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