Section 1
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mean value theorem
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Mar 1, 2020
Cards (131)
Section 1
(50 cards)
mean value theorem
if f(x) is continuous on [a, b] and differentiable on (a, b), the slope of tangent line equals the slope of the secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)
f(x) above x-axis is
accumulation is positive
f(x) has a relative minimum
If f '(x) = 0 and f"(x) > 0,
Second derivative test for local maximum at x=a
f'(a)=0 and f"(a)<0
Formal definition of derivative
First derivative test for local maximum at x=a
f(a) exists and f'(x) changes from f'(x)>0 to f'(x)<0 at x=a
Instantenous Rate of Change
Slope of tangent line at a point, value of derivative at a point
Area under a curve
A=∫ f(x) dx over interval a to b if f(x)>0 on a to b
Area under velocity-time graph
displacement
absolute max/min
Only occur at critical points or end points of a continuous function (guaranteed by the EVT)
Derivative of an Inverse Function
g'(x) = 1/ f'(g(x)) where g(x) is the inverse of f(x)
Finds horizontal asymptotes of f(x)
Find limit as x approaches infinity of f(x) and limit as x approaches negative infinity of f(x)
Second derivative test for local minimum at x=a
f'(a)=0 and f"(a)>0
When f '(x) changes from negative to positive, f(x) has a
possible relative minimum
Slowing down
Velocity and acceleration act in opposite directions
right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
average value of f(x)
= 1/(b-a) ∫ f(x) dx on interval a to b
Linearization
use tangent line to approximate values of the function near the point of tangency
Quotient Rule
(uv'-vu')/v²
definite integral
has limits a & b, find antiderivative, F(b) - F(a)
When f '(x) changes from positive to negative, f(x) has a
possible relative maximum
Particle is moving to the right/up
velocity is positive
When f '(x) is decreasing, f(x) is
concave down (f"(x)<0)
When f '(x) is negative, f(x) is
decreasing
When a limit does not exist
1. f(x) approaches a different number from the right as it does from the left as x→c 2. f(x) increases or decreases without bound as x→c 3. f(x) oscillates between two fixed values as x→c
First derivative test for local minimum at x=a
f(a) exists and f'(x) changes from f'(x)<0 to f'(x)>0 at x=a
Definitions of a Derivative at a Point x=a
f'(a) = lim h->0 of [f(a+h)-f(a)]/(h) f'(a) = lim x->a (f(x)-f(a))/(x-a)
Velocity is increasing
Acceleration is positive
Velocity is decreasing
Acceleration is negative
Chain Rule
f '(g(x)) g'(x)
left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
instantaneous rate of change
The derivative of a function at a point
When f"(x) changes sign at x=a, f(x) has a
possible point of inflection at x=a (f(a) must exist and f must be continuous at x=a)
Derivative process for a function raised to a function (f(x))^(g(x))
use logarithmic differentiation. Start process by taking Ln of both sides, use log rules to stretch, take derivative, isolate derivative. (used also for messy derivatives)
Definition of Continuity
1. lim x→c f(x) exists. 2. f(c) exists. 3. lim x→c f(x) = f(c)
Speeding up
Velocity and Acceleration act in the same direction
Average Rate of Change
Slope of secant line between two points, use to estimate instantanous rate of change at a point.
trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
indefinite integral
no limits, find antiderivative + C, use inital value to find C
Speed
absolute value of velocity
f(x) below x-axis is
accumulation is negative
When f '(x) is increasing, f(x) is
concave up (f"(x)>0)
When a function is not differentiable
cusp, vertical tangent, discontinuity
Product Rule
uv' + vu'
When f '(x) is positive, f(x) is
increasing
Particle is moving to the left/down
velocity is negative
Area of a trapezoid
[(h1 + h2)/2]*width
f(x) has a relative maximum
If f '(x) = 0 and f"(x) < 0,
Finds vertical asymptotes of f(x)
Unbounded behavior where denominator is equal to zero. Take limit as x approaches a from the left or right = infinity or neg infinity
Limit exits if
the limit from the left is equal to the limit from the right
Section 2
(50 cards)
indeterminate forms
0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰
second derivative of parametrically defined curve
find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative and divide by dx/dt
Area of an equilateral triangle
A=(√3)/4)s²
Indeterminate power
Take Ln of both sides, use log rules, change to quotient, use L'H, raise answer to base e.
∫ u dv =
uv - ∫ v du
Integral Test
If integral converges, series converges If integral diverges, series diverges must meet 3 conditions to use the test (positive, decreasing, continuous)
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
To draw a slope field,
plug (x,y) coordinates into differential equation, draw short segments representing slope at each point
nth term test
If lim a[n] is not equal to 0, the series diverges. If lim a[n] =0 must use anther test to see if it converges or diverges
Power Series
polynomial with infinite number of terms, includes general term
converges conditionally
alternating series converges and positive form diverges with another test
If g(x) = ∫ f(t) dt on interval a to x, then g'(x) =
g'(x) = f(x)
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
alternating series test
lim as n approaches zero of general term = 0 and terms decrease, series converges
Elementary Series for e^x
Methods of integration
simplification, substitution, integration by parts, partial fraction decomposition
volume of solid of revolution - no washer
π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution
Taylor expansion
Area of Trapezoid
converges absolutely
If both the alternating series its positive form converges with another test
Euler's Method
Alt. Series Error:
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
given rate equation, R(t) and inital condition when t = a, y(a) = y₁ find y when t = b
y(b)=y₁ + Δy Δy = ∫ R(t) over interval a to b
Indeterminate products
Change to a quotient and use L'Hopitals rule
Find radius of convergence
use ratio test, set <1 and solve absolute value equations, radius = 1/2 (length of interval)
Lagrange Error
To find a particular solution to a differential equation, dy/dx = x/y
separate variables, integrate + C, use initial condition to find C, solve for y
given v(t) find displacement
∫ v(t) over interval a to b
area between two curves
∫ (f(x) - g(x))dx over interval a to b, where f(x) is top function and g(x) is bottom function
Average Rate of Change
logistic differential equation, M = carrying capacity)
dP/dt = kP(1 - P/M)
logistic growth equation
P = M / (1 + Ae^(-kt)) A=(M-P0)/P0
Elementary Series for cos x
Second Fundamental Theorem
Find interval of convergence
use ratio test, set < 1 and solve absolute value equations, check endpoints
Area of an isosceles right triangle with hypotenuse as a base
s²/4
use partial fractions to integrate when
integrand is a rational function with a factorable denominator ( use LD first when needed)
given velocity vectors dx/dt and dy/dt, find total distance travelled
TD=∫ √ (dx/dt)² + (dy/dt)² dt over interval from a to b
given v(t) find total distance travelled
∫ abs[v(t)] over interval a to b
derivative of parametrically defined curve x(t) and y(t)
dy/dx = dy/dt / dx/dt
Fundamental Theorems of Calculus
∫ f(x) dx on interval a to b = F(b) - F(a)
given velocity vectors dx/dt and dy/dt, find speed
speed =√(dx/dt)² + (dy/dt)² not an integral!
Indeterminate difference
Factor or Change to a quotient.
Area of an isosceles right triangle with side as a base
s²/2
L'Hopitals rule
used to find indeterminate limits(0/0, ∞/∞). Find derivative of numerator and denominator separately then re-evaluate limit
use substitution to integrate when
a function and it's derivative are in the integrand
Elementary Series for sin x
use integration by parts when
two different types of functions are multiplied and substitution does not work
Limit Comparison Test (LCT)
if lim as n approaches ∞ of ratio of comparison series/general term is positive (not equal to zero) and finite, then series behaves like comparison series
Section 3
(31 cards)