))/h〗
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What must be true for a limit to exist?
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LH Limit = RH Limit
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What must be true for continuity?
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f(c) = LH = RH
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How do we know when the limit = infinity?
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When nothing else works, plug in really close numbers to determine sign.
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What are the rules for Limits at Infinity?
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Powers = limit is coefficients.
Power > in Num. = Infinity Power > in Dem. = 0 |
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What is the Intermediate Value Theorem?
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If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k.
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Rate of Change
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Derivative
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Average Rate of Change
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Slope Formula
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Instantaneous Rate of Change
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Derivative
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d/dx [ f ( g(x) ) ]
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f ' (g(x)) * g'(x)
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chain rule
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What do you need to write the equation of a tangent line?
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Point & Slope
Derivative gives Slope |
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What action do I do to find the Summation of an Amount
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Integral
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What action do I take to find the Accumulation of Amount
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Initial Amount + Integral
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Critical Number
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Where derivative equals zero.
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What do you do if looking for relative max or min?
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Make a sign chart and write a sentence. Do not forget endpoints.
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What do you do if looking for absolute max or min?
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Find critical points. Make a table, calculate values, and write a sentence. Do not forget endpoints.
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Mean Value Theorem
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Must be continuous & differentiable.
f ' (c) = ( f(b) - f(a) ) / b - a Derivative = slope of endpoints |
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What does the First Derivative tell you?
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Where function is increasing and decreasing.
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What does the Second Derivative tell you?
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Concavity:
Concave up like a cup, Concave down like a frown. |
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What is the 2nd Derivative Test?
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If 1st Derivative = 0 &
2nd Derivative > 0, point is a min. 2nd Derivative < 0, point is a max 2nd Derivative = 0, test fails |
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Monotonic
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All increasing or all decreasing.
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Normal Line
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Perpendicular to the tangent line.
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Perpendicular Slopes
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Negative reciprocals
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Parallel Slopes
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Parallel slopes are the same.
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How do you know if speed is increasing?
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Velocity & Acceleration have the same sign.
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How do you know if speed is decreasing?
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Velocity & Acceleration have different signs.
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Possible Points of Inflection
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Where 2nd Derivative = 0
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lim┬(x→0)〖(sin x) / x〗
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1
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lim┬(x→0)〖(1 - cos x) / x〗
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0
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Vertical Asymptotes
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Where denomiator = 0 after canceling
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Hole in Function
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Where numerator and denominator cancel
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How do you find the Domain of a function?
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Where denominator = 0
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How do I find the X - Intercepts of a function?
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Where numerator = 0
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Y - Intercept
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Find f(0)
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How do I prove an Even Function?
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Change x-value and get same answer.
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How do I prove an Odd Function?
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Change x- & y-values and get same answer.
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Cos 0
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1
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Sin 0
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0
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Cos π
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0
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Sin π
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0
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cos π/6
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√3 / 2
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sin π/6
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1/2
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cos π/4
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√2 / 2
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sin π/4
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√2 / 2
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cos π/3
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1/2
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sin π/3
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√3 / 2
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Trig Formula that = 1
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sin^2 x + cos^ x = 1
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sin 2u
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2 sin u cos u
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cos 2u
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cos^2 u - sin^2 u
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sin ( u ± v)
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sin u cos v ± cos u sin v
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SC Cool State
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cos ( u ± v)
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cos u cos v - sin u sin v
cos u cos v + sin u sin v |
cos stutters and has issues
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How do you go from velocity to position or location?
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Integral
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How do you go from velocity to acceleration?
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Derivative
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How do you evaluate the derivative of an integral with a function as a limit?
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Derivative & intergral cancel, function goes into "t" times the derivative of the function.
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2nd Fundamental Theorem of Calculus.
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How do you find the integral from a graph?
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Use geometry formulas to determine area between the graph and the x - axis.
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What is the sign of the area of the portion of the graph below the x - axis?
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Negative
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What is the sign of the area of the portion of the graph above the x - axis?
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Positive
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Area of a Trapezoid?
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h/2 * (b1 + b2)
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Width of Equal Intervals in RH or LH Riemann Sum?
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(b - a) / n
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What is the formula for finding the sum by the Trapezoid Rule with equal intervals or subdivisions?
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(b - a) / 2n * [ f(x0) + 2*f(x1) + ...+ 2*f(x[n-1]) + f(xn)]
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How do you use the Trapezoid Rule with unequal intervals or subdivisions?
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Use Geometry. Find the area of each separate trapezoid using the area formula. There is no short cut here!
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How do you find the LH or RH Riemann Sum with unequal intervals?
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Use Geometry. Find the area of each separate rectangle using the area formula. There is no short cut here!
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How do you find the sum using the Midpoint Rule with equal subdivisions?
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(b - a) / n * [ f(midpt1) + f(midpt2) +...+ f(midpt n)]
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How do you find the sum using the Midpoint Rule with unequal subdivisions?
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Use Geometry. Find the area of each separate rectangle using the area formula. There is no short cut here!
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When integrating by substitution and you have limits, what must you do?
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Change the value of the limits from x to u values.
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ln e
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1
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ln 0
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DNE
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What is the sign of
ln (fraction)? |
negative
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ln (negative number)
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DNE
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ln 1
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0
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ln (ab)
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ln a + ln b
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ln (a ^ n)
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n ln a
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ln (a/b)
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ln a - ln b
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f ^( -1) ' (x)
Derivative of an Inverse Function |
1 / ( f ' ( f ^ (-1)(x) ) )
1 over the derivative evaluated at the value of the inverse point |
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What is true about the slopes of inverse functions?
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They have reciprocal slopes.
3 and 1/3 |
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ln (e ^ x)
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x
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e ^ (ln x)
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x
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What does anything raised to the zero power equal?
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1
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d/dx [a ^ u]
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(ln a)(a ^ u) u'
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d/dx [log(base a) u]
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{1/ [ (ln a) * u] } u'
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a ^ [log (base a) x]
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x
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log (base a) a ^ x
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x
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Write y = a ^ x as a log function.
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The base stays the base and exchange the x and y.
log (base a) y = x |
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Exponential Growth & Decay formula
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y = Ce ^ (kt)
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Formula for Volume using the Disk Method
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π ∫ [ f(x) ]^2 dx with limits of a and b
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Formula for Volume using the Washer Method
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π ∫ [f(x)]^2 - [g(x)]^2 dx with limits of a and b. Use absolute value signs if you do not know which function has the larger radius.
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Formula to find the Area of a Region
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∫ [f(x) - g(x)] dx with limits of a and b. Use absolute value signs if you do not know which function is the larger.
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Formula for Volume using Cross-Section
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∫ [Area of Shape] dx with limits of a and b.
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How do we draw the radius when rotating a region to calculate volume?
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The radius is drawn perpendicular to the axis you are rotating around
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What do you do when rotating around a line that is not an axis?
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Subtract the line from f(x) and g(x) prior to squaring.
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