Section 1
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y = sin(x), y' =
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Cards (153)
Section 1
(50 cards)
y = sin(x), y' =
y' = cos(x)
Quotient Rule
(uv'-vu')/v²
definite integral
has limits a & b, find antiderivative, F(b) - F(a)
right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
indefinite integral
no limits, find antiderivative + C, use inital value to find C
y = e^x, y' =
y' = e^x
y = ln(x), y' =
y' = 1/x
When is a function not differentiable
corner, cusp, vertical tangent, discontinuity
y = csc(x), y' =
y' = -csc(x)cot(x)
When f '(x) is positive, f(x) is
increasing
left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
area above x-axis is
positive
rate
derivative
If f '(x) = 0 and f"(x) < 0,
f(x) has a relative maximum
Intermediate Value Theorem
If f(1)=-4 and f(6)=9, then there must be a x-value between 1 and 6 where f crosses the x-axis.
Instantenous Rate of Change
Slope of tangent line at a point, value of derivative at a point
y = tan(x), y' =
y' = sec²(x)
When f '(x) changes from negative to positive, f(x) has a
relative minimum
absolute value of velocity
speed
Chain Rule
f '(g(x)) g'(x)
y = x cos(x), state rule used to find derivative
product rule
y = log (base a) x, y' =
y' = 1/(x lna)
y = a^x, y' =
y' = a^x ln(a)
Alternate definition of derivative
limit as x approaches a of [f(x)-f(a)]/(x-a)
Linearization
use tangent line to approximate values of the function
Formal definition of derivative
Average Rate of Change
Slope of secant line between two points, use to estimate instantanous rate of change at a point.
Particle is moving to the left/down
velocity is negative
Particle is moving to the right/up
velocity is positive
y = cot(x), y' =
y' = -csc²(x)
When f '(x) is negative, f(x) is
decreasing
Product Rule
uv' + vu'
When f '(x) is increasing, f(x) is
concave up
area under a curve
∫ f(x) dx integrate over interval a to b
trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
To find absolute maximum on closed interval [a, b], you must consider...
critical points and endpoints
y = sec(x), y' =
y' = sec(x)tan(x)
If f '(x) = 0 and f"(x) > 0,
f(x) has a relative minimum
y = cos⁻¹(x), y' =
y' = -1/√(1 - x²)
y = cos(x), y' =
y' = -sin(x)
When f '(x) changes from positive to negative, f(x) has a
relative maximum
y = cos²(3x)
chain rule
y = cot⁻¹(x), y' =
y' = -1/(1 + x²)
y = tan⁻¹(x), y' =
y' = 1/(1 + x²)
y = ln(x)/x², state rule used to find derivative
quotient rule
When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a
point of inflection
When f '(x) is decreasing, f(x) is
concave down
y = sin⁻¹(x), y' =
y' = 1/√(1 - x²)
[(h1 - h2)/2]*base
area of trapezoid
mean value theorem
if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)
Section 2
(50 cards)
second derivative of parametrically defined curve
find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt
L'Hopitals rule
use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit
use substitution to integrate when
a function and it's derivative are in the integrand
ratio test
lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges
Fundamental Theorem of Calculus
∫ f(x) dx on interval a to b = F(b) - F(a)
indeterminate forms
0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰
find interval of convergence
use ratio test, set > 1 and solve absolute value equations, check endpoints
Volume of Shell
given v(t) find total distance travelled
∫ abs[v(t)] over interval a to b
slope of vertical line
undefined
nth term test
if terms grow without bound, series diverges
slope of horizontal line
zero
volume of solid of revolution - no washer
π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution
To find particular solution to differential equation, dy/dx = x/y
separate variables, integrate + C, use initial condition to find C, solve for y
∫ u dv =
uv - ∫ v du
To draw a slope field,
plug (x,y) coordinates into differential equation, draw short segments representing slope at each point
geometric series test
general term = a₁r^n, converges if -1 < r < 1
area inside one polar curve and outside another polar curve
1/2 ∫ R² - r² over interval from a to b, find a & b by setting equations equal, solve for theta.
given velocity vectors dx/dt and dy/dt, find total distance travelled
∫ √ (dx/dt)² + (dy/dt)² over interval from a to b
average value of f(x)
= 1/(b-a) ∫ f(x) dx on interval a to b
volume of solid of revolution - washer
π ∫ R² - r² dx over interval a to b, where R = distance from outside curve to axis of revolution, r = distance from inside curve to axis of revolution
area inside polar curve
1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta
given velocity vectors dx/dt and dy/dt, find speed
√(dx/dt)² + (dy/dt)² not an integral!
derivative of parametrically defined curve x(t) and y(t)
dy/dx = dy/dt / dx/dt
length of parametric curve
∫ √ (dx/dt)² + (dy/dt)² over interval from a to b
length of curve
∫ √(1 + (dy/dx)²) dx over interval a to b
methods of integration
substitution, parts, partial fractions
limit comparison test
if lim as n approaches ∞ of ratio of comparison series/general term is positive and finite, then series behaves like comparison series
given v(t) and initial position t = a, find final position when t = b
s₁+ Δs = s Δs = ∫ v(t) over interval a to b
Product rule Derivatives
converges conditionally
alternating series converges and general term diverges with another test
dP/dt = kP(M - P)
logistic differential equation, M = carrying capacity
P = M / (1 + Ae^(-Mkt))
logistic growth equation
area between two curves
∫ f(x) - g(x) over interval a to b, where f(x) is top function and g(x) is bottom function
alternating series test
lim as n approaches zero of general term = 0 and terms decrease, series converges
6th degree Taylor Polynomial
polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative
Volume of Disc
area below x-axis is
negative
Taylor series
polynomial with infinite number of terms, includes general term
use integration by parts when
two different types of functions are multiplied
find radius of convergence
use ratio test, set > 1 and solve absolute value equations, radius = center - endpoint
given rate equation, R(t) and inital condition when t = a, R(t) = y₁ find final value when t = b
y₁ + Δy = y Δy = ∫ R(t) over interval a to b
converges absolutely
alternating series converges and general term converges with another test
given v(t) find displacement
∫ v(t) over interval a to b
Volume of Washer
use partial fractions to integrate when
integrand is a rational function with a factorable denominator
integral test
if integral converges, series converges
volume of solid with base in the plane and given cross-section
∫ A(x) dx over interval a to b, where A(x) is the area of the given cross-section in terms of x
If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) =
g'(x) = f(x)
p-series test
general term = 1/n^p, converges if p > 1
Section 3
(50 cards)
Arc Length Cartesian
Polar Conversion for y
Elementary Series for e^x
Geometric series test
Total Dist.
Check for turning points too!
Integral of u'/u
Mean Value Theorem
Elementary Series for ln x
Alt. Series Error:
Polar Conversion for x
Trapezoidal Rule
Euler's Method
derivative of arctan u
Integral of a^x
Parametric Derivatives
Integral of csc^2 x
Alternating series test
terms decrease in absolute value means convergence
Integral test
Whatever integral does, series does
Speed
Taylor expansion
Integration by parts
Limit definition of derivative with delta x
Integral of tan x
Polar Area
Polar Conversion for r^2
Area of Trapezoid
Logistics Equation
Arc Length Parametric
p-series test
nth term test
Integral of csc x cot x
Volume of Cross Section
Integral of sec x tan x
Integral of cos x
Lagrange Error
Limit definition of derivative with h
Inst. Rate of Change
Average Rate of Change
Polar Conversion for theta
Integral of sin x
Arc Length Polar
Integral of cot x
Integral of sec^2 x
Logistic differential
Average Value of a Function
Second Fundamental Theorem
derivative of arcsin u
Elementary Series for sin x
Elementary Series for cos x
Intermediate Value Thm
A function f that is continuous on [a,b] takes on every y-value between f(a) and f(b)
Section 4
(3 cards)