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